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Paradox and Foundation

Zach Weber

Submitted in total fulfilment of the requirements of the

degree of Doctor of Philosophy

May 2009

School of Philosophy, Anthropology and Social Inquiry

The University of Melbourne

This is to certify that- the thesis comprises only my original work towards the PhD,- due acknowledgement has been made in the text to all other material used,- the thesis is less than 100,000 words in length.

Preface

Dialethic paraconsistency is an approach to formal and philosophical theories inwhich some but not all contradictions are true. Advancing that program, this thesisis about paradoxes and the foundations of mathematics, and is divided accordinglyinto two main parts.

The first part concerns the history and philosophy of set theory from Cantorthrough the independence proofs, focusing on the set concept. A set is any col-lection of objects that is itself an object, with identity completely determined bymembership. The set concept is called naive because it is inconsistent. I arguethat the set concept is inherently and rightly paradoxical, because sets are bothintensional and extensional objects: Sets are predicates in extension. All consistentcharacterizations of sets are either not explanatory or not coherent. To understandsets, we need to reason about them with an appropriate logic; paraconsistent naiveset theory is situated as a continuation of the original foundational project.

The second part produces a set theory deduced from an unrestricted compre-hension principle using the weak relevant logic DLQ, dialethic logic with quantifiers.I discuss some of the problems involved with embedding in DLQ, especially relatedto identity and substitution. Then I outline how the basic toolkit of standard settheory may be developed in this logic from the naive set concept alone, up throughthe theories of ordinal and cardinal numbers, including Cantor’s theorem. The in-famous paradoxes are just theorems, as are the existence of objects like a universalset, unrestricted compliments, and the full set of ordinals. This furnishes the startof a purely paraconsistent foundation for mathematics, providing recapture of clas-sical theorems. It is further demonstrated that paraconsistent naive set theory isable to establish strong results such as the existence of large cardinals, the axiomof choice, and a decision on the generalized continuum hypothesis. The founda-tional reduction to sets, then, is both philosophically illuminating and technicallyrewarding.

To conclude, some mathematical results are used to answer outstanding ques-tions about infinity and the transconsistent—demarcating the line between trans-finite and absolute infinity, in the same way that Dedekind demarcated the linebetween finite and infinite: by turning a paradox into a definition.

3

4 PREFACE

The main original contribution of this piece is an operational artifact of para-consistent machinery. The elemental chapter 5 capturing the theories of ordinaland cardinal numbers is the central moment. A reader already convinced of theintrinsic interest of a robustly inconsistent paraconsistent set theory may wish tobegin there.

∗ ∗ ∗

A note on names. To token that some contradictions are true, Routley andMeyer had been using variations of dialectic, until 1981 when Routley and Priestco-coined the neologism dialeth(e)ism, or two-way truth, inspired by a remark ofWittgenstein’s. (DL, then, originally stood for dialectical logic.) According tofirst-hand accounts, Priest and Routley then forgot to agree how to spell the ism,and it has appeared since both with (in Priest) and without (in Routley) the extra‘e’. I use the e-free form uniformly, both for aesthetic preference and in honorof Routley, who inaugurated dialethic set theory in his [Rou77]. Further, in themid-eighties Routley changed his surname to Sylvan, and both names can be foundintermittently distributed in the literature. As all the works I cite by him werepublished as Routley, that epithet is the only one used here.

∗ ∗ ∗

Thanks are due foremost to my supervisor, Graham Priest, who’s own work andpatient hours of discussion have been invaluable. Greg Restall co-supervised andoffered constructive discussion and comments. Audiences at the Melbourne LogicGroup and the University of Melbourne Postgraduate Colloquia have listened andcontributed over the years. Similar thanks are due the Melbourne-Adelaide LogicAxis, the Melbourne Postgraduate Logic Conference 2006, the APPC 2004, andthe AAP in Australia and New Zealand 2004 - 8. Ross Brady has gone aboveand beyond in checking details and suggesting strategies for improving the proofs.Anonymous referees for [Webng], which is incorporated here into chapter 4, alsoprovided a great deal of help. My office mates, in particular Conrad Asmus, havebeen constant sources of consultation and challenge, as have been many of thepostgrads here. And Vicki Macknight has been loving, listening, and patientlyencouraging throughout—and read the manuscript, twice.

This work was funded by an International Postgraduate Research Scholarshipand a Melbourne International Research Scholarship. My thanks go to the Aus-tralian government and its taxpayers, who I am sure are eagerly awaiting the riseof inconsistent set theory.

Melbourne, May 2009

Contents

Preface 3

Introduction 71. The Set Concept 72. Definitions and Diagonals 113. Dialethism and Paraconsistency 134. A Place to Stand 16

Chapter 0. Into the Dialethic Fields 191. Paraconsistent Set Theory in C=

1 192. Dialectics 213. Inconsistent Sets 28

Part One — Paradox 31

Chapter 1. First Foundations—The Nineteenth Century 331. Introduction 332. Cantor: From Transfinite Numbers to Sets 353. Dedekind 464. Frege: From Logic to Number 485. Paradox and Prospect 51

Chapter 2. The Absolute 571. Introduction 572. Equivalents of the Axiom of Choice 593. The View From Below 644. The View from Above 685. Conclusion 71

Chapter 3. Foundations in the Twentieth Century 731. Introduction 732. On Method 763. Paradise Lost, Paradise Regained 774. Hierarchy 82

5

6 CONTENTS

5. Of Comprehension 946. Conclusion 97

Part Two—Foundation 101

Chapter 4. On A Logic for Naive Set Theory 1031. Introduction 1032. Logics of Formal Inconsistency 1043. A Paraconsistent Logic 1094. Relevant Identity 1145. Restricted Quantification 1186. An Iatrogenic Disorder 122

Chapter 5. Elements 1271. Introduction 1272. Logic 1283. Axioms 1324. Basics 1335. ZF 1366. Ordinals 1397. Global Choice 1488. Cardinals 151

Chapter 6. Reflection and Large Cardinals 1591. Introduction 1592. Reflection Theorems 1623. Ultrafilters 1654. Axioms of Infinity 1665. The End of the Ordinals 1716. Reflecting the Universe 1747. Conclusion 176

Chapter 7. On the Transconsistent 1771. Characterizing the Absolute 1782. Characterizing the Transconsistent 1813. Characterizing the Mathematics 184

Conclusion 195

Bibliography 201

Introduction

1. The Set Concept

The concept of a set is simple to state: A set is any collection of objects that isitself an object, with its identity completely determined by its members. Many settheory textbooks open by claiming that ‘set’ cannot be formally defined, because itis too primitive, but this isn’t so; we’ve just had a fine definition. This is the naiveset concept, and can be completely characterized in the language of first order logicby the principles of abstraction and extensionality,

x ∈ z : Φ(z) ↔ Φ(x),

x = y ↔ (∀z)(z ∈ x↔ z ∈ y),

respectively. These clauses fix the meanings of ∈ and =, the only non-logical partsof the vocabulary of set theory; they look very much like analytic definitions ofpredication and identity. From abstraction immediately follows the principle ofcomprehension,

(∃y)(∀x)(x ∈ y ↔ Φ(x))

(Later we will state these officially, specifying a language.) The clauses underpinbroader mathematical needs in a natural way: The comprehension principle pro-vides for existence, and the extensionality principle governs uniqueness, of objects.A set is the unique extension of an arbitrary predicate.

The reason textbooks claim that there can be no definition of set is not thatthe concept is somehow opaque or ambiguous. The issue is that the set conceptis formally inconsistent. The concept is not indeterminate, or underdetermined;it is overdetermined. I will argue that the inconsistency—a paradox, since it is acontradiction inside a tautology—is not accidental, but inherent to the concept.

A dialethia is a contradiction following from true premises by valid inferencerules, a true inconsistency. Dialethism is the thesis that some theories comprisedof inconsistent sentences are still true; some of the sentences about sets are incon-sistent, and they are still true. Insofar as sets are among the most basic, primitiveobjects that can be described at all, the presence of contradiction in them is trans-fixing.

7

8 INTRODUCTION

Roughly, a set is a multiplicity that also has unity, or, in more antique terms,a many that is also a one. How can a many also be a one? It sounds inconsistentfor something to be both singular and plural—and it is. But this way of puttingthe paradox has a mysterious ring to it; happily, modern logic allows us to to applysophisticated tools to old problems, to make the meaning of the phrase, and so itsdeep tension, precise.

Generally, a system is extensional if coincidence of members, or of some otherobjects (e.g. truth values) obtained from a list is sufficient to establish an identity.Logicians then often call a system intensional just if it is not extensional. But anintension, and so an intensional system, has an independent meaning: Identity andother relations are intensionally determined by properties or predicates, by defini-tions. A basic assertion here is that extensional systems and intensional systemsare not exclusive categories; intensionality does not rule out extensionality.

Set theory is considered the theory of extensions par excellence, and rightly so:Sets are extensions. On the full, naive set conception, sets are further naturallythought of as predicates in extension, suggested at least by Peano’s choice of the εsign to denote predication, from the greek verb εστιν, ‘to be’. In this conception,the ‘is’ of predication becomes a matter of membership; in usual model theory,predicates are still interpreted as sets. Without qualification, this is a simple ex-planation of what it means to be a set, a collection encompassed by a predicate.And this means we will face a collision of extensionality with intensionality. Suchwas Cantor’s original statement in 1883 (from e.g. [Hal84](33)):

By a ‘manifold’ or ‘set’ I understand generally any multiplicitywhich can be thought of as one (jades Viele, welches sich als Einesdenken lasst), that is to say, any totality of definite elementswhich can be bound up into a whole by means of a law.

Similarly, Frege’s system construed sets as the ontology of predication. In hisGrundgesetze, he stated the set concept in a single axiom, his fifth basic law, theinfamous equivalence

x : Φ(x) = x : Ψ(x) ↔ (∀x)(Φ(x)↔ Ψ(x)).

Frege’s axiom looks like a tautology: The set of Φs is identical to the set of Ψsexactly when the Φs are all and only the Ψs. That is obvious to the point ofbanality. Sets and concepts, or properties, or predicates-in-extension, are all muchthe same thing. The key to the appeal of the comprehension scheme is that itilluminates collections in a way that extensionality alone cannot. There is, in aword, more than what Forster [For82](2) calls “the most tough-minded expression”of extensionality, that “the only thing a set theorist can know about = is that it

1. THE SET CONCEPT 9

is a congruence relation with respect to ∈.” We can know that = unites identicaldefinitions.

Here then is the nexus of the paradox. Sets are bound up into a whole bya law, a concept, a definition; the comprehension principle induces sets that areintensional. Meanwhile the extensionality principle governs all sets. Since these to-gether characterize sets, sets are both intensional and extensional. This is unstable,as witnessed by the ensuing paradoxes; this is unavoidable because it is analytic.This is the reason that the set concept is dialethic.

The naive position is tenable for two reasons. First, there are logics available inwhich the naive set concept may be studied in full without falling into incoherence.Any such logic is called paraconsistent ; so paraconsistency insures that the naiveposition, even if surprising in places, is still sensible. Second, and most emphasizedhere, the axiomatized set concept can be used to provide the same resources asstandard axiomatizations, as well as answer questions that no other system has.

The set concept is axiomatic, in the old sense of being self-evident. The conceptjust affirms that sets are the last word on collections. If the concept were not so intu-itive, even inalienable, then its inconsistent consequences would be reason to rejectit. Like the naive concept of truth, it “seem[s] forced upon us in such a way,” writesSlaney, “that we should in all intellectual honesty take [it] seriously.”[Sla89](472)As as a consequence of being axiomatic, the set concept is not just inconsistent, butdialethic—a paradox forced upon us. And this account completely explains the settheoretic antinomies in an immediate, satisfying way: The paradoxes are theorems.Our task becomes not to explain the contradictions away, but rather, as has beenthe case with past surprises in science and mathematics, to come to grips with thephenomenon, in the service of trying to understand.

In the presence of classical logic, the comprehension principle is outright in-coherent. This has been cause of much surprise and consternation. The mostprevalent response has been to maintain classical logic, and to adopt Zermelo’s1908 selection of some instances of the comprehension principle, and abandon therest. This is not the only possible response. Given a theoretical surprise, a para-dox in the informal, etymological sense of thwarting δoξα, belief, we have a choice.We can treat the result as a mistake and revise the assumptions that brought theparadox about. Or, if the force of the assumptions was strong enough, and moreif the rejection of the assumptions is at least as paradoxical as their consequences,we can recast the surprise as a new truth, and instead go about revising the waywe think about these facts. Set theory has been moulded almost exclusively inthe past century by the first strategy; this has resulted in much fine mathematics,philosophical controversy, and the persistence of seemingly intractable questionsabout the nature of sets themselves. Classical logic makes comprehension absurd;

10 INTRODUCTION

the consternation, then, is due to the deep sense that comprehension is not absurd.So we may instead maintain comprehension, and adopt, say, Routley and Meyer’s1976 selection of logical axioms with which to reason about it.

The dialethic paraconsistent approach recasts how we think about old phe-nomena. Forster, introducing his own unorthodox work on Quine’s set theory NF ,writes

In the ZF [Zermelo-Fraenkel set theory] world, ... the para-doxes are viewed as large holes in the ground that one might fallinto. ... However, it is always a mistake to think of anything inmathematics as a mere pathology, for there are no such thingsin mathematics. ... One should think of the paradoxes as super-natural creatures, oracles, minor demons—on whom one shouldkeep a weather eye in case they make prophecies or by some othermeans divulge information from another world not normally ob-tainable otherwise. One should approach them as closely as issafe, and from as many different angles as possible.[For95](11)

The question turns on just how close is still a safe distance. In the first foundationsof the nineteenth century, we will find paradoxes, see how and why they form,and what they mean. Then we will build a paraconsistent foundation, giving someprecision to Wittgenstein’s imagery:

Why should Russell’s contradiction not be conceived as some-thing uber-propositional, something that towers above the propo-sitions and looks in both directions like a Janus-head? . . . Mightone begin logic with this contradiction? And as it were descendfrom it to propositions. The proposition that contradicts itselfwould stand like a monument (with a Janus-head) over the log-ical propositions.[Wit56](III.59)

The whole can be summarized by paraphrasing Frege’s 1902 reply to Russell:With the loss of a full comprehension principle, the sole possible foundation seemsto vanish. It still ought to be possible to reason coherently with such a principle.

Ultimately this is a work on set theory; since it is the case that set theory isinconsistent, this is a work on dialethic set theory; since not everything is true, thisis a work on paraconsistent set theory. Dialethic paraconsistency is a motive and amethod for the rational study of theories containing contradictions. Dialethism isthe motive: Because contradiction is embroiled in truth, it is incumbent upon us totry to understand contradiction. Paraconsistency is the method, insulating truthfrom explosive inferences, from ex falso quodlibet. This is to speak of potentials,

2. DEFINITIONS AND DIAGONALS 11

and this much is already known. What is not known is how much work dialethicparaconsistency can accomplish.

The aim of the present work is to begin to find out.

2. Definitions and Diagonals

An otherwise disparate collection is encircled by a predicate, a property, adefinition. Birds become a flock, stairs a flight, people a crowd (a Menge). Themost general instance: Objects become a set. The naive concept captures, in sloganform, sets as the metaphysics of definitions.

Because he emphasized intelligibility, Socrates spoke of definitions. Definitionsare the key to, perhaps synonymous with, intelligibility; and the Socratic hope isthat better definitions will lead to more intelligibility, to the better life. From theEuthyphro 6e [SMC00],

Tell me what is the nature of this idea, and then I shall havea standard to which I may look, and by which I may measureactions, whether yours or those of any one else, and then I shallbe able to say that such and such an action is just, such anothernot just.

Large abstraction schemas are what is wanted, of the form

x is Φ if and only if Ψ,

whereby it can be further known that x is not Φ if and only if not Ψ. The Socraticcall for intelligibility leads inexorably to such general abstraction schemes. Suchgeneral abstraction schemes lead inexorably to inconsistency, by diagonalization.Consider the definition

x is Λ if and only if x is not x,

and its diagonal:

Λ is Λ if and only if Λ is not Λ.

Then by excluding the middle,

Λ is Λ and Λ is not Λ,

a contradiction. Truth and sets both lead to such unsettling self-referential state-ments, to outright contradictions.

Diagonals have been and continue to be a very fertile source of information.Diagonal arguments find the limits of definitions. It is suitably sweeping to say that“the history of Western thought can be regarded as a history of our relationshipto the diagonal argument.”[Kin93] A very early appearance of diagonalization isEuclid’s proof that there are infinitude of primes: Take any finite set of primesk0, ..., kn and consider

∏ni=0 ki plus one; this is either a prime not on the list, or

12 INTRODUCTION

implies the existence of a new prime by the prime factorization theorem. In thisinstance, diagonalizing leads only, harmlessly, to more objects. When diagonalsmeet totality, though, then fixed points result (c.f. §4 below).

These fixed points can do, and have done, a lot of philosophical work; forexample, Descartes famously used a doubt-operator, and came upon his cogito1:

I doubt(p I am q) iff I am.

In 1874 Cantor proved that the concept of countable infinity has a diagonal, andso showed the existence of uncountable cardinal numbers. The most importantappearance of a diagonal was as the hinge of Godel’s 1931 incompleteness proof, inthe form

pΞq is not provable iff Ξ.

With “the development of mathematics toward greater precision,” as Godel wrote,he proves that no consistent system capable of expressing anything as interestingas arithmetic can go on to establish all truths. The impact of inconsistency in suchotherwise innocuous notions has left a crater in the Socratic/Cartesian rationalproject. Inconsistency was not the projected or expected outcome, and after manydecades of our getting used to incompleteness, it is important for us to recall justhow bad Godel’s news was. It was to be in mathematics that proof and truthperfectly coincide; there was to be no ignorabimus. Responding appropriately toGodel’s discovery remains an outstanding task for human thought.

One response has been that these discoveries show our intuitions are bankrupt.On this despondent reaction, there is no formal or precise concept of set, proof,or truth with which to work [Woo03](159). We can only work scientifically withstripped down fragments of these original ideas. This attitude takes contradictionto be the worst thing that can happen—worse than abandoning hope of a precisetheory of sets, proof and truth. Paraconsistency can be taken as the doctrine thata contradiction is not the worst thing that can happen. In fact, since contradictiondoes seem to be the sort of thing that happens, it is rather unhelpful to panic oraccept rational chaos when they do. Consider the folly of an explosive inference(discussed by Weir in [PBAG04](406)): “War is wrong. Therefore, if we go towar, it is okay to bomb civilians.” At the least, this is not the way we want ourleaders to reason. This is not the way important philosophical concepts like truthshould be reasoned about.

The response to Godel has not been good. Received wisdom is now that theworld is too complex, too difficult, too big to understand as a whole. There is nosuch thing as everything. We will be studying a strain of scientific optimism as it

1Barwise and Moss similarly gloss the cogito as a circular phenomenon [BM96]. Treatment ofthe cogito as a full diagonal, not just left to right, is [Boo83](285).

3. DIALETHISM AND PARACONSISTENCY 13

flourished at the conclusion of the nineteenth century, and how it collapsed in thetwentieth. In broad terms, the old project of fully and rationally understandingthe world has been decided a failure. Certainty and completeness are lost. But thelast century’s was only the first response to the emergence of dialethias. Beginningwith Descartes, the Enlightenment project was to fix a total theory of a uniquelyquantifiable world, and, I will maintain, its proponents were verging on such atheory—complete with the strange and wonderful surprise that a total theory willbe paradoxical. Tarski states a precise schematic definition of truth,

TpΦq↔ Φ.

And in the presence of sentences like

The first self-referring sentence of [Web09] is not true,

the ancient liar paradox follows as a theorem. When a system is complete, it willcontain paradox.

In 1944, Tarski wrote of the liar paradox:

In my judgment, it would be quite wrong and dangerous fromthe standpoint of scientific progress to depreciate the impor-tance of this and other antinomies, and to treat them as jokesor sophistries. It is a fact that we are here in the presenceof an absurdity, that we have been compelled to assert a falsestatement....[Tar44]

And in 1990, Vann McGee accords:

There are scarcely any philosophical problems of greater urgencythan the liar paradox, for there are scarcely any concepts morecentral to our philosophical understanding than the concept oftruth. ... Quite unmistakably, our present way of thinking abouttruth and reference is inconsistent.[McG90](vii)

The liar is only the easiest to state of the paradoxes. That these are serious para-doxes is not in doubt; they call for a response.

3. Dialethism and Paraconsistency

Some propositions of the form Φ∧¬Φ are true; some true propositions have truenegations. Classical logic in turn is wrong for the most basic reason: Its consequencerelation is not truth preserving. A paraconsistent logic remains appropriate to thenature of truth, by using some modern technology. A logic is a pair 〈L,`〉, whereL is a set of formulae and ` is a relation between subsets of L and members ofL. Distinguishing between logics is a matter of the relation `, the consequencerelation. In classical logic, the consequence relation `C is explosive: Where Φ,Ψ

14 INTRODUCTION

are formulae,Φ,¬Φ `C Ψ,

a rule also called ex contradictione quodlibet. Since Ψ can be anything at all, di-alethism in classical logic is nonsensical. Any logic is paraconsistent when explosionfails over the consequence relation.

An immediate consequence of dialethism is arrestingly simple. Dialethism isitself false, because all contradictions are false. So dialethism itself is a dialethia[Pri79](238); see chapter 0 below. And Tarski was clear that the problem ofcontradiction is not ex falso, but rather falsity:

I do not think that our attitude toward an inconsistent theorywould change even if we decided for some reason to weaken oursystem of logic so as to deprive ourselves of the possibility ofderiving every sentence from any two contradictory sentences.It seems to me the real reason for our attitude is a differentone: We know (if only intuitvely) that an inconsistent theorymust contain false sentences; and we are not inclined to regardas acceptable any theory which has been shown to contain suchsentences.[Tar44](368)

Any theory which includes contradictions will include falsities. The question thenturns on which way we incline—toward completeness or consistency. Priest agreeswith Tarski:

Truth and falsity come inextricably intermingled, like a constantboiling mixture. One cannot, therefore, accept all truths andreject all falsehoods... [Pri06b](100)

In the “concluding self-referential postscript” to his 1979 paper, Priest notes that“the subject of paradoxical assertions is one full of surprises. However, that itshould be so is not particularly surprising.”[Pri79](240) To wit, by maintaining fullabstraction schemes, dialethic paraconsistency recasts the force of counterexamples.Both ∀xΦx and ∃x¬Φx can hold at the same time, in the same way (and then bythe usual rules, ¬∀xΦ(x) and ¬∃xΦx would both hold, too). The cartesian methodis relaxed: While that which can be doubted is not certain, it may also still becertain. Some certainties are dubitable, just as some truths are also false. Herein dialethic paraconsistency, paradox is still playing its traditional role of inspiringnew developments. Here is also the departure and so a way forward.

The thesis of dialethism remains contentious, but I generally assume it withoutprotracted argument, for two reasons. The first and main reason is that the initialbattles have already been fought in the canonical works, e.g. [Pri06b]. To advance

3. DIALETHISM AND PARACONSISTENCY 15

from here does not require a rehearsal of abstract polemics about the law of non-contradiction. Necessarily along the way, some of these debates will surface; butwhoever is yet unmoved is unlikely to be swayed by repetition, while whoever doeshave some sympathy will be ready for something fresh—for progress. While thereare no vertebrate arguments against dialethism, to interrupt the inertia of classicismrequires some very strong compulsion to dialethism, lest we appear to be paradox-mongering. So as my second reason for letting the assumption stand, what is vitallyimportant now is to show that dialethic paraconsistentcy forms a fertile ongoingresearch field. A project emerging from mathematics and logic, after all, can onlybe vindicated when it produces some good, true mathematics and logic. Then,these should nourish some good philosophy.

Most new ideas require their fitting into pre-existing conceptual categories.Some ideas are novel enough to require new categories altogether. Dialethic para-consistency is not like either of these. Dialethism is like gravity, in that it shapes thevery arena in which it acts. Dialethism changes the configuration of our concepts;it changes the way we think, and tries to leave the concepts themselves untouched.

By analogy, in non-euclidean, elliptic geometry, we are to imagine that anytwo straight lines, even two lines both perpendicular to a third, have a point ofintersection. This is not what is usually understood from straight lines, right angles,and so forth; but an assumption, the parallel postulate, has been dropped, and theconcepts realign coherently. Elliptic geometry can be studied fruitfully once onemakes some adjustments, not to the notion of straight line, but to the way onethinks about straight lines. So here we recast the law of non-contradiction, werecast the idea that implication is material or that ` is classical, and begin to seein a new way. Contrary to common perception, dialethism tampers neither withtruth, nor negation, nor even with contradiction; it shows only how these conceptsare to fit if the world is to be intelligible. There are very few certainties to be had,so it would seem incumbent upon us that we learn to think about them well.

Paraconsistent naive set theory shows why there are paradoxes, and how afoundation can be had from them. A set is both an extensional entity, determinedcompletely by membership, and also an intensional object, a unity out of multi-plicity bound by a law. Respecting closure on both these aspects, the universe ofsets has a dialethic boundary. I will therefore indicate that the infamous paradoxesof naive theory are all due to a single, powerful truth: An inconsistent fixed pointstands where none can be, at the ineluctable, impossible meeting of extension andintension, at what Cantor called absolute infinity.

The aim is to make philosophical and mathematical sense of the fixed points,and moreover, to put them to work.

16 INTRODUCTION

4. A Place to Stand

Total provability operators, total truth predicates, total collectivizing proper-ties, cannot be free of paradox. So teaches Godel: No consistent scope is wideenough to see everything. Put broadly, to understand the world completely wouldbe to see it from the outside—a vantage which is impossible to occupy. “In orderto draw a limit to thought,” writes Wittgenstein, “we should have to think bothsides of this limit (wir mussten also denken konnen, was sich nicht denken lasst).”We find the other side both thinkable and not.

On the dialethic view, incompleteness phenomena show only that, becausewe can conceive of the world as a whole, the intelligibility we find when chasingSocratic definitions will harbor paradox. One can only ask that a theory accountfor what the data offers. The data is inconsistent; so will the theory be. In this way,dialethism provides a solution to the outstanding antinomies in the foundations ofmathematics, the contradictions of set theory, the contradictions at the limits ofthought in general. As Russell suggested, and as is shown by Priest [Pri02a], all theparadoxes of self-reference—Russell’s, Konig’s, Grelling’s, Berry’s, Mirimanoff’s,Burali-Forti’s, not to mention the liar and its cognates—are instances of a singleschema, the inclosure:

There is a set, W = x : Φ(x); and Ψ(W ).There is a diagonal function δ, such that if y ⊆W and Ψ(y),then δ(y) 6∈ y, called transcendence,and δ(y) ∈W , called closure.

The contradiction comes because the whole is a part, W ⊆ W , entailing thatδ(W ) ∈ W and also δ(W ) 6∈ W . All the dialethias we will be concerned with inthis work are inclosure paradoxes. (It is my suspicion, as yet unsubstantiated, thatthe only true contradictions are inclosure paradoxes.)

The central motif is that paradoxes are the limit, the horizon of the world;they are both inside and outside, defining the world’s edges; and paradoxes areaccessible—part of naive truth, concepts, and language. Paradoxes, then, are apossible way to see the world from the impossible exterior vantage. Euclid’s [Euc56]thirteenth definition tells us that a figure is that which is given by limits.

A limit is that which is an extremity of anything.A figure (σχεµα) is that which is contained by any limit or limits.

Without the limit of a figure, there is no figure at all. Intelligibility issues fromthe edges. And as with geometry, so it is with the world: To know the world as awhole, we need to know its extremity. If paradoxes, true contradictions, demarcatethe limits of truth, then we have by Euclidean lights a figure, an overall shape, oftruth. The motto is from Archimedes:

4. A PLACE TO STAND 17

δoυ µoι πoυ στω, και κινω την γην.

“Give me a place to stand, and I move the world.”My hypothesis is that paradox offers an Archimedean point, and I intend to

investigate this hypothesis through the foundational fragment of set theoretic math-ematics, using the modern rendering of Socrates’ scheme, a full comprehension prin-ciple, inconsistent fixpoints and all. A call for definitions provokes the question,and, as I aim to show, its very schematic form provides the answer.

CHAPTER 0

Into the Dialethic Fields

De omnibus dubitandum. — Descartes

The origin and testing ground of dialethic paraconsistency is in mathematicsand set theory. Throughout this work we will be focused on set theory, and its naive,paraconsistent rendering; so a concrete example and discussion of methodology isimportant to have up front. In this short inaugural excursion, I quote a theoremand question whether, in addition to being true, it is also false. Since the possibilityof overlap between truth and falsity is the defining feature of dialethism, I use theopportunity to meet some standard criticisms, all of which have that dialethismis defective in some way or another. The aim is to air questions about the forceof dialethic paraconsistent truths, coherence and absurdity, and reasoning aboutinconsistent theorems.

For most of this work we will be in the paraconsistent tradition of Routley,Priest, Brady, et al—the Australian relevance tradition. Another tradition stemsfrom the work by Newton C.A. da Costa and his school in Brazil; in those papers wehave some of the more direct mathematical engagement with inconsistent structure,summarized in [dCKB04]. Here we will use some of that Brazilian mathematicsas a case study.

1. Paraconsistent Set Theory in C=1

I take as a starting point Newton da Costa’s set theoretic investigations [dC00]using his extensional paraconsistent logic C=

1 . (The results here were first obtainedby Arruda and Batens [AB82], in the context of a much weaker logic P , which isa fragment of almost any other logic, including Routley’s DK; see ch.4.) To flagthat this system is distinct from the theory of part two, I will follow da Costa’suse of the signs ⊃ and ≡ here for the conditional and biconditional. For da Costathe negation operator is divergent [PR89b](163, 175) too, but to avoid needlessconfusion in the discussion to follow, the standard ¬ is used.

With read as ‘behaves consistently,’ the sentential fragment of C=1 has the

following axioms; see [dC74].

Φ ∧Ψ ⊃ Φ.Φ ∧Ψ ⊃ Ψ.

19

20 0. INTO THE DIALETHIC FIELDS

Φ ⊃ Φ ∨Ψ.Ψ ⊃ Φ ∨Ψ.Φ ⊃ (Ψ ⊃ Φ).(Φ ⊃ Ψ) ⊃ [(Φ ⊃ (Ψ ⊃ Υ)) ⊃ (Φ ⊃ Υ)].Φ ⊃ (Ψ ⊃ Φ ∧Ψ).(Φ ⊃ Υ) ⊃ [(Ψ ⊃ Υ) ⊃ (Φ ∨Ψ ⊃ Υ)].Ψ ⊃ [(Φ ⊃ Ψ) ⊃ ((Φ ⊃ ¬Ψ) ⊃ ¬Φ)].Φ ∧Ψ ⊃ (Φ ⊃ Ψ) ∧ (Φ ∧Ψ) ∧ (Φ ∨Ψ).¬¬Φ ⊃ Φ.Φ ∨ ¬Φ.

There is one rule: modus ponens, Φ,Φ ⊃ Ψ ` Ψ.For identity postulates, we have

x = x,

x = y ⊃ [Φ(x) ≡ Φ(y)].

Also assumed are the usual ZF axioms, including extensionality,

x ⊆ y ∧ y ⊆ x ⊃ x = y,

with x ⊆ y defined as (∀z)(z ∈ x ⊃ z ∈ y).While da Costa’s student Arruda found that his principle systems trivialize

full comprehension [Arr89], by taking some of the facts about naive set theory asaxiomatic, the system can be used for fruitful non-trivial research. Da Costa’s workis conducted in Church’s version of ZF with added existence assumptions aboutRussell’s set, R, and a universe, U ; he postulates that

(∃R)R = x : x 6∈ x,

(∃U)U = x : x = x.

To gain a sense of the system’s properties, we first follow da Costa in proving thatthe Russell set is closed under some set-forming operations.

Given singletons x, note that by definition

y ∈ x ≡ y = x.

To prove that R contains all the singletons of its members, let x ∈ R. Eitherx ∈ x or x 6∈ x. In the second case, x ∈ R, and we are done. But thefirst case gives x = x.1 If x = x then by substitution of equals for equals,

x ∈ R ⊃ x ∈ R.

1In Aczel’s set theory [Acz88], the set x such that x = x is unique; foregoing his anti-foundationaxiom, the identity or difference between a = a and b = b here remains undecided.

2. DIALECTICS 21

Given any two elements x, y of R, the pair x, y is in R, too. For, by thepairing axiom, from x, y we have x, y, and if this pair is not self-membered, it isin R. If the pair is self-membered, then either x = x, y or y = x, y, and againby substitution we conclude that

x ∈ R ∧ y ∈ R ⊃ x, y ∈ R.

The argument generalizes: If x, y, z, ... ∈ R then x, y, z, ... ∈ R, too.Now for a lemma leading to the main theorem: For any x,

x,R ∈ R.

The proof again considers the case where x,R ∈ x,R, since given non-self-membership the result is immediate. So x,R = x,R, and thereforex = R = x,R. And R ∈ R, so by substitution the lemma holds.

Finally, Arruda and Batens make the following observation: Every x is in theunion of the Russell set, ⋃

R = U

where⋃R = x : (∃y)(x ∈ y ∧ y 6∈ y), the union of R. Again by pairing, we can

always consider x,R for any x; and again, if x,R 6∈ x,R, then the pair is inthe Russell set and so x ∈

⋃R. If x,R is self membered, then either R = x,R

or x = x,R. In the first case, since R ∈ R, by substitution x,R ∈ R and thenx ∈

⋃R. In the second case, x = x,R. By the lemma above, x,R ∈ R.

Substituting, x ∈ R, and therefore x ∈⋃R. This shows (∀x)(∃y)(x ∈ y∧y ∈ R),

that (∀x)(x ∈⋃R).

Not being a relevant logic, C1= includes the axiom that Φ ⊃ (Ψ ⊃ Φ). From

this it follows that (∀x)(x ⊆⋃R) and (∀x)(x ⊆ U). This is sufficient for

⋃R = U

as desired.Let us call this the Russell-union theorem, or RU for short.

2. Dialectics

Now let us stand back and enquire about the nature of this mathematics.

2.1. First we make a general observation about negation. The discussion is onthe sentential level so we use variables p, q, r... etc.

Paraconsistent logic is often described in terms of negation. Generally, theoperator ¬ can be characterized by the basic semantic clauses:

p is true iff ¬p is false,p is false iff ¬p is true.


Classical logic views ¬p as indistinct from p ⊃ ⊥, where ⊥ is, say, the absurdconjunction of all propositions. (See ch.4 for an exact formulation.) If we are tostudy non-trivial inconsistent theories in the presence of a detachable conditional→, this equivocation must be incorrect.

The notions of denial and rejection are teased apart. Following Frege, denialcan be construed as the assertion of negation2: ¬p expresses the denial of p. Mean-while, p → ⊥ can be construed as the rejection of p; Curry called this absurditynegation and we will write −p. To accept p is to walk into incoherence, so p is tobe rejected. This is a strong notion, far stronger than asserting ¬p.

Dialethism claims that it may happen that both p and ¬p are true (and soboth p and ¬p are false (and so p is both true and false (and ¬p is both trueand false))). By relaxing the explosion rule, we separate denial from rejection,¬p from −p. The former only leads to the latter via ex contradictione quodlibet.The other direction, from −p to ¬p, is (literally) trivial. Classical mathematicsmarks no difference between denial and rejection. As our concern is paraconsistentmathematics, novel questions emerge.

In a paraconsistent logic with contraposition, denying a proposition p by reduc-tio is unproblematic. But even with contraposition, rejecting a claim by reductioad contradictione is not possible. Supposing that p entails q ∧ ¬q, still it may bethat p is true. Even if p entails ¬p, the possibility of p is not ruled out; witness theRussell set, where R ∈ R→ R 6∈ R. Contradiction is not grounds for rejecting theassumptions at the start of a line of reasoning. So the question becomes: What, ifanything, is?

A further question is on the force of a proof: Suppose p to be proved. That is,we are contented to agree that p is true. If there is no way to ensure that p is notalso false, then what have we really proved?

2.2. Apropos: How do we know that the Russell-union theorem is true only?Glut logics, the sort appropriate for dialethic paraconsistency, tell us that some

propositions are true; some are false; and some are both. Any interesting theoryshould have propositions in each category; and presumably, as suggested by Brady’snon-triviality result for weak relevant logics [Bra89], naive set theory has some true(and not false) theorems. But we as yet lack any mechanism for deciding, uponproof of p, whether ¬p is also provable, or whether p is true and true only.

As the axiom list shows, da Costa’s systems have a unary operator that denotesconsistency at the propositional level. He writes p0 for ¬(p∧¬p). Then a strongernegation can be defined, ¬0p for ¬p ∧ p0. These devices are meant to capture ourability to insist that a proposition is true and true only.

2I am diverging in terminology and theory from [Pri06a](103 - 115).

2. DIALECTICS 23

There is much to say on this question, which Berto calls the “exclusion problem”[Ber07], the question of expressing consistency. Rather than remain in the torturedabstract, though, let us see first if in fact we can determine the consistency of thetheorem in question, perhaps in order to utilize . For, in the case of RU , we havea simple argument using elementary logic. But there is little reason prima facie totake the theorem as true only; in fact, since it makes a claim about a notoriouslyinconsistent object, the Russell set, there is a strong likelihood that the theoremis also false. I should emphasize that I do not dispute the proof of RU, at least inprinciple. Assuming the lemmas above really can be delivered in da Costa’s logic,I think that RU is a theorem. I am concerned, rather, with the force of saying thatit is a theorem. Let us test the negation of RU , and see whether any more light isshed.

If RU is also false, then⋃R 6= U is true. By the axiom of extensionality, the

sets differ with respect to membership:

x 6= y ≡ (∃z)(z ∈ x ∧ z 6∈ y) ∨ (∃z)(z 6∈ x ∧ z ∈ y).

Something must be in one that is not in the other. We will first ask if⋃R is

lacking: Are there any objects in the universe which are not in the Russell union?3

There would have to be some x such that for no y ∈ R is x ∈ y. Using this asa collection criteria for a set,⋃

R = x : (∀y)(x ∈ y ⊃ y 6∈ R).

(That⋃R =

⋂R is, as it will turn out, an undesirable feature peculiar to some

systems and absent from our theory.) Now,⋃R cannot be empty to be a coun-

terexample; so we christen a member: Suppose a ∈⋃R.

Fact one: (∀y)(a ∈ y ⊃ y ∈ y), because a ∈ x : (∀y)(x ∈ y ⊃ y 6∈ R).From this fact it follows that, since a ∈

⋃R, by modus ponens

⋃R ∈

⋃R.

Fact two: Because a ∈ a, by fact one, a = a.Now let b be an arbitrary set, and consider the pair b, a. Since a ∈ b, a, by

fact one it follows that b, a ∈ b, a. Thus either b = b, a or a = b, a. Fromeither of these disjuncts, unpleasantness follows, as we now see.

If b = b, a, then either b ∈ b (since b ∈ b, a = b) or b = a. Since b isarbitrary, then everything is either self-membered or identical to a. To trivializethis we define ∅ = x : ⊥ and pick it for b. Instantiating, ∅ ∈ ∅ ∨ ∅ = a. In thefirst case, ⊥. In the second case, since a = a, by transitivity of identity follows∅ = ∅; but since ∅ ∈ ∅, substitution implies ∅ ∈ ∅, and ⊥ again. Ergo, ⊥,triviality.

3The following was developed in discussions with Conrad Asmus.


Alternatively, if a = b, a, then also a = b, a, since a = a. Sinceb ∈ b, a, by substitution of equals for equals, b ∈ a. Then b = a, and ⊥ again.

Alternatively, if there were some x ∈⋃R but x 6∈ U , then by substitution, still

some x 6∈⋃R, and we are back to ⊥.

All lines lead to ⊥. And so this seems sufficient cause to declare RU as true andtrue only. (If the system we are working in is not trivial, then RU must be exclu-sively true.) But very much is called into question by dialethic paracosnsistency—enough, at least, to require analysis of some of the dialectical norms in this chain ofreasoning. Since some of the usual principles of mathematical practice have beensuspended, we need a principled way to interpret these consequences.

2.3. Since the start of dialethism, amongst both its adherents and serious crit-ics, there has been a nagging concern. Routley flags it in [Rou80]. More recently,several authors in the landmark collection [PBAG04] bring it up: How, dialecti-cally, is rejection possible?

The dialethist asserts p; but this does not entail a rejection of ¬p, since a trueproposition can have a true negation. Thus someone wishing to disagree with thedialethist’s assertion of p may feel unable to register disagreement by imploring¬p. Does this not constitute an infringement of the rules of rational discourse, byfrustrating any attempt at refutation?

The issue takes prototypical form when the object of discussion is dialethismitself. An opponent of dialethism wishes to say that, no, no contradictions are true.The dialethist will agree. No contradictions are true (only), at least not accordingto the logics we will be using. The law of non-contradiction is itself a dialethia.Shapiro explains,

It is difficult for the critic to know what to say. One can refuteother opponents by showing that their views lead to contradic-tion, especially if the inconsistency comes by the opponent’s ownlights ... This cuts no ice against the dialethist. [PBAG04](337)

By its own admission, the dialethic thesis is false. What more can be done to forceretreat, the critic asks, than to prove the falsity of a claim?

There is a constellation of arguments around the idea that dialethism is a non-starter, because by its very nature it flouts the norms of dialectics. In 1999, DavidLewis sent this note to the editors of [PBAG04]:

I’m sorry; I decline to contribute to your proposed book aboutthe ‘debate’ over the law of non-contradiction. My feeling isthat since this debate instantly reaches deadlock, there’s nothingmuch to say about it.

2. DIALECTICS 25

Nearly identical recoiling is widespread: “A common response to those who questionthe law of non-contradiction is that it is impossible to debate such a fundamentallaw of logic,” note Otavio Bueno and Mark Colyvan, because “there is a minimalset of logical resources without which rational debate is impossible.” [PBAG04](156)

Common ground is essential. Let us see what sort of common ground is about.

2.4. The dialectical worries are essentially tied to the question of what mech-anism decides the true contradictions from the false (only) ones. An exact answeris provided in chapter 7 using the full mathematical apparatus. But by addressingthis question now, we will delimit a clearer map of the dialethic landscape, whatdialethism shares with the wider community, and thereby assess the work done onthe Russell-union theorem.

While it would seem to make a radical claim about truth, in fact dialethismderives from very commonsense commitments. A few facts are indubitable, suchas ppq is true iff p, and what validly follows from truth is also true. Negation,conjunction, and truth all are as before. Further, in most paraconsistent logics onehas the inference of modus tollens,

p→ q,¬q ` ¬p.

Along with the theorem ¬(p∧¬p), which these logics also carry, then, the dialethistcan ascertain the truth of a negation by reductio ad contradictione, like anyone else.So some propositions, which happen to have the form p ∧ ¬p, are true, in just thesame way as any other true proposition is true. And p∧¬p is false in the same wayas any other propositon is false, too.

Thus the short answer as to how true contradictions are discerned from untruecontradictions is instructive, if a bit dissatisfying. Philosophers have yet to devise afailsafe device to sort truth from falsehood. (By classical lights, the construction ofsuch a device even for relatively simple systems would solve the decision problem,and is provably impossible.) Nevertheless, we have imperfect means; and whateverthese means are, are basically the same means the dialethist uses.4

Less quickly, a useful appeal is made to absurdity. An analogy, that consistencyis to contradiction as coherence is to absurdity, is helpful. The inference of reductioad absurdum would be stronger than reduction to a contradiction; this is a means

4Priest’s standing answer is similar. “I often find myself being asked the following question: ‘Sinceyou believe some contradictions, but not all, you must have a criterion for deciding between those

that are true and those that are not. What is it?’ In reply I usually point out that the questionerbelieves some things are true, but not all, and ask them what criterion they have ... The answer,

I think, is the same in both cases. Nice as it would be to have a criterion of truth, to expect one

would seem utopian. One has to treat each case on its merits, whether the proposition concernedis a contradiction or some other thing.”[Pri06a](56)


to reject. And certain notions are, indeed, absolutely absurd. Positions which arecommitted to absurdities are to be rejected. The conjunction of all propositions ⊥ isan absurdity. (What else might count as absurd is an open philosophical question.)So a mechanism is provided for rejection: If p entails ⊥, then reject p. Beingconcerned with truth, like everyone else dialethism does not admit absurdities;this is why so much work must be poured into finding appropriate paraconsistentlogics. If anything, I am compelled to dialethism exactly because abandoning someprinciples would deviate so far from common sense as to be an absurdity.

Some (e.g. [Sor01](77)) claim that dialethism changes the meaning of ‘con-tradiction’: Since dialethists don’t think contradictions are absurd, they can’t betalking about contradictions. This is relies on the bald assertion that the role acontradiction plays in reductio ad absurdum proofs is constitutive of the meaningof ‘contradiction’, as an “absolute stopping point”. This is to equivocate. A con-tradiction is a proposition conjoined to its own negation. It can be the stoppingpoint in a modus tollens inference; absurdity is grounds for total rejection. Insistingthat contradiction is synonymous with, or entails absurdity, is either a change interminology or begging the question.

The only clogging point for our rejection device via reductio ad absurdum wouldbe the so-called thesis of trivialsim, ‘everything is true.’ If trivialism is true, it wouldseem, then ⊥ is not absurd and there is no route to reductio.“The question,” Prieststates, “of why one should not accept everything is one an honest dialetheist willask themself.”[Pri06a](56) Frederick Kroon pushes more in [PBAG04], asking,“What makes the claim that everything is both true and false incoherent?” andinsisting that dialethists owe an answer. For both Priest and Kroon, I take thereasoning to be that, previously, contradiction announced a point of no logicalreturn; but the dialethic paraconsistentist ventured farther; so some new line mustbe drawn.

Trivialism is a red herring. It serves only as a useful point of comparison, asan example of an absolutely untenable and defective thesis. Dialethism is unliketrivialism, and a dialethist is under no more obligation to redress it than any otherphilosopher. Dialethism is independently motivated by natural paradoxes and is, onthe whole, conservative. The same considerations that lead to acceptance and re-jection of more pedestrian theses lead also to dialethism, and likewise could lead toits rejection. For example, if a compelling and consistent solution to the paradoxesof self-reference were produced, and concomitantly a theory of unrestricted quan-tification, then I would very seriously reconsider the thesis. By contrast, nothingcan challenge trivialism (or defend it, for that matter), a priori. Trivialism entails,since it entails everything, that trivialism is by far the most heinously incoherentthesis that can ever be formulated. Dunn calls it “that than which nothing sillier

2. DIALECTICS 27

can be conceived,”[PRN89](476) and a moment’s thought will show this is not anexaggeration: Trivialism says all this of itself, adding that there can never be anycogent engagement with trivialism. “Surely,” writes Weir, “one is under no moreobligation to reason with those who reject rational argumentation than to reasonwith a rabid dog.”[PBAG04](386)

Contradiction is not the reason for rejecting trivialism; formal inconsistency,I think, is not the reason sensible people reject most of the things they reject.Rather, like the vast majority of grammatical strings, what trivialism asserts isjust nonsense (quite literally: If everything is true, then it is true that BA!!!BA

is a well-formed sentence, and a true well formed sentence at that; so trivialismasserts certifiable gibberish (and asserts that it does so)). A few, perhaps even alot, of contradictions being true, is categorically unlike the thesis that everythingis true. The latter is only a bizarre form of madness.

Trivialism provides a rejection device. And the general form of the followingproof by cases, which the trivialist would consider valid, will be our last word onthat topic. Above, we found that the consequences of negating RU led to triviality.So it seems right to conclude that

⋃R = U is true, and true only, at least as far

as C=1 goes. The possibility of it being false is excluded, because either da Costa’s

set theory is trivial, in which case it proves that⋃R is universal only, or else the

set theory is not trivial, in which case⋃R is just universal.

2.5. How to argue against dialethism? Logically, reduce it to absurdity. Rhetor-ically, produce consistent solutions to the logical and set theoretic paradoxes. Then,like any philosophical position (and unlike the dialectically defective trivialism), di-alethism would be rebuked.

Aristotle’s arguments in Metaphysics Γ for the law of non-contradiction fail[Pri06a](7 - 42), which suggests that one cannot be so quick to dismiss dialethism;on the other hand, because dialethists follow dialectical rules, the failure in Γ is notenough to establish the truth of dialethism, because rational agents bear burdens ofproof. Dialethism, again, follows standard dialectical rules—importantly, rejectingincoherence and absurdity. The purpose of this whole thesis is an indirect argument,beyond defense against challenges, for dialethism.

All up, the dialethic dialectic is more difficult, but far from impossible. It doesnot do simply to prove the negation of a proposition p in order to rule out p. Nordoes it do to demonstrate an arbitrary contradiction. But it has always been thecase that such inferential steps are expressions of faith, not constructive demon-strations; this is detailed in chapter 3. The dialethist requires serious engagementwith the components of a theory, a thoroughgoing understanding of the underlying


principles, and articulation of what is absurd or unacceptable. These are the mak-ings of far stronger grounds for debate than unreflective acceptance that the worldis like a sudoku puzzle.

This discussion has aimed to show, then, that there is rational space for cogentdialethic investigations, in particular inconsistent mathematics. I have not given anexplanation of the meaning of inconsistent theorems, save to say that the force ofa proof is that its conclusion is to be regarded as a theorem. As with the excursionhere, better to go out and do some real mathematical work, rather than try toanswer such questions a priori. The question of truth and the bearing our theoremshave on the rest of mathematics, will be taken up at the end of this thesis, chapter7, once we have some theorems to examine.

3. Inconsistent Sets

There are difficult and unanswered questions in the field of paraconsistent settheory. Some questions are philosophical and methodological, and these themeshave been touched upon. There are also straightforwardly mathematical questions,directed at new theorems on the structure of inconsistent sets, of their own intrinsicmerit. Indeed, it is most urgent to get to these, since otherwise there is no suchthing as paraconsistent set theory. And this I believe deserves investigation for itsown sake.

It would be as interesting to study the inconsistent systems as,for instance, the noneucledian geometries: we would obtain abetter idea of the nature of certain paradoxes, could have a betterinsight on the connections amongst the various logical principlesnecessary to obtain determinate results, etc. [dC74](498)

To highlight the basic richness and novelty of the field, this opening excursion endsby observing a strange topology on the Russell set.

3.1. Even though everything is self-identical, since R both is and is not amember of itself, R 6= R. (This is proved in detail in chapter 5, prop 5.7.) By thecontrapositive of the extensionality axiom,

R 6= R ⊃ ¬(R ⊆ R ∧R ⊆ R)

⊃ R 6⊆ R ∨R 6⊆ R

⊃ R 6⊆ R.

Let P (y) = x : x ⊆ y, where

x ⊂ y ↔ x ⊆ y ∧ (∃z)(z ∈ y ∧ z 6∈ x).

3. INCONSISTENT SETS 29

We use the eccentricities of R to prove

(∀n)[Pn+1(R) ⊂ Pn(R)],

where, by finite recursion, P 0 = P and Pn+1 = PPn. That is, each successiveiteration of the powerset is a proper subset of the previous.

First, Arruda in [dC00] has found that

. . . PPP (R) ⊆ PP (R) ⊆ P (R) ⊆ R.

To see that P (R) ⊆ R, suppose x ∈ P (R). If x 6∈ x, then x ∈ R. If x ∈ x, thenx ∈ R because x is a subset of R. So P (R) ⊆ R. Now suppose x ∈ PP (R). Thenx ⊆ P (R), so x ⊆ R by transitivity. Therefore x ∈ P (R), and thus PP (R) ⊆ P (R).The argument can be repeated.

This is Arruda’s finding. To strengthen it, we employ contraposition at eacharrow. For P (R) ⊂ R, recall that R ∈ R and R 6∈ R; this implies R 6⊆ R, and allup, (∃x)(x ∈ R ∧ x 6⊆ R). And R 6⊆ R ∧ R ∈ R gives P (R) ⊂ R. For PP (R) ⊂ R,R ⊆ R but R 6∈ P (R) so R 6∈ PP (R). Again the argument may be continued:

...PPP (R) ⊂ PP (R) ⊂ P (R) ⊂ R.

As it happens, since C=1 does not have contraposition on inconsistent propositions,

the above argument probably cannot be carried out there. But a version of it wouldbe formalizable in the logic DLQ of part two.

Could one go on to define by recursion on the ordinals the ultimate ‘innermodel’:

R0 = R,

Rα+1 = P (Rα),

Rλ =⋂κ∈λ

Rκ,

where λ is a limit ordinal? The many combinatorial questions, and metaphysicalquestions, too, that can be asked about a structure like this, will be raised in chapter6.

3.2. What follows will present a view of sets—a proposal about the nature ofthe world in so far as it is comprised of collections. I claim that sets are inconsistentbecause they are both extensional and intensional; this claim follows from the ax-ioms of extensionality and the naive comprehension principle, respectively. For thesake of honesty, it is important to reiterate up front that naive comprehension isaxiomatic. Indirectly I argue for comprehension by showing that no other assump-tion is adequate for a foundation, and that comprehension is a good description of


pervasive mathematical practice. Still, this amounts to assertion that sets are oneway and not another.

The claim that a set is intensional, for example, might pique some questions.If a collection is intensionally bound, who is binding it? Neither Cantor nor Frege,we are about to see, wanted the binding to be a personal, psychological matter;Cantor nominated God, and Frege adopted a brute realism. In practice these cometo the same thing—a primitive observation about the nature of structured objects:that they are structured. This will be our view, too. There are some facts, namelyabout mathematical structure, and we are interested in their nature. Specifically,we want to know which theory records the facts about sets correctly. I openedwith Socrates’ call for intelligibility, which presupposes an organized world, proneto intellection. A brute fact about reality is that it is structured, bound by someappropriate intension about which not much more can be said. If this is not so,then there is no reason to investigate.

There will not be, then, a full plunge into the philosophy of mathematics—inparticular, the sea trenches of metaphysical debate over platonic realism, nominal-ism, or logical pluralism. The hope is to keep the technical results as philosoph-ically neutral as possible, the arguments simply and recognizably valid. Now, aneo-meinongianism of the sort found in Routley’s jungle [Rou80] or Priest’s re-exploration thereof [Pri05] dispenses with the more confusing aspects of realismand leaves a commonsense view in place. The idea is to quell undue anxiety aboutexistence per se, to see ∃ as without existential weight. I imagine that this is notthe only means of interpreting the results. (Compare Tarski’s great set theoreticskill, despite avowedly not believing in sets, to Godel’s, who avowedly did.) Howone reads the ∃ claims will probably strongly influence how much one believes. Asfar as possible, then, the mathematics is presented on its own terms. This is thestuff of axioms—brute facts—about which there cannot be more justification.

Dialethic paraconsistency, again, will not sink or swim on the basis of aca-demic dissection of Aristotelian liturgy. The program will carry on if its practice isadopted, for the sociological reasons programs are adopted: intrinsic interest andbeauty; explanatory and elegant advantage over competing programs; and, mostelusively, a ring of truth.

The original purpose of the exercise with Arruda and Batens’ theorem wasto see if any indication of ‘truth-only’ could be had by negating the theorem andfollowing out its results. Although our paraconsistent logic provides shelter fromexplosion, it is clear that such investigations tread at a precipice, without safetynetting. Truth is dangerous; at least it is inconsistent. I am moved by this surprisingand ineluctable fact and now endeavor to make a study of this unexplored tract,the transconsistent.

Part One — Paradox

Set theory as foundation of mathematics must satisfy at least two criteria.• Formally, it must provide and describe an ontology for mathematics.• Philosophically, it must be a clear and natural explanation of mathematical

truth.A foundation is a means and an end.Classical drives at foundation have faltered on both counts, because at the

heart of set theory is a paradox, the absolutely infinite, the impossible point ofconvergence between intension and extension. Aristotle was correct: The infinitealways was and always will be contradictory. The infinite is also indispensableto quantitative thought. In 1786, the Berlin Academy of Sciences announced acompetition:

The usefulness of mathematics, the esteem in which it is held,and the honourable name of ‘exact science par excellence’ rightlygiven it are all due to the clarity of its principles, the rigour ofits proofs and the precision of its theorems.In order to ensure the perpetuation of these precious merits inso beautiful a part of our knowledge, we seek a clear and precisetheory of what is called Infinite in mathematics.

The need to understand the infinite and thereby secure an unequivocal foundationhas met severe obstacles and is still outstanding. As Erdos has quipped, though,“Problems worthy of attack prove their worth by fighting back.” Advances in futureresearch as much as past missteps depend crucially on “the definitive clarificationof the nature of the infinite,” as Hilbert proclaimed in 1925 [Hil67]—“not merelyfor the interests of the individual sciences, but rather for the honour of the humanunderstanding itself.”

I will argue by tracing paradox’s history from Cantorian roots to modern ax-iomatization, showing that the problem is everywhere acknowledged but nowheresolved. The absolute was brought from the nineteenth to the twentieth century asthe doctrine of limit on size, a brace on the iterative conception of sets and shelterfor the contrivance of proper classes. Once we show that the absolute presents an

31

32 PART ONE — PARADOX

intractable and unpurged contradiction, then its derivatives cannot be classicallymaintained.

The first chapter covers basic themes in foundational history. The second chap-ter gives sustained attention to Cantor’s idea of the absolute and how it plays intoshaping the transfinite. The third chapter concerns the muddy effect of paradoxeson the mathematics of last century. The central and persistent claim is that thenaive set concept just is the set concept—the analytic constitution of the meaningof the word. When this intuition is resisted, confusion and vengeful paradox results.The path of least resistance puts bends in a stream.

∗ ∗ ∗

And life itself confided this secret to me:“Behold,” it said.“I am that which must always overcome itself.”— Nietzsche, Zarathustra

CHAPTER 1

First Foundations—The Nineteenth Century

After introducing the idea of set theoretic reduction and its foundational merits,we examine the works of Cantor, Dedekind, and Frege on this basis. By isolatingthe set concept of nineteenth century foundations, the potentials and problems ofthe reduction are drawn out.

1. Introduction

Set theory demonstrably provides a language, a toolkit, for mathematics. Thismuch is lexography. Since its inception, set theory has also contended as an on-tology for mathematics, a foundation. If this ontology were provided, then all ofmathematics could be reduced to set theory, showing the objects of mathematicalstudy are all just sets, and, even more presumptuously, these just part of logic. Asalways, this grandiose possibility is rife with both hope and hindrance.

One hindrance to a reduction is conceptual clarity. Mathematical objects suchas numbers, functions, and metric spaces are reasonably well understood. To le-gitimate set theoretic reduction, sets must be conceptually clearer, better under-stood objects. Otherwise there is no point—indeed, serious loss of perspicacity—inrephrasing mathematics in terms of sets [Hal84](299). Frege’s critique of past foun-dations for arithmetic may have been scathing in his 1884 Grundlagen, but wouldhave been no more than intellectual vandalism without the development of his ownelegant analytic system. Hausdorff in 1914 warned against definitions of obscu-rum per obscurius, defining the obscure by the still more obscure [Hau57](11). Afoundation as reduction requires both formal and philosophical vision.

A second raft of troubles suggests that the reduction is not even possible.Frege’s logicist program sought to build up arithmetic from logic alone, but hisown system exploded under ex falso quodlibet. And a weightier second attemptat this project, Russell and Whitehead’s Principia Mathematica 1910 - 13, wasforced to assume rather than prove important facts, such as the axiom of infinity.(On this, “One might well think that something more like utter exasperation withRussell’s procedure is called for,” writes Boolos [Boo98](256).) The axiom of re-ducibility, too, has been generally condemned; so Frege’s project was inconsistent,

33

34 1. FIRST FOUNDATIONS—THE NINETEENTH CENTURY

and Russell’s repair was not analytic, leading to widespread scepticism about logi-cism’s potential. More, if reduction means, as it now usually does, reduction to theiterative conception of sets, then there is mathematics (e.g. category theory, notto mention the programs sketched in [Rou80] or [Mor95]) that fall outside thescope of the cumulative hierarchy. And to reiterate the first worry, which we willbe substantiating over this and the next chapters, the iterative conception of sets isnot able to answer basic questions about the nature of sets, making the reductionone to confusion rather than clarity.

These obstacles stand or fall depending on our conception of set. And here isthe hope. A set is a multiplicity that forms a unity, a many that is also a one; butthis is not a definition; this is a mystery. Here are birds. Here is a flock. How isa many also a one? If we can clear the fog by giving a firm characterization ofwhat a set is, and what makes its unity hold, then we can simultaneously make anintellectually satisfying reduction of mathematics, and rework the reductive projectso that it does not founder on classical limitations. The unrestricted understandingof sets endorsed here goes back to Bernard Bolzano’s 1837 Wissenschaftslehre,

I permit myself, then, to call any group you please, in which thenature of the connection among the parts is to be regarded asan indifferent matter, a set [Inbegriff ]....[Bol73](128)

A natural, intuitive, and complete explanation of the set concept—of sets as theunique extensions of arbitrary predicates—can vindicate reduction and facilitatereconstruction. We will in part seek this explanation in the narrative of its unfold-ing.

This chapter considers the conceptual roots and early developments of thesubject, working through the history of the set concept in the nineteenth century,the romantic period of Cantor, Dedekind and Frege. I give considerable attentionto Cantor and his tangled, beautiful first sketch of the full set theoretic universe, byfollowing his efforts to justify his transfinite numbers and naming the underlyingassumptions that are still alive in modern set theory. I will also briefly describeFrege’s system, which in many ways is the clearer, and so more clearly contradictory,edifice. A few comments on Dedekind’s excellent 1888 monograph are also made.None is intended as primary exposition, which is ably done elsewhere; nor is this ahistory. Discussion is guided to areas of dialethic gravity.

The nineteenth century set concept is not uniformly stated, or even particularlyclear in all points of detail. But by the end we will have found the seeds of a naiveapproach, of intension and extension together, waiting in the fertile soil.

2. CANTOR: FROM TRANSFINITE NUMBERS TO SETS 35

2. Cantor: From Transfinite Numbers to Sets

Georg Cantor (1845-1918) had many interests, but one passion: to foster histransfinite numbers into the intellectual world. While he sought clarification in thefoundations of analysis; while he developed a conception of number that drew onEuclidean themes; and while he discussed the nature of mathematical truth, aboveall was an uncompromising and urgent need to explain the transfinite and make itsarithmetic a part of accepted mathematics.

The transfinite is capable of manifold formations, specifications,and individuations. In particular there are transfinite cardinalnumbers and transfinite ordinal types which, just as much asthe finite numbers and forms, possess a definite mathematicaluniformity, discoverable by men.[Hal84](21)

Cantor stated in a variety of correspondence that he regarded his discovery as noless than revealed world of God.1 Establishing the transfinite’s legitimacy was amatter of missionary fervour.

Of his many writings, there are a few milestones. In 1874 Cantor publishedproof that the real numbers are uncountable, signalling the start of a science of theinfinite. The six papers titled “Uber unendlich, lineare Punktmannigfaltigkeiten”(exposited in [Dau79]) culminated in his Grundlagen einer allgemeinen Mannig-faltigkeitslehre of 1883, which presented and defended transfinite numbers on bothmathematical and philosophical grounds, discussed in [Tai00]. His “Principieneiner Theorie der Ordnungstypen”, written in 1884 but first published almost acentury later, clearly articulates set theoretic reductive ideas. Crucially, the de-finitive Beitrage zur Begrundung der transfiniten Mengenlehre, published into twoparts in 1895 [Can95] and 1897 [Can97], purged almost entirely of philosophi-cal comment, is the only work popularly available in English [Can15] and is themature expression of Cantor’s theory.

During his most active periods, Cantor suffered psychologically and emotion-ally from lack of recognition, and from what he perceived to be hostility fromthe mathematical powers in Berlin—Wierstrauss, Kummer, and especially LeopoldKronecker. (The biographical sketch by E.T. Bell in his influential and entertain-ing Men of Mathematics is largely embellished or outright fabricated; but the morescholarly sources, as well as Cantor’s own voluminous epistles, confirm in broadoutline the story of a passionate, troubled man in real strife against the academicestablishment.) Remaining isolated in Halle for decades, Cantor ascended to fameonly in the last years of his active life, when it was in many ways too late. So the

1“I entertain no doubts about the truth of the transfinite, which with God’s help I have recognizedand studied in its diversity, multiformity, and unity more than twenty years.”[Hal84](11)


transfinite and its place in mathematics was a stikingly personal matter. In thissection we will follow Cantor’s attempts to mount campaigns in service of his cru-sade, each successively clearer and more successful than the last, which culminatein set theoretic reductionism.

2.1. Actual Infinity. The first obstacle for Cantor to overcome was the long-standing resistance to actual infinities. An actual infinity is complete, extant, andstable quantity; a potential infinity, by contrast, is a modal notion of a processthat can go on indefinitely, and is not a fixed quantity. Polemical, Cantor (from[Ruc82](46)) writes

The fear of infinity is a form of myopia that destroys the pos-sibility of seeing the actual infinite, even though in its highestform it has created and sustains us, and in its second transfiniteform occurs all around us and even inhabits our minds.

On this, though he realized that he was placing himself “in a certain opposition toviews widely held concerning the mathematical infinite and to opinions frequentlydefended on the nature of numbers,” [Dau79](96) Cantor held strong convictions,which he did not attempt to disguise.2 The Grundlagen involves, in the wordsof a reviewer at the time, large sections which “are exclusively philosophical andare intended to obviate the quarrel with philosophers who have denied transfinitenumbers since Aristotle.” [Dau79](95) The story of the Cantorian triumph is oftentold; but it is important to recall the extreme tension inherent in Cantor’s initiative,and the legacy that remains thereof. The resistance he faced was well entrenchedand well motivated by a suspicion: that the infinite is inconsistent.

In passing I compare Cantor’s situation to the dialethic paraconsistent tra-dition, particularly in having to argue against a prejudice dating from Aristotle,and particularly in arguing that the Aristotelian dogma is simply question-begging[Pri06a]. The finite stands to the transfinite in much the same way that theconsistent stands to the paraconsistent, a thought behind Priest’s suggestive (butthereafter forgotten) coinage, the transconsistent, detailed in my chapter 7. Wemove to a new genus of entities, be they transfinite or transconsistent, and main-tain cogency by adapting the arithmetic and logic we bring to these novelties.

All the so-called proofs against the possibility of the actual infi-nite are faulty....From the outset they expect or even impose allthe properties of infinite numbers upon the numbers in question,while on the other hand, the infinite numbers if they are to be

2“I am so in favour of the actual infinite that instead of admitting that Nature abhors it, as is

commonly said, I hold that Nature makes frequent use of it everywhere, in order to show more

effectively the perfections of its Author. Thus I believe that there is no part of matter which isnot—I do not say divisible—but actually divisible....” [Dau79](124)


considered in any form at all, must (in their contrast with the fi-nite numbers) constitute an entirely new kind of number, whosenature is entirely dependent upon the nature of things and isan object of research, but not of our arbitrariness or prejudices.[Dau79](125)

I lay aside these comparisons for now, to recall a few pre-Cantorian theories ofinfinity.

The poet and philosopher Parmenides offered sophisticated thoughts on thebehaviour of negation, and more importantly, used these insights to support hismonism [SMC00]. Parmeidean arguments to the unity of the many are a firstarticulation of the problem in the set concept: What is the relation between amany which is also a one? Centrally for us, Parmenides’ student Zeno of Elea putforward devious arguments in the form of his various paradoxes: On one reading,Zeno demonstrates that the space between any two points is infinitely divisible,and concludes that there are infinitely many points between any two. It beingimpossible, this gloss of the argument goes, to cross an infinite number of points inany finite amount of time, motion is impossible.

Zeno’s arguments are both very clever and obviously empirically wrong. Withthe syllogistic technology available at the time, the manifest falsity of Zeno’s con-clusion provoked a rejection of one of his premises: Aristotle fixed on the claim thatthere is an actual infinity of points between any two. While the process of divisioncan go on indefinitely, reasoned Aristotle, it does not complete at some infinitestage—indeed, assuming some such infinite stage is just to smuggle the conclusioninto the premise. So there is no resultant totality of points numbering beyond thefinite, Aristotle thus decreed, no problem with potential infinities, and likewise noactual infinities. Conciliatory, in the Physics III:7, Aristotle:

Our account does not rob the mathematicians of their science,by disproving the actual existence of the infinity ... In point offact they do not need the infinite and do not use it. They postu-late only that straight lines may be produced as far as they wish.

This last would become Euclid’s second postulate [Moo82].Cantorian philosophy notwithstanding, the actual infinite’s eventual resurgence

was not due to any sound defeat of Aristotelian dogma, nor even a putative solutionof Zeno’s puzzle in the concept of convergence. The mathematicians, rather, reallydid not need the infinite and did not use it, until roughly Newton’s and Leibniz’sadvances in analysis, at which time they then did need and use it. Clarificationof ideas underlying trigonometric series and real functions was exactly Cantor’s


point of entry to transfinite research and the background social legitimation fortalk about actually infinite sets. It is in the details, and not the dogma, that theserious points of contention lie.

Foreshadowing, there is a steady relocation of troubles in the progression ofthe calculus: from infinitessimals to reals, from reals to sets of reals, from sets ofreals to transfinite sets. This puts the trouble—contradiction—at enough distancefor working mathematicians to work without inhibitions; but it will be the aim ofespecially the next two chapters to show that the latent contradictions are not inany satisfactory way, because they cannot be, eliminated. The most obvious, andmost serious, concern about quantitative treatment of the infinite is that it disobeysarithmetic rules. Using the lemniscate ∞, it seems fairly clear that

∞+ 1 =∞

is true; but with the usual cancellation laws, this leads to absurdities, since sub-traction of ∞ from both sides yields 0 = 1. Confusion and the threat of inconsis-tency is enough to keep ideas in abeyance for a period (however unscientific andself-defeating it is not to investigate a topic due to lack of knowledge). By theEnlightenment, though, the ban on actual infinities was growing stale, and, himselfno friend to Aristotelian orthodoxy, Galileo began to think seriously about infinitesets. In 1638 he noted in his Two New Sciences that , on the one hand, there aremore natural numbers than even numbers, but on the other hand there seem to bejust as many, at least because both sets are infinite, and more because the two setscan be put into a one-one correspondence.

At this idea, then, we glimpse the modern technology required to make trans-finite numbers rigorous, and the means to overcome the problems Aristotle couldnot solve. In a turn that is a dialectical motif, the paradox identified by Galileobecomes the modern definition of infinity: A set is (dedekind) infinite if and onlyif it can be put into a one-one correspondence with a proper part of itself. Anda set is (dedekind) finite iff there is no such correspondence. (In the presence ofthe axiom of choice the ‘dedekind’ qualification can be omitted.) This redirectswonder that infinity is not arithmetically well behaved to wonder at the arithmeticbehaviour of the infinite.

With the development of a coherent theory of the infinite, the Aristotelianprohibitions went slack. Bolzano in his [Bol51] made important initial observations,including use of one-one correspondences, but his work did not receive attentionuntil later, from Husserl and Kazimierz Twardowski, and also Frege. It is Cantorwho has been credited with an epochal advance. In Russell’s idiom,

The Infinitessimal Calculus, though it cannot wholly dispensewith infinity, has as few dealings with it as possible, and contrives


to hide it away before facing the world. Cantor has abandonedthis cowardly policy, and has brought the skeleton out of itscupboard. He has been emboldened in this course by denyingthat it is a skeleton. Indeed, like many skeletons, it was whollydependent on its cupboard, and vanished in the light of day.[Rus37](304)

Russell’s approval at the time of writing was not yet a consensus view. Cantor’sideas remained controversial for decades after his initial observations of 1874 thatthere are differing sizes of infinite cardinal. Some attention to the fundamentalconcept of his theory is due.

2.2. Wellorder and Counting: The one-one correspondence. Accord-ing to Dauben, few theories in any science so clearly bear the mark of the personalityof their creator as does Cantor’s Mengenlehre. While the idea of a one-one corre-spondence describes something ubiquitous in mathematics, nevertheless Cantor’semphasis on it was distinctive as much for its prominence as its audacity. And,as we will be seeing, this feature of Cantor’s thinking is swollen with his meta-physics: Many in the pure sciences have talked with varying degrees of literalnessabout understanding the mind of God; Cantor was absolutely serious about this[Dau79](147) and so, without irony or scorn I think that Cantor believed that themind of God is organized into perfectly structured, inductive lines.

As the focal notions in set theory, wellorder and one-one correspondence carrythe full weight of both the interest and problems. Cantor’s theory fixates on the ideathat any two sets can be compared in size using pairing functions. This requires notonly that all sizes are comparable, but that all sets have some size; and thereforethat all sets can be primed for one-one mappings. While there are preludes inBolzano [Bol51], the seminal, idiosyncratic moment in Cantor’s thought is in 1883,when any set at all can be arranged so as to allow all its members to be countedoff. From [Hal84](155),

The concept of well-ordered set is fundamental for the wholetheory of manifolds. It is a basic law of thought [Denkgesetz ],rich in consequences and particularly remarkable for its generalvalidity, that it is always possible to bring any well-defined setinto the form of a well-ordered set. I will return to this law in alater memoir.

(A proof of Cantor’s law of thought without use of the axiom of choice is found inmy chapter 5, theorem 5.50.) A wellorder presents an inductive structure: When aset is wellordered, every subset has a least element; and all the elements stand in a


linear connection. Cantor’s original presentation, which states a base case and aninduction step, makes the priming obvious:

We call a simply ordered set F ‘well-ordered’ if its elements fascend in a definite succession from a lowest f1 in such a waythatI. There is in F an element f1 which is lowest in rank.II. If F ′ is any part of F and if F has one or many elements ofhigher rank than all elements of F ′ then there is an element f ′

of F which follows immediately after the totality F ′, so that noelements in rank between f ′ and F ′ occur. [Can15](137)

Any two inductive structures can be naturally put into a one-one correspondence.Helping ourselves to the existence of functions which pair off uniquely and

completely all the members of sets of any size solves Galileo’s paradox for a start,and accordingly leads to the idea that collections which cannot be wellordered haveno size at all, and so are not even sets. Cantor was keenly aware that this is allvery presumptuous and accordingly directed the majority of his intellectual powerat a dense thicket of problems: the wellorder theorem, the aleph theorem, and thecontinuum hypothesis. These are, respectively, that: every set can be wellordered;every set can be assigned a distinct cardinal power; and no cardinals fall betweenthe powers of the naturals and the reals.

The former problems, most histories agree, were to be solved in service of thelatter; failure to secure theorems that every set has a cardinal, and therefore thatevery cardinal is tied to an aleph, would likewise cast into extreme doubt whetherthe continuum hypothesis can be answered at all. Now, Cantor did not himself fearfor the integrity of his system.3 But he did hope to persuade his colleagues of thelegitimacy of his system, and further was deeply perplexed about the power of thecontinuum. A clear elucidation of wellorders was urgent.

While the motivations concern cardinals and specifically the cardinal of thecontinuum, then, the route to cardinals is in wellorder: Cantor descended on histheory of ordinals. These prototypes of wellorders can be described from a varietyof angles—as equivalence classes, as ordertypes of wellordered sets, as von Neumannordinals, the recursive notion that an ordinal is just the set of all preceding ordinals.What is important is that the ordinal line is a series of discernable cairns leadingfrom zero to absolute infinity, or to take up another metaphor, that the ordinalsact as the central load-bearing column of the set theoretic architectonic (explicitly

3“My theory stands as firm as a rock; every arrow directed against it will return quickly to itsarcher. How do I know this? Because I have studied it from all sides for many years; because I

have examined all objections which have ever been made against the infinite numbers; and aboveall because I have followed its roots, so to speak, to the first infallible cause of all created things.”In [Dau79](289)


discussed in the opening chapter of [Hal84]). The ultimate success or failure of theCantorian initiative and set theory in general rests on the extent and consistencyof the ordinals—which, we will see, turn out to be very much the same thing.

The question that comes out of wellordering goes to some basic concerns aboutactual infinity. For in some literal sense, which we have today in von Neumannordinals coupled with the axiom of choice, Cantor thought that infinite sets can becounted, and he gave expression to this thought in the wellordering principle. Thisis not to use ‘countable’ in the specific sense of ‘denumerable,’ meaning ω-size, butrather in the primitive sense of quantitative discernment. Zeno put pressure on theidea of traversing an infinity of points, a pressure largely relieved by advances incalculus. But—even without Aristotelian reservations—why would anyone presumethat an infinite set, even points on an infinite line, can be completely counted? Therepeated locution ‘to put in a one-one correspondence’ is highly constructive in itsphrasing, even while it is beyond contention that no human can or ever will be ableto put infinite sets, even small ones, into any correspondences at all (Zeno-esque‘hurry-up’ processes notwithstanding). We will take up the issue of metaphorsin mathematics in chapter three; for now I insist that Cantor did not have anymetaphor in mind. Counting indicates a process of discerning discrete units insuccession. God is the counter.

To purge the idea of its constructive and theological phrasing, the notion of‘putting into correspondence’ can be understood, not as a process like counting,but stating a rule. For example, N is put into a one-one correspondence with thesquares by the rule f(n) = n2. Without God, we have baldly stated the existence ofa function. What is the justification for such an existence statement? One naturalthought is ruled out: Existence is not a definition by recursion on the ordinals. Itwould be viciously circular to invoke any such principles at this stage, as Poincarein 1908 righty saw: The development of set theory and one-one correspondencesis itself intended to justify the complete induction. Yet some justification, someunderlying principle, is warranting the invocation of one-one correspondences andother mappings. The needed explanation is that we are invoking a broad com-prehension principle. A law or property binds extensions of arbitrary size. Noidealized, infinite being is required. The required function is a set, that exists byvirtue of a law (c.f. chapter 5, theorem 5.47).

To make more sense of this we grapple closely with the assumptions behindCantor’s mathematics, his medieval metaphysics.

2.3. The Domain Principle and the Set Concept. Having cleared con-ceptual space for the possibility of actual infinities, Cantor needed to present reasons


why there are transfinite quantities, and how their existence and manipulation isto be understood.

Cantor’s justification for the transfinite is a far-flung ontological optimism,which is vividly suggested in the metaphor of a road:

Apart from the journey which strives to be carried out in theimagination [Phantasie] or in dreams, I say that a solid groundand base as well as a smooth path are absolutely necessary forsecure travelling or wandering, a path which never breaks off, butone which must be and remains passable wherever the journeyleads. Thus every potential infinity (the wandering limit) leadsto a Transfinitum (the sure path for all wandering), and cannotbe thought of without the latter. [Dau79](127)

From image we pass to ideas. The domain principle, so named by Hallett, isCantor’s master argument to the existence of the transfinite.

The principle states that quantification presupposes a stable domain of quantifi-cation; modulo the transfinite, that “every potential infinity presupposes an actualinfinity.” In 1886 [Hal84](25),

In order for there to be a variable quantity in some mathematicalstudy, the domain of its variability must strictly speaking beknown beforehand through a definition. However, this domaincannot itself be something variable since otherwise each fixedsupport for the study would collapse. Thus, this domain is adefinite, actually infinite set of values.

This is essentially a comprehension principle—in the contentious direction: If x isΨ, then there is a set of all the Ψ objects, Ψ(x)→ (∃y)x ∈ y; and from this followsthe standard first order comprehension schema:

(∃y)(Ψ(x)→ x ∈ y).

(The other direction, that given a set all its members have some property in com-mon, is tautological.) To put it again but the other way around, if there were noappropriate domain of quantification, there would be no meaning to Ψ(x), becausethere would be no way to ground the variable. And

There is no doubt that we cannot do without variable quan-tities in the sense of the potential infinite; and from this canbe demonstrated the necessity of the actual infinite. . . . Eachpotential infinite, if it is rigorously applicable mathematically,presupposes an actual infinite.[Hal84](25)


Priest calls the domain principle “a formulation of the Kantian insight that total-ization is conceptually unavoidable.” [Pri02a](124) Jane goes further, remarkingthat “the domain principle is more than an existence principle—it is a principle ofintelligibility as well.”[Jan95](385) Putting the principle in the contrapositive, hereads Cantor as saying that any sequence is unacceptable in mathematics unlessthe terms of the sequence have been determined, unintelligible if the totality hasnot been previously defined.(386) Cartwright has re-identified the domain principleas the all-in-one principle: Any domain of discourse constitutes a set, or set-likeobject [Car94]. These are elucidating, and essentially repetitious assertions ofthe same idea. The general axiom, that there must be sets to serve the needs ofquantification, simply stands: as axioms do.

Behind Cantor’s formulation of the domain principle lies a thorny theologicalmetaphysic, which may be summarized in a nod: The transfinite exists because itis known to God. Cantor in 1886 states,

That an infinite creation must be assumed to exist can be provedin many ways. ... One proof stems from the concept of God.Since God is of the highest perfection one can conclude that it ispossible for him to create a transfinitum ordinatum. Therefore,in virtue of his pure goodness and majesty we can conclude thatthere actually is a created transfinitum.[Hal84](23)

To structure his conception, Cantor distinguishes between immanent and transientreality, which are enjoyed by objects when they are clearly defined in our thoughts,and are a reflection of mind-external reality, respectively. This emphasizes the con-nection Cantor saw between mind and ontology. But without any further argument,these are little more than unapologetic existence claims—in modern discourse, un-abashed mathematical platonism, seasoned with the godelian conviction that wecan apprehend platonic objects.

The details of Cantor’s thought are not of as much concern to us as the ideasunderlying them. In the domain principle and its correlate metaphysics is Cantor’sanswer to our initial question, as to how collections of diverse objects are joinedinto a unity. Given a multitude of objects, God perceives them all together; theunity has abstract existence erkettnis, in the mind of God, and so fall under a singlecompass. A plurality becomes singular when they are under a purview; a commonproperty, which is at the least the property of being thought about, provides therelations absent in sheer multiplicity. (Again, in a bare realist setting, conceptsdo not require a conceiver; they just are.) Be it numbers or sets, abstract objectsboth, Cantor rests his theory on an ontology of logical objects.


Drake, albeit while trying to discredit the notion, neatly links the domainprinciple to excluded middle.

The notion that every abstraction term should give a set seems toarise if we think of a complete universe given, and a set as beingany partition of the universe (so that a set is only somethingabout which we can say, of anything in the universe, either thatit belongs to the set or that it does not).[Dra74](14)

To complete the naive characterization of sets in Cantor, compare the following tothe Socratic schema given in the introduction:

An aggregate of elements belonging to any sphere of thoughtis said to be ‘well defined’ when, in consequence of its defini-tion and the logical principle of the excluded middle, it mustbe intrinsically determined whether any object belonging to thissphere belongs to the aggregate or not. [Can15](46)

If the identity of a set a is defined materially, a = a iff (∀z)(z ∈ a ≡ z ∈ a), thenthis last is classically equivalent to (∀z)(z ∈ a∨ z 6∈ a); that is, Cantor’s connectionbetween an exhaustive universe and a set being ‘well defined’ can be seen to endorsethe axiom of extensionality.

As with any robust naive principles, the Cantorian set concept soon leads tocontradictions. In the next chapter we will look at the most important of these: Ifa multiplicity becomes a unity because God thinks of it or sees it to be so, then themultiplicity of all unities must itself be a unity—

In the absolute mind the entire sequence is always in actual con-sciousness, without any possibility of increase in the knowledgeor contemplation of a new member of the sequence. [Dau79](143)

—triggering the paradoxes of the absolute.

2.4. Reduction and Abstraction. Until 1883, Cantor seemed to take histransfinite numbers as sui generis, and argued for them as such. As the foundationsof his theory became cemented, however, he saw that the numbers were in factclosely tied to sets. The unpublished Principien of 1884 can now be regardedas a turning point in his thought. A cardinal power, realized Cantor, is a powerof something, and that something is, in a term picked up from Bolzano’s earlierpioneering work, a Menge [Dau79]. (Also following Bolzano, and perhaps moretellingly etymologically, Cantor alternately used the word Inbegriff.) The reductionof mathematics to sets had begun.

Indeed the move to sets was Cantor’s best hope for the acceptance of transfinitenumbers. For if these numbers are just properties of sets (or even, as it would laterbecome, nothing more than sets), then belief in the transfinite could be reduced to


belief in sets. And this was an easier battle to fight, sets being at least a primafacie primitive mathematical notion. By the end of his productive mathematicalcareer, Cantor had a sufficiently precise and palatable exposition of his theoryto insure, in a way he previously had been unable, the continued interest of andwidespread acceptance by the mathematical community. Over twenty years ofsustained attention culminated in the two part Beitrage, a near-text book of oneof the most original pieces of mathematics in two millennia.

The Beitrage articles of 1895 and 1897 contain the most oft-quoted Cantorianfragments: his definitions of set, cardinal, and ordinal. Common to each of theseis an appeal to the mental faculty of abstraction, by which we are to arrive at thedesired concept.

Unter einer ‘Menge’ verstehen wir jede Zusammenfassung M vonbestimmten wohlunterschiedenen Objecten m unsrer Anschau-ung oder unseres Denkens (welche die ‘Elemente’ von M genanntwerden) zu einem Ganzen.[Can95](31)By a set we are to understand any collection into a whole M ofdefinite and separate objects m of our intuition or our thought.These objects are called the elements of M . [Can15]

Cantor earlier, in 1883, had given a stronger form, one which should leave no doubtof a naive comprehension principle; we have seen it already in the introduction:

By a ‘manifold’ or ‘set’ I understand generally any multiplicitywhich can be thought of as one, that is to say, any totality ofdefinite elements which can be bound up into a whole by meansof a law.[Hal84](33)

And in stark rebuke of later revisionist reconstructions that would have Cantoradvocating a highly modern ‘combinatorial’ view of sets (see chapter 3), he appendsto this definition the remark: “By this I believe I have defined something relatedto the Platonic ειδoς or ιδεα.”

The binding laws are left rather indeterminate, but the ones he has in mindwill instantiate transfinite numbers. From 1895,

We will call by the name ‘power’ or ‘cardinal number’ of M thegeneral concept which, by means of our active faculty of thought,arises from the aggregate M when we make abstraction of thenature of its various elements m and of the order in which theyare given.

Similarly, an ordinal number is “the general concept which results from M if weonly abstract from the nature of the elements m and retain the order of precedenceamong them.”


Abstraction wedded the transfinite to sets, and incorporated reduction to sets asa permanent fixture in Cantorian mathematics. The abstraction operation as statedis ultimately underdetermined and attracted criticism; Cantor and Frege’s debateon the matter can be followed from Frege’s more caustic remarks on abstraction—“Iffor example one finds a property of things upsetting, one abstracts it away...Possessingsuch magical powers, one is not very far from omnipotence”—in [Dau79](222), toCantor’s sometimes agreeing [Hal84](121), sometimes acerbic [Hal84](127) com-ments on Frege. Further discussion is in [Dum91](49). Glossed as the intensionalapplication of some explicit predicate Φ, abstraction is just a comprehension pro-cess. Abstraction would hold as a respectable explanation of mathematical objectslong enough to carry the actual infinite into the next century.

3. Dedekind

The approach of abstraction is not unique to Cantor. In 1888, Julius WilhelmRichard Dedekind (1831-1916) published his short masterpiece, Was sind und wassollen die Zahlen? [Ded88]. In what is now reconstructed as a second-ordertheory, he too endorsed the general use of abstraction as an operation for arrivingat numbers. Dedekind:

If, in considering a simply infinite system N , ordered by a map-ping s, we entirely disregard the particular nature of its elements,retaining only their discriminability from each other, and havingregard only to the relations to one another imposed by the map-ping s which orders them, then these elements are called naturalnumbers...

He supposed that a number, or any mathematical object, is a “free creation of thehuman mind.” This can be cast in retrospect as a species of early structuralism[Sha97].

Dedekind is often situated between Cantor and Frege. Like Cantor, his concernswere allied with pure mathematics; his philosophy was ardent but inarticulate; hisproofs were in natural language rather than Frege’s unimpeachable formalism; andhis ontology was considerably richer than Frege’s. For example, in his fantasticallyinfluential essay on continuity also in [Ded88], Dedekind uses actually infinite setsin his famous method of cuts to produce the reals. Dedekind even infamously provesthe existence of an infinite set using a direct ‘mental construction. Proof of theorem66: The totality of my thoughts is infinite....” (See McCarty’s “The Mysteries ofRichard Dedekind,” in [Hin95].) We will give this theorem a precise form in parttwo.

Dedekind’s set concept is perhaps closer to what is now called mereology, ac-cording to Potter, as evinced by his discarding of an empty set, and elision between

3. DEDEKIND 47

subset and membership [Pot04](23). Peano also, according to Potter, exhibitedsome confusion about the difference between a set and its singleton, although he isusually credited for introducing distinct symbolism for parts and members in 1889.Regardless, Dedekind went much further than Frege in giving a characterization ofthe natural numbers [Pot00]. In 1889, Peano stated axioms for arithmetic [vH67],but these were based on Dedekind’s monograph of the previous year, and can bestated as follows:

0 is a number,and is not the successor of any number.The successor of every number is a number, too,and is unique.These are all the numbers (mathematical induction).

Within his naive class theory, Dedekind proves the recursion theorem (theorem126), that for any function g from sets M to M , with some m ∈ M , there is aunique function f from the numbers N to M such that

f(0) = m

f(n+ 1) = g(f(n))

for every n ∈ N . This provides for recursive definition; but more, it paves the wayto showing any function satisfying the peano axioms is unique up to isomorphism.That is, if the numbers N are no more than an infinite system under an appropriatesuccessor function s, then the existence and uniqueness of the recursive f show thatsecond order arithmetic is categorical : “Theorem 132. All simply infinite systemsare similar to N ...” This is a much more robust provision for arithmetic foundationthan Frege’s, in that the intended model is pinned down; insofar as it makes senseto say so, these really are the numbers and what they are for.

To my mind, Dedekind’s 1888 monograph is the most appealing product of theromantic period in naive set theory. The work exemplifies exactly what was sounusual about nineteenth century science: intuitive yet sophisticated, concise yetwide-ranging, surprising and yet obviously, luminously true. As rope-and-pulleystagecraft at the time was at its zenith, soon to be overtaken by electronics, sois Dedekind 1888 the height of pure mathematics just before the onset of the un-certainty and complexity that characterize the digital age. Wedging Dedekind be-tween Cantor and Frege—myself here partaking in the guilt—does a disservice tohis impressively clear and directed, and highly successful, answer to his eponymousquestion. A good foundation for mathematics would look much like Dedekind’s,drawing on the full resources of our intuitions with evident rigour. This said, weforge ahead.


4. Frege: From Logic to Number

Friedrich Ludwig Gottlob Frege (1848-1925) in almost a single stroke inau-gurated the foundational program in mathematics, and the analytic movement inphilosophy. Frege had an idea: to ground the proofs, and thereby the truths, ofarithmetic in the apodictic certainties of pure logic. He called for the deduction,or construction, of arithmetic from logic alone, showing that arithmetic is a partof logic, and therefore is just as certain. Logicism is the thesis that mathematicaltruth is logical truth. If successful, the logicist project would, in Shaprio’s idiom,make mathematics “maximally immune to rational doubt.”[Sha91]

Frege wrote two central books, the Grundlagen (1884) and the Grudgesteze(1893/1902), the first in prose and the second a formal counterpart in the notationof his Begriffschrift (1879). Both aimed to derive arithmetic a priori from first-principles and, against Kant, prove it to be analytic. As a test for the successof logicism we have the axioms for arithmetic given by Peano. Whatever Frege’sapproach—and to this day the same goes for any foundationalist—it must show atthe end that the Peano postulates are true.

The first question for Frege is simple: What is arithmetic about? The answeris simple, too: numbers. But now follows an opaque question: What is a number?By analogy, astronomy is about stars, and a star is a mass of burning gas. Anumber by comparison is much harder to fix; nevertheless, Frege committed togiving an answer of the same form: “The number 3 is x.” And x is to be replacedby some object. A Grund for arithmetic, then, would be a precisely defined bunchof objects which behave according to Peano’s postulates. And Frege was clear thatthese objects are not psychological artifacts,4 echoing platonism and foreshadowingGodel’s unabashed realist imperatives.

A judgeable content is not the result of an inner process or theproduct of some human being’s mental operation, but somethingobjective, which means something that is exactly the same forall rational beings, for all capable of grasping it, just as the Sun,say, is something objective. [Dum91](49)

Frege’s bold realism was to meet an adequacy criterion:

It is impossible to ascribe to each person his own number one;for it would then have to be investigated how far the propertiesof these ones coincided. And if one person said one is one, and

4Wittgenstein once told Peter Geach, “The last time I saw Frege, as we were waiting at the

station for my train, I said to him, ‘Don‘t you ever find any difficulty in your theory that numbers

are objects?’ He replied, ‘Sometimes I seem to see a difficulty—but then again I don’t see it.’”[Wri83]

4. FREGE: FROM LOGIC TO NUMBER 49

another one is two, we could only register the difference and say:your one has that property, mine has this.(ibid)

To give his solution, his foundation, Frege introduces concepts and extensions.A concept is a property or predicate, under which every, some, or no things fall;an extension is the collection of all those things. Aside from some idiosyncraticterminology, this is very ordinary, little more than scaffolding for subject/predicatediscourse. That is, this is just logic, the logic of predication. What is most impres-sive is that this simple distinction leads to a ground for mathematics. In a wayCantor never explicitly did, Frege drove back to a primal level of thought, which hasless to do with the particulars of mathematics so much as universals or attributes(called ‘classes’ in the Middle Ages), and found the numbers there.

The use of sets here is also more natural than in Cantor’s abstractionist reduc-tion. If numbers are to be objects, then we need objects to found the numbers. Andsets are very general sorts of objects. Basic propositional logic is often said to beontologically neutral, but since the stuff of maths must come from somewhere, sinceFrege is demanding a logical ontology for arithmetic, sets are a good choice; sets areprimitive; Frege closes the space between concepts and extensions—sets—to nearlyflush.

What is pleasing indeed about the Fregean analysis is that, in explicating num-bers, much more general and natural technology has been provided. We have, ineffect, naive set theory. Frege invokes a basic connection in his fifth law, which hetook to be a logical truth; in modern notation (with the material biconditional—afoible to be remedied in my chapter five),

x : Φ(x) = x : Ψ(x) ≡ (∀x)(Φ(x) ≡ Ψ(x)).

Read: Two extensions are identical iff exactly the same objects fall under theextensions’ respective concepts. (“It sounds obvious,” Boolos remarks, “doesn’tit?”[Boo98]) This assumption, basic law five, includes both the axioms of compre-hension and extensionality.

Most importantly, Basic Law V provides Frege’s answer to how sets arise outof multitudes. Concepts, or properties, or predicates, depending on one’s predilec-tion, cast the requisite net over diverse objects that warrants their being regardedtogether. Frege’s intensional view of collections, like Dedekind’s, does differ fromCantor’s. But it is the broad points of agreement in this period of development thatconcern us. On the basis of Basic Law V, the membership relation ∈ is (second-order) definable:

x ∈ y ≡ (∃Φ)(y = z : Φ(z) ∧ Φ(x))

As always with precise notation, all the assumptions, for better and worse, are ondisplay for inspection.


Frege avoids the mists of Cantor’s theology and abstractionism by bald platon-ism. He posits (in some unfortunate terminology) a dritte Reich of abstract objectsand that is the end of it. Perhaps this is no more philosophical help than Cantor’sUbermind, but it is at least more honest in dispensing with any distractions, and,depending on how the ∃ quantifier is understood, unproblematic. Intensionalityplays a vexed role here, since ‘the number 2 I am thinking about’ should be thesame number no matter who is speaking. Like Cantor echoed Socrates, so Fregedefends an extensional objectivity.

Are we to suppose that the law of excluded middle does not holdfor classes? ... In [that] case we should find ourselves obliged todeny that classes are objects in the full sense; for if classes wereobjects, then the law of excluded middle would have to hold forthem. [Fre03](128)

With such scientific zeal, Frege’s call for an object to play the part of a number is in-tended exactly to remove any psychological or subjective doubt. “If [numbers] are tobe anything at all,” Russell writes approvingly in 1903, “they must be intrinsicallysomething; they must differ from other entities as points from instants, or colorsfrom sounds...”[Dum91] Concepts and extensions, linked by Frege’s abstractionprinciple, are just this intrinsic something, a fixed and objective horizon. Frege’sprovision was simple and sufficient: Sets are comprehended extensions. Frege cannow proceed with the logicist reduction.

Here is a nice example from the Fregean system. Making use of one-one corre-spondences, two sets M,N can be put in a 1-1 correspondence iff M is equinumerousto N , writing

M ∼ N.

Now let |M | denote the cardinal of the Ms, the set of all the sets which are thesame size as M :

x ∈ |M | ≡ x ∼M

Then we can derive the principle of equinumerosity, now called Hume’s principle(because Boolos refers to Frege’s reference to a passage from Hume). That is, weprove

|M | = |N | ≡M ∼ N.

First let |M | = |N |. Then x ∈ |M | ≡ x ∈ |N |. Since M ∈ |M |, by hypothesis M ∈|N |. And therefore M ∼ N . Now let M ∼ N . By definition, x ∈ |M | ≡ x ∼ M ;so suppose K ∼ M for any K. Then again by hypothesis K ∼ N by transitivity,so again by definition, K ∈ |N |. Repeating mutatis mutandis for K ∼ N anduniversally generalizing, we have x ∈ |M | ≡ x ∈ |N |, as desired.

5. PARADOX AND PROSPECT 51

In part two of this work we will produce Hume’s principle from within naiveset theory. (The proof just stated is too informal and reliant on extensional, ma-terial inferences to establish a paraconsistent theorem.) It is worth showing thenaturalness of the original argument here, however, in case the idea that Frege’soriginal system is essentially absurd or uninhabitable (as is suggested by Dummettwhen he uses imprecations like ‘archaic’ at the outset of [Dum91]) needs furtherdiscrediting. That it is contingently absurd in contexts like classical logic is not atissue.

From Hume’s principle Frege goes on to derive second-order Peano arithmetic.Following Wright [Wri83] neo-logicism observes that this fragment of Frege’s sys-tem is consistent, and calls this fact Frege’s theorem. The question qua old logicismis whether or not Hume’s principle is analytic [HW01]. Since we will be providingproof of Hume’s principle in naive set theory, though, entering into this debatewould be a pointless diversion; it is analytic if comprehension is. Suffice to saythat Frege’s original system boasts many of the features of a good foundation: Hisaccount of cardinality is conceptually clearer than the original notion, vindicatingthe reduction; the system is formally and philosophically rich. No such system,however, can be embedded in classical logic and survive.

5. Paradox and Prospect

In his taxonomy of reactions to paradox, Woods titles the most severe Frege’sSorrow. For Frege, upon receipt of an otherwise laudatory note on “just one diffi-culty” from one Mr. Bertrand Russell, most memorably wrote:

Hardly anything more unwelcome can befall a scientific writerthan that one of the foundations of his edifice be shaken afterthe work is finished. ... And even now I do not see how arith-metic can be scientifically founded, how numbers can be con-ceived as logical objects and brought under study, unless we areallowed—at least conditionally—the transition from a concept toits extension. ... Solatium miseris, socios habuisse malorum. Itoo have this solace, if solace it is; for everyone who in his proofshase made use of extensions of concepts, classes, sets, is in thesame position. It is not just a matter of my particular method oflaying the foundations, but of whether a logical foundation forarithmetic is possible at all. [Fre03](127)

This echoes his repy to Russell: “With the loss of my Rule V, not only the founda-tions of my arithmetic, but also the sole possible foundations of arithmetic, seemto vanish.” Woods notes that “it may strike us as an extreme response, a trifle onthe hysterical side.” [Woo03](11) However, since he took his Begriffschrift to be


the correct description of logic, and that description is classical, Frege is correct.His edifice is destroyed.

Russell had hit upon his paradox, not by studying Frege’s system in depth, butrather while pondering Cantorian arithmetic. His correspondence to Frege was yearsafter he noticed the paradox, an act prompted mainly by his own inability to solveit [Lav94](61). Russell was explicitly committed to a first-order comprehensionaxiom, and reasoned as follows. Since there are cardinal numbers, there is a setof all the cardinals, C. This set has some cardinal. Cantor’s eponymous theorem,though, shows that all the subsets of any set M together are cardinally greaterthan M . So there must, and yet cannot be, some cardinal beyond the cardinal of C.Moore fills in the historical details of Russell’s thinking in his essay in [Hin95]. Thisis Cantor’s paradox, known to Cantor himself probably a decade before Frege—acardinal form of the paradox of absolute infinity, to be studied next chapter.

The same contradiction applies to Frege’s system, for deep reasons. It is builtinto Frege’s assumptions. Concepts are the same when, and only when, they havethe same extension. The tie between concepts and extensions leads to unexpectedconsequences, in the form of paradoxes. On the one hand, there is a concept forevery extension, and an extension for every concept (even if for most concepts theextension is just empty). Basic Law V states that the relation between extensionsand concepts is functional and one-one. On the other hand, there are more con-cepts than extensions. The general comprehension principle inside Frege’s axiomrequires there be strictly more concepts than extensions, since there are conceptslike ‘extension’. Formally, this is to consider the totality of either the Cantorian orFregean universe, and to see simultaneously that the universe is one-one with allits subsets, and yet there be more subsets than members. Zalta explains the pointat length in his [Zal07]. Any attempt to work with concepts in extension, then,will face this surprising, powerful and beautifully paradoxical fact: that there is aninconsistent fixed point where concept and extension inevitably, impossibly meet.

Dummett [Dum91] asks how the serpent got into the garden: Why is Frege’sset theory inconsistent? A technical and anachronistic answer, near but not quiteDummett’s, is that Frege took concepts to be second-order, and ranged over themwith corresponding second order variables, while taking the extensions of conceptsto be first order. This restates Zalta’s diagnosis that there both needs to be andcannot be the same number of extensions as concepts. Naive comprehension is infact stronger than positing equipollence of extensions and concepts: It is a statementof equivalence between extension and concept. For instance, an (apparently) secondorder principle such as mathematical induction

(∀F )[F (0) ∧ (∀n)(F (n)→ F (n+ 1))→ (∀n)F (n)]


in the presence of the set F = x : F (x) becomes just a first-order statementabout sets:

(∀F )[0 ∈ F ∧ (∀n)(n ∈ F → n+ 1 ∈ F )→ (∀n)n ∈ F )]

Much has been said about the superiority of second-order theories, e.g. [Sha91]and it is widely rumored that mathematical truths are second order; Potter re-peatedly emphasizes that first-order set theory is guided by and gains legitimacyfrom the second order variant, and c.f. Tait in [DO98] on reflection theorems.Endorsing naive comprehension is an act of agreement with these notions, but byimporting higher-order, intensional entities into a first-order extensional theory. Ittakes properties to be what they are: objects. (This is the core thesis of Routley’s[Rou80] opus: Everything is an object.) Quantifiers ranging over, say, sets of sets,are still ranging over just sets.

The problem returns us to the question asked at the start: How does a manybecome a one? We have seen that, covertly for Cantor and overtly for Frege, theanswer is through law-like grasping. On the other hand, overt for both men is thatsets are fully extensional, identified by no more than the determination of theirmembers. Frege’s Basic Law V does us the service of perspicuously saying both atthe same time. Cantor in effect does the same by giving an abstractionist, and sointensional, account of all his most important objects, and also asserting the solelyextensional nature of sets, as in [Hal84](301). Dedekind’s ‘systems’ are somewherebetween the two.

As remarked from the outset, I cannot claim that the set concept I will beusing, as encoded in the first order axioms of comprehension and extensionality,is unequivocally the same as the concept used by the first founders. There is toomuch confusion, both internal to their respective theories and externally betweenthem, to expect a static notion to emerge. Dedekind and Peano may have beenthinking about mereological fusions. Frege’s own signature metaphysics have beennoted, and Cantor’s ideas are wild. Such is always the case in human endeavors,particularly in early days. I do, however, claim to have extracted the commonthreads of extensional members intensionally bound, as predicates in extension,and do not think it too contentious to say that this is, retrospectively, an explana-tion or rational reconstruction, with minimal mutilation, of the nineteenth centuryapproach.

The set concept, then, is inconsistent. I started by saying that our ability toexplain the set concept directly affects the point or possibility of reducing mathe-matics to sets. We would be straying too far into the twentieth century to begindiscussion of the Zermelodic axioms here, but it is worth pointing out now their


philosophical import: total abandonment of conceptual foundation, and so aban-donment of a true foundation. Bernadete (in [Woo03](334)) understands that

what we want to know is precisely what it is about sets as such,sets qua sets, that precludes there being [a Russell set]. Insightbeing above all what we seek, Zermelo would seem to be the leastpromising source to consult; for he was positively ostentatious ininsisting that this was no more than a home remedy, suited tothe needs of the working mathematician on the most pedestrianlevel.

This view is emphatically asserted by Hallett:

Axiomatization went hand in hand with the divorce from anyattempt to understand what sets are or what conceptual rolethey play. We cannot say with any kind of conviction what sortof things sets are, so we attempt a type of ostensive definition ofthem through axiomatization or ‘listings’. [Hal84](303)

At least at the end of the nineteenth century, and as I will argue in the nextchapters, over the twentieth century’s cumulative hierarchy, the set concept wasabandoned by the mathematicians into the hands of metaphysicians. Hallett’sis no exaggeration; von Neumann says: “One understands by ‘set’ nothing butan object of which one knows no more and wants to know no more than whatfollows about it from the postulates.”[vN67] Potter characterizes this approachto axiomatics somewhat glibly, as intent on retaining “as many as possible of thenaive set theoretic arguments which we remember with nostalgia from our days inCantor’s paradise.”[Pot04](34)

Zermelo’s ad hoc method alludes to the true dilemma of set theory: The naiveversion is both irreplaceable and classically untenable. In a footnote to the com-ments just made, von Neumann admits

There is, to be sure, a certain justification for the axioms in thatthey go into evident propositions of naive set theory if in themwe take the word ‘set’, which has no meaning in the axiomati-zation, in the sense of Cantor. But what is omitted from naiveset theory—and to circumvent the antinomies some omission isessential—is absolutely arbitrary. [vN67](396)

What we see is a general persistent sense that set theory, the naive version, isstill in essence correct, its main ideas easily preserved, but rather mysteriouslyattached to contradictions. “It seemed unworthy of a grown man to spend histime on such trivialities, but what was I to do?” Russell (in [Lav94](61)) wonders.


“There was something wrong, since such contradictions were unavoidable on ordi-nary premises.” By the outset of his 1903 landmark advance of logicism, Russellis utterly distracted by the import of his contradiction: that the concept of setrequires clarification.5 Because the reduction is not conceptually satisfying, theomission of an analytic concept of set was and remains an important contribut-ing reason for the collapse of the logicist program. What is initially reckoned tobe a minor omission has blossomed into a major problem, a blatantly irresolvabledilemma: How can we both practice set theory as it is intuitively understood, andbe perfectly consistent?

Sets, as the meeting of properties and objects, are inconsistent. War is vio-lent, love is blind, and to be explicable the set concept involves contradiction: amultiplicity which is also a unity. The three chief founders of set theory, Cantor,Dedekind and Frege, as well as precursors like Bolzano, each essentially employednaive sets. Holding the naive view retains out the possibility of extending and com-pleting the maps of the conceptual universe in a way that any consistent theorycannot. “Philosophers have said so often that mathematics is extensional that manymathematicians have come to believe it,” Meyer and Routley remark, reemphasiz-ing a theme from Meyer’s [Mey71]. “With respect to any reasonable construal ofextension, we cannot think of any doctrine more evidently false. Mathematics dealswith such stuff as dreams are made on...”[MR77](368)

A dialethic paraconsistent theory picks up reduction where Frege left off. Forthe objections to logicism are these [PRN89](524): The reduction is not concep-tually illuminating; the reduction is impossible; a complete reduction is necessarilyinconsistent. This last is undeniable and the motive for going paraconsistent. Thefirst is assuaged by returning to the naive idea of a set. As for the possibility of thereduction, this difficult task is carried through in part two.

That the paradoxes of infinity intertwine with the tension between intensionsand extensions is surprising and important. The underlying structural link is dis-played in the inclosure schema, and will be discussed at length in chapter 7. Wellin advance of identifying the inclosure, in 1979, Priest writes:

...Our familiar world disappears and we find ourselves in strangenew surroundings. The new terrain clearly needs to be explored.Where it will lead is not yet clear. Yet one consequence for thehistory of mathematics already stands out. The discovery by

5”In the case of classes, I must confess, I have failed to perceive any concept fulfilling the con-

ditions requisite for the notion of class. And the contradiction discussed in Chapter X provesthat something is amiss, but what this is I have hitherto failed to discover. For publishing a

work containing so many unsolved difficulties, my apology is, that investigation revealed no near

prospect of adequately resolving the contradiction or of acquiring a better insight into the natureof classes.” [Rus37](preface).


Russell of a set which was both a member of itself and not amember of itself, is the greatest mathematical discovery since√

2. [Pri79](240)

In contrast to Russellian disappointment, Zermelodic formality, or especially Frege’ssorrow, “the view that there is no concept of set,” Woods identifies the dialethicreaction as Routley’s serenity [Woo03](159): The axioms prove contradictions, andthese are worthy of study.

Now that the foundational set concept and reactions to it are reasonably clear,the next chapter draws out the paradoxical aspects of the concept at length, arguingthat these properties are not mere spandrels of an otherwise consistent notion, butformative and essential.

CHAPTER 2

The Absolute

This chapter picks up the crucial tension remaining from early set theory, theinconsistent absolute, and shows in some detail its place in the transfinite. Ratherthan simply being a forerunner to the iterative conception of set as a limit on size,Cantor’s doctrine of the absolute captures the rich and paradoxical nature of theset concept. On the naive view, the absolute is an inconsistent set. After expositinginherent properties of the absolute as it bears on wellorder and choice, we examinethe absolute from two perspectives—as viewed from below (as the set of all sets),and as viewed from above (as reflective).

1. Introduction

For over a century, Cantor’s principle of absolute infinity has remained animportant guide to set theoretic investigations. Insofar as set theory charts theuniverse of sets at the limits of consistency, set theory just is the study of Cantor’sabsolute. The absolute is a boundary condition and as such it shapes the universe;it is also correct to characterize the absolute as the universe itself. To say that aboundary, what it incloses, and the far side of the ultimate bound are all in somesense identical, is nearly to have articulated the paradox. The absolute is a fixedpoint where there can be none, a paradigmatic and powerful dialethia.

Cantor was among the first to see how infinite sets can be dealt with coherently.He was also among the first to see precisely a problem with overlarge collections.This is in good keeping with his general philosophical attitude: The transfiniteis that which is extendible, while the absolute is that which is not. Cantor (in[Ruc82](10)):

The actual infinite occurs in three contexts: first when it is re-alized in the most complete form, in a fully independent other-worldly being, in Deo, where I call it the absolute infinite or sim-ply absolute; second when it occurs in the contingent, createdworld; third when the mind grasps it in abstracto as a math-ematical magnitude, number, or order-type. I wish to make asharp contrast between the absolute and what I call the transfi-nite, that is, the actual infinities of the last two sorts, which are

57

58 2. THE ABSOLUTE

clearly limited, subject to further increase, and thus related tothe finite.

Pre-theoretically, these aspects are all the same—infinity is infinity is infinity, thechild knows. Cantor’s great accomplishment was to isolate the properties of theinfinite that may be studied meaningfully, to find “an intricate and beautiful formin an area that had hitherto been thought formless.”[Pri02a](113)

Along with the core property of being incapable of further increase, Cantorcharacterizes the absolute by turns as inconsistent, incomprehensible, and ineffable.He also associates the absolute with God. Understanding these features is the aim ofthis chapter. Cantor thought that what can be counted and contained is transfinite,and what is mathematically ineffable is the absolute, both the last outpost and farside of intelligibility. Cantor’s legacy is a continuing tension between the transfiniteand absolute aspects of the infinite—the problem of finding a precise demarcationof a limit and a far side amidst their perpetual intertwining. A metaphor suggestingwhat our philosophy and mathematics will find is that of a glacier, cutting across thelandscape and coming to rest at the top of the world, leaving transfinite morainesand sediment in its linear wake: The transfinite is the residue of the absolute.

Looking ahead, the naive view has that all collections are sets. The naivedialethic approach, then, is to regard all infinite sets as transfinite, and some trans-finite sets as absolute, too. This will take on more detail once it is seen thatoverlarge or absolute sets are extendible, too. All this is rather paradoxical in it-self, as we will see here and reappraise in chapter 7. What it means for now is thatwe will be more interested in the absolute in itself than what exactly Cantor saidabout it; if the absolute really is dialethic, but Cantor was attempting a consistentexplanation of it, then his remarks will be inevitably mislead in places. In §2 belowwe face the most challenging aspects of the doctrine with God and the axiom ofchoice. In §3 and §4 below we then review the main two approaches to the abso-lute’s indirect study. A limit has a near and a far side—a view from below, and aview from above. As an approach to understanding set theory, the more common isfrom the bottom up, using generalizing generating principles under a limitation onsize; more recently the top down method has gained prevalence, through reflectionprinciples.

In his erudite study [Jan95], Jane argues that there are two distinct periods inCantor’s thinking about the absolute. (Lavine [Lav94] argues for a similar transi-tion on Cantor’s thinking overall.) From the 1883 Grundlagen until 1896, Cantorbelieved in the actual existence of the absolutely infinite. In his late work, on theother hand (with the 1895 - 97 Beitrage silent on the matter), Cantor took theabsolute to be “irreducably potential”—that is, something close to the Aristoteliannotion of infinity, the απειρoν. The two directions of study in this chapter are

2. EQUIVALENTS OF THE AXIOM OF CHOICE 59

linked to Jane’s distinction: The view from above is tied to the early view of theactual absolute, where “the totality of all ordinals is assumed to be given...not re-ally created through the application of the generating principles—generation beingonly a suggestive way of describing the matter.”(383) The view from below is anexpression that the absolute can only be approached through generating principles,“in such a way as the generation process is absolutely incompletable.” In sum,

Actually conceived, the absolute bounds the entire ordinal se-quence, whereas potentially conceived, the ordinal sequence isabsolutely unbounded.[Jan95](383)

There is a good deal of tense energy in this formulation; it shows that there aretwo ways of understanding the absolute. Both ways are correct. Cantor held bothat different times. Which perspective Cantor occupied at which time is, I think, amatter of Cantor biography, in the same way as which aspect of the necker cubeone sees at any given time is just a matter of empirical reporting. Both aspects arereal.

The work here is ultimately gauged at understanding and accepting the meetingof oppositions, at occupying the point of collision that is the dialethic absolute, andcarrying on study directly, with neither mysticism nor superstitious angst.

2. Equivalents of the Axiom of Choice

A set M is infinite iff M does not map to any finite ordinal, and finite otherwise.The axiom of choice makes infinity equivalent to dedekind infinity (defined at prop.5.59). The axiom of choice makes precise the boundary between the finite and theinfinite.

“Someone once said to me,” Forster writes, “‘Doing set theory with a universalset is like believing in God.’ This arresting simile expresses a sound intuition....”[For95](11) Some have said that set theory is no more than precise theology, echo-ing comments of the editor of Mathematische Annalen to Hilbert, “Das ist keineMathematik, das ist Theologie!”1 To make a very imprecise comment, in the worksof some set theorists, past and present, are the tones of unabashed divine thinking.

Now, the naive set concept is of an extensional intension, or perhaps an in-tensional extension, but it is not theological. There is no need of that hypothesis.Belief in a set of all sets does not induce questions of theodicy; the ordinals did notcreate the universe; and offering up a prayer to x : x = x is obviously absurd.The truth in Forster’s intuition is due to the fact that some of the more potentaspects of the naive set concept, e.g. infinite magnitude, have also been attributedto God.

1Recorded in Max Noether’s obituary of P. Gordan in Mathematische Annalen 75, 1914, page 18.

60 2. THE ABSOLUTE

Nevertheless, moderns tend to dampen the extent to which metaphysics influ-enced Cantor’s thought, wishing to see in the absolute no more than a forerunnerof modern iterative set theory based on the limit on size. This is to ignore entiretectonic plates of what Cantor says. Before we go on, then, and in light of Can-tor’s theological proclivities—“...the transfinite have existed from eternity as ideasin the Divine intellect”[Hal84](21)—relations between the absolute and theologymust be cleared up. We consider the absolute in itself, before describing its relationto the transfinite numbers. In particular we follow threads from the last chapteron the axiom of choice and the wellordering principle, and their relationship to de-ific heuristics. We will see that the axiom of choice has inherited the metaphysicalghosts of Cantorian theory; that the ghosts were noticed, and not entirely exorcised;but will allude to the fact that paraconsistent set theory simply shows the axiomof choice to be a consequence of naive comprehension (theorem 5.56 of chapter 5),and therefore a consequence of the (god-free) naive set concept.

A basic mode in which Cantor’s doctrine is given is through inconsistency ; inhis well-known 1899 letter to Dedekind [vH67](114), Cantor spells out the principleas a fact about collections:

For a multiplicity (Vielheit) can be such that the assumptionthat all of its elements ‘are together’ leads to a contradiction, sothat it is impossible to conceive of the multiplicity as a unity, as‘one finished thing.’ Such multiplicities I call absolutely infiniteor inconsistent multiplicities. As we can readily see, the ‘totalityof everything thinkable,’ for example, is such a multiplicity. . . Ifon the other hand the totality of elements of a multiplicity can bethought of without contradiction as ‘being together,’ so that theycan be gathered together into ‘one thing,’ I call it a consistentmultiplicity or a ‘set’.

The 1899 letter indicates Cantor’s awareness of the paradoxes underlying his fledglingset theory; he had thought about absolute paradoxes at least since 1895 [Lav94](55).The absolute is, foremost, a name for insoluble paradox.

In 1897, Peano’s student Caesar Burali-Forti published the first of the set the-oretic antinomies. While he was working with a mistaken definition of wellorderedset, nevertheless Burali-Forti provided enough for the mathematical community,preeminently Russell, to see that naive sets point into the inconsistent. Burali-Forti considers the set of all ordinals, On, and finds that this set is wellordered. Onmust therefore have an ordertype, and this type is by definition within On—exceptthat, by wellorder, the ordertype must be greater than any member of On. Burali-Forti believed himself to have disproved the trichotomy of ordinals [vH67]. I believe


he proved that understanding the ordinals as a whole is beyond the resources ofclassical logic.

Very notably, in the 1899 letter Cantor uses the inconsistency of On to providean argument that every cardinal is an aleph. The alephs

ℵ0,ℵ1, ...,ℵω,ℵω+1, ...,ℵε, ...

are linearly ordered transfinite cardinals, indexed by the ordinals. Assuming thewellordering principle, Cantor argues that any set with cardinal not an aleph mustbe large enough to contain all the ordinals—that is, be inconsistent—that is, abso-lutely infinite—and so has no cardinal at all. Of this letter, which Zermelo wrote“operates with ‘inconsistent’ multitudes, indeed possibly self-contradictory con-cepts,” Moore [Moo82](53) remarks that

Cantor exhibited no alarm over the state of set theory in his let-ter [to Dedekind]—in sharp contrast to Gottlob Frege’s dismay... Cantor did not treat these apparent difficulties as paradoxesor contradictions, but as tools with which to fashion new math-ematical discoveries.

One (anachronistic) reconstruction of Cantor’s proof is as paraconsistent (ch.5, thm.5.72).

The proof of the aleph theorem in the 1899 letter, aside from using overlargesets, assumes that the ordinals can be “projected” into a set. How can the entiretyof the ordinals be projected into a set? It is not a coincidence that the absolute asa name for inconsistency guides a proof utilizing a massive wellorder. We saw inthe last chapter Cantor’s fixation on the notion of one-one mappings, and that hisfaith that all sets have comparable powers derived from the certainty that all setscan be wellordered. For Cantor, the wellordering principle was a law of thought.This is to say that even very large collections can be, more or less literally, counted.And we saw that the justification for this belief resides in God; or, what is largelythe same, that its truth was an article of faith. An important component of thisfaith is that it puts no pressure on, and so calls no attention to, the distinctionbetween the quasi-constructive image of putting a set into a wellorder, and thepure existential notion of a set being wellordered. Or again, for Cantor, whether aset can be dynamically counted, or is statically quantified, is immaterial; the effect,and the certainty that it has been effected, is the same.[Hal84](154) Whether thisimproves the credibility of the constructive metaphor or reduces the plausibility ofthe existential statement, I shall return to next chapter.

In a very explicit way, Cantor invests the absolute with the full retinue ofmetaphysical and theological mystery.

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What surpasses all that is finite and transfinite is not a “Genus[supremum]”; it is the single, completely individual unity inwhich everything is included, which includes the “Absolute,” in-comprehensible to the human understanding. This is the “ActusPurissimus” which by many is called “God.” [Dau79](290)

Zermelo saw that Cantor’s projection procedure without further argument failedthe standards of proof. Zermelo made explicit an assumption that has now beenrevealed as ubiquitous in mathematics, making such a projection possible without,apparently, any overt theology. It is equally baroque and interesting, then, thatto the Rubins’ compendium [RR63] can be added another entry: Robert Meyerand Richard Routley separately assert that the axiom of choice is equivalent tothe existence of God. Routley presents the claim in his [Rou80](133); Meyer’s isfound in “God Exists!” in [Mey87]. Here is the proof. Suppose the there to becauses and effects, these naturally arranged into partially ordered chains. And,Meyer argues, there is some reason, related to Aquinas’ cosmological argument, tothink that every chain has a lower bound. Now suppose that the axiom of choice istrue. Then (a downward version of) Zorn’s lemma is true, whereby the whole nexusof lower-bounded causes and effects must have a minimal element—a first cause.And this all men call God. Conversely, assume God exists; an argument to theaxiom of global choice is left as an exercise. The same authors elsewhere say that“set theory is, at best, an exercise in philosophical and mathematical imaginationanyway.”[MRD78].

In Cantor’s wake, Zermelo sought to place transfinite set theory on a rigorousbasis and simultaneously to answer in a mathematically respectable way the out-standing open problem of Cantor’s career. To this end he published a proof in 1904and then a proof with background axioms in 1908 (both in [vH67]), which reducesthe problem of wellorder to our acceptance or rejection of the axiom of choice. “Themethod by which a wellordering necessarily results,” Hausdorff writes, “is basicallyvery simple, although it places something of a burden on the abstract thinking ofthe reader.”[Hau57](66) In retrospect, Zermelo’s is a simple algorithm for orderinga set, M , by transfinite induction with a choice function:

mβ =

a choice from M − mα : α < β, if there is one;

stop otherwise.

Tellingly enough are the initial reactions to Zermelo’s postulate, as detailed in chap-ter two of Moore’s fine history [Moo82]. For these, as I’ll briefly recall, call keyproperties of the absolute into question: existence, coherence, and comprehensibil-ity, among others.


The interested French—Baire, Borel, Lebesgue, Hadamard—generally voicedconstructivist or empiricist concerns, registering the now usual complaint that theaxiom offers no rule for making a univocal choice, and so does not provide enoughinformation about the choice process to be mathematically respectable. This is tooutright deny the existence, if not the coherence, of a general choice function. Arecurrent theme in this reaction to set theory is a doubt about whether intuitionsgrown on finite collections, e.g. of thinking of sets like a “bag of marbles,” can besensibly extended to the transfinite.

Elsewhere, German Cantorians did not doubt the choice principle but wereexercised that the Burali-Forti paradox might afflict the consistency of Zermelo’swellorders. Bernstein, for example, saw no safeguard against a set of chosen el-ements being identical to the set of all ordinals On, and therefore carrying theknown paradoxes of that overlarge collection. Burali-Forti’s original proof, afterall, had been based on the wellorder of On; Zermelo proves the wellorder of any setat all. The Germans thus questioned the classical mathematical comprehensibilityof unspecified choice sets—or more precisely, the consistency of such sets. That thiswas the salient concern for Bernstein, Schoenflies, Hausdorff et al who neverthelessmade use of the axiom, shows an early recognition of the problematic relationshipand indiscernible boundary between the transfinite and absolute.

For their part, the Cambridge mathematicians Russell and P.E. Jourdain2, inconversation with G.H. Hardy, simply set about trying to prove the axiom frommore primitive principles. The impression was that shared by many since: thatthe axiom is true, and should be a theorem of any natural set theory; yet is alsoelusive, beyond the ken of established mathematics.

These reactions all show the incipient troubles with respect to choice, trou-bles closely allied with its metaphysical origins. Further anomalies, such as Vitali’snon-Lebesgue measurable set or the Hausdorff-Tarski-Banach decomposition of asphere, only upset the scene further. A century of controversy has judged in favourof Zermelo’s method, much for pragmatic reasons. “No doubt the axiom of choicewill always be desirable,” writes Dana Scott in 1974. “If only it could be deducedfrom some more primitive principle!” Moore concludes his study in echo: “The

2On a sad biographical note, Jourdain met with frustrated and persistent failure on this, until

his eventual decline. Less than a week before his death in 1919, Jourdain’s wife wrote to Russellin despair: “He [Jourdain] is now quite unable to talk or see anyone, but just lies in a semi-unconscious state. Why didn’t you make an effort to come a little sooner? You have made him

so unhappy by your inability to see his well-ordering.”[Moo82] Indeed, the last nine entries inJourdain’s bibliography all bear the same impossible title: “A proof that every aggregate can be

wellorded.” The last was published posthumously, with a disclaimer from a sympathetic Mittag-

Leffler, that Acta Mathematica “will not to any further extent be at the disposal for papers of thesame kind.”

64 2. THE ABSOLUTE

axiom of choice is surely necessary, but if only there were some way to make it self-evident as well.”[Moo82](310) The philosophical questions about choice, owing toan impoverished set concept, have never been answered. The axiom of choice willbe provable from the naive comprehension principle. This in some ways vindicatesZermelo’s axiomatic reconstruction. Our proof is of the wellordering theorem, fromwhich choice follows, so it is Cantor’s suspicion of a Denkgesetz that is most em-phatic for the naive theory. The existence of God, hopefully, is not provable for us,recalling Euler’s rebuff of Diderot in the court of Tsarina Catherine: “Monsieur,(a+ bn)/n = x, therefore God exists!”

The absolute is no pariah in Cantorian thought, but a leitmotif. This case isstrengthened now as we turn more concrete matters in the ordinal ascension.

3. The View From Below

Here we approach the absolute from beneath, the most common method ofintroducing and explaining the transfinite. We mean to establish the transcendencecondition of the inclosure schema.

3.1. Generalization. Transfinite counting is effected by two generating prin-ciples, found in the 1883 Grundlagen. In modern terms, these are the two casesneeded to define a transfinite recursion, at successor and limit ordinals:

If α is a number, then α+ 1 is a number.For any sequence of numbers, there is a least number greaterthan all of them.

As stated, these are bald existence claims. To justify this continued climb, twentiethcentury ZF requires interplay of the infinity axiom and Fraenkel’s replacementscheme, which begins with a set and a function, and posits the existence of a range.

The broad thought is of generalization, harkening back to Cantor’s method ofabstraction. On this view, the natural numbers suggest the existence, and then arejust an instance, of a form, a (small) initial segment of a self-similar chain. 0 is alimit ordinal over which ω stands qualitatively higher. And ω is a limit ordinal,too; so there should be something that stands in the same transcendent relation toit. That is, once the first ω-sequence is in place, we generalize; else ω would be theonly one of its kind, the only number that cannot be reached by finite iteration.Generalization posits that ω is but one among many.

The higher and higher cardinals are further explicable in terms of generaliza-tion, too. Consider the heuristic notion that most of the finite natural numbersare very, very large. Pick an n ∈ N; no matter how large n be, almost every othernumber is bigger. The vast tracts of higher set theory captures this same sense ofspace. In Devlin’s modern text we meet just this argument:

3. THE VIEW FROM BELOW 65

The cardinal ℵ0, being both regular and a limit cardinal, is verymuch larger than any of its predecessors. Neither the replace-ment axiom nor the cardinal successor function gets up to ℵ0

from below. But our set theoretic universe should surely possessa uniform character! The cardinal ℵ0 should not be so unusual.There ought to be a proper class of such cardinals. [Dev79](118)

Whether or not one suspects a median ground between an entity being ‘unusual,’and there being a proper class of like entities, the lure of upward generalizationis strong. It is the clearest sense in which we extend, step-by-step, our intuitionsabout “bags of marbles” into the infinite.

How long does the generation go on?

3.2. Paradox. Generalization gives us perhaps the most trodden entry tothe transfinite ordinals, as recapitulations of the structure of the natural numbers.Generalization also brings us to an explicit contradiction, which was alluded tolast chapter. Cantor is committed by the domain principle to the existence of aset of all transfinite numbers. This collection is the absolute itself. (Or: Cantor’sjustification for how a set arises from a multitude is found in the mind of God: “Allof these particular modes of the transfinite have existed from eternity as ideas inthe Divine intellect.” [Hal84](21) The multitude of all transfinite numbers is surelyknown to the Divine intellect.) Stating the domain principle with respect to ordinalgeneration,

The transfinite with its plenitude of formations and forms nec-essarily indicates an Absolute, a ‘true infinite’ whose magnitudeis capable of no increase or diminution, and is therefore to belooked upon quantitatively as an absolute maximum. [Hal84](44)

The beauty and danger lies in the fact that, like the natural numbers, thereare always more ordinals; they are always increasable. The unstoppable diagonaldrive is set in motion, and with it we can anticipate the waiting inclosure paradox.For the existence of upwardly generalized numbers is philosophically granted bythe domain principle. We quantify over the transfinite, for example in stating thegenerating principles, which is predicated upon a stable domain of quantification.The absolute is just this stable domain.

The structure of the generating principles that breaks through every barrierthen shows its Janus-head. The domain principle prompts us to consider On as atotality:

If α is an ordinal, then there is a set On of all ordinals.

But the generating principle that triggers the domain principle powers on, insistingthat there must be more ordinals, a least greater than all of them. In broad point

66 2. THE ABSOLUTE

of principle, and specifically as witnessed by the Burali-Forti argument, this is acontradiction. This may be what is at issue when Boolos expresses discomfortwith the limiting points that arise, cardinals like ℵλ = λ, a “teensy” cardinal bycomparison to even the least large cardinal, in his “Must we believe set theory?”reprinted in [Boo98]. Cantor’s doctrines involve an outright contradiction—allthe ordinals may be apprehended because they inhabit a stable domain, and theordinals all together may not be apprehended because their domain is overlarge[Hal84](47). The absolute manifests quantitatively, impossibly, as the final limitof the iterable.

For Cantor, the transfinite numbers were to be treated as far as possible likethe finite numbers, a tendency Hallett dubs ‘finitism’ [Hal84](7). Before Cantor,mathematicians ventured to the edge of the infinite, but no further; Cantor calledthis edge merely ω, and showed that there are many more gulfs yet to bridge. ButCantor does recognize a closure, and more, that the closure is unstable. It is inthis vein that Cantor proscribes the absolute from being a set, insisting that it isbeyond all mathematical or rational approach:

The Absolute can only be recognized (anerkannt), but neverapprehended (erkannt), even approximately. . . . The state ofthings is like that described by Albrecht von Haller: “ich ziehsie ab und Du liegst ganz vor mir [I count them off and theylie completely before me].” The absolutely infinite sequence ofnumbers seems to me to be, in a certain sense, a suitable symbolof the Absolute. [Can15](62ff)

Yet the absolute is recognized. It follows from the most basic principles of the the-ory. It is recognizably inconsistent. The process of succession ends at the absolute,and does not. In the straightforward sense that ∞ = ∞ + 1, the absolute is asteadfast fixed point.

Having posited grounds for a continued climb, Cantor insists that there is anend of the process—an end beyond the end, what Jaspers in [Kau75] calls theencompassing,

not a horizon within which every determinate mode of Being andtruth emerges for us, but rather that within which every particu-lar horizon is enclosed as in something absolutely comprehensivewhich is no longer visible as a horizon at all.

Because all quantity falls into three categories—finite, transfinite, and absolute—Cantor requires there to be a demarcation at which the transfinites fades and onlythe absolute remains. That point of demarcation is necessary, but paradoxical.And this brings us to the main attempt to drive out the contradiction.

3. THE VIEW FROM BELOW 67

3.3. Limit On Size. In terminology that arose soon after the discovery ofthe antinomies, the doctrine of limit on size is an attempt to control the absolute.Sometimes called the central heuristic overseeing modern set theory, limit on sizeis an interpretation of Cantor’s absolute, though it took insight on Russell’s part,in his 1905 presentation to the London Mathematical Society, to draw it out, andvon Neumann to baptise as a law. Axiomatically, von Neumann states that no setmay contain as many objects as the universe. (In NGB set theory, that On is a‘proper class’ is equivalent to the axiom of choice.) In a previously untranslatedbut otherwise transparent passage,

Aber sie sind nur dann als Elemente bei der Bildung andererMengen verwendbar, wenn sie nicht “zu gross” sind. . . . Einegenaue Definition: Eine Menge ist dann und nur dann “nichtzu gross,” wenn sie von kleinerer Machtigkeit ist als die Mengealle Dinge uberhaupt (d.h. wenn sie nicht so abgebildet werdenkann, dass diese vollkommen Uberdeckt wird).A nice definition: A set is “not too big” if and only if it is dis-tinctly smaller than the cardinal of the set of all things.[vN76](494)

The intuition, now made eine genaue Definition, is that problematic sets are toobig; ergo, we forbid any set which is too big, zu gross, and the problem ostensiblyvanishes along with the problematic sets. By the time of the Tarski Symposiumin 1971, limit on size was so entrenched that Church, in his paper “Set Theorywith a Universal Set,” went as far as to claim that only two heuristic principlesgovern set theoretic progress: comprehension, and limit on size: “We must avoidaxioms leading to sets which are too large, especially sets having a one-to-one ormany-to-one relation with the universal class.”[Hen71](296)

While the other set theorists of his day saw the comprehension principle asinviolable and therefore moved to make ad hoc restrictions on the set of all ordinals,(see Moore chapter three and my next chapter), Zermelo moved to weaken capacityto form sets, by giving his separation axiom (Aussonderung),

(∀x)(∃y)(∀z)(z ∈ y ↔ z ∈ x ∧ Φ(z)).

(See Zermelo 1908b in [vH67].) Aussonderung is sometimes considered to be alimit on size in itself: New sets can be extracted only from previously given sets,and if a is not overlarge, then any subset b of a will not be overlarge. This beingmuch like an induction clause, the limit doctrine has an iterative character. If anyoverlarge set did exist, though, separation would not be an adequate block againstcontradiction [Hal84](251).

Limit on size is called up by the otherwise overtly inconsistent drive of thegenerating principles. On a prevalent view, limit on size and generalization together

68 2. THE ABSOLUTE

are a consistent extraction from Cantor’s theory. Challenging this view is the taskof the next chapter, but we can say now that without any interpreting, fixating onthe view from below seems just to identify the contradiction generating principlesengender, especially in conjunction with the domain principle. Since the absolute isthe set of all sets, a limit on the absolute is just the absolute again. Having statedthe transcending iterative conditions, then, we view the absolute from the otherdirection.

4. The View from Above

While the absolute confirms that consistent counting, or even just conceptualiz-ing, does enjoin paradox at some point, the absolute does not show how to mark offa clear boundary—at least, not before the point of inconsistency is already reached.The absolute has so far only served as a name for a problem, and not a solution.We do not yet have enough information about the absolute for it to serve Cantor’sintended purpose, of demarcating the bounds of the consistent universe. Havingonly gazed up from below, we are missing much of the picture; so in this part weobserve the panorama from above, using reflection principles: Any properties of theabsolute are also properties non-absolute sets. That is, the absolute, which Can-tor wrote as Ω, reflects. In this mode the absolute represents not inconsistent butthe qualitatively inaccessible, or even ineffable, aspects of infinity. This is alreadyinconsistent with respect to expressing the inexpressible; see [Pri02a] chapter 8.

Cantor urges that absolute infinity is incomprehensible. “In a certain sense ittranscends the human power of comprehension, and in particular is beyond math-ematical determination.”[Hal84](13) The reflection principle states that the ab-solute is beyond reckoning, because any attempt to comprehend it will inevitablyresult in only capturing a smaller subset. Formally, with Φ a predicate,

Φ(Ω)→ (∃X)[X ⊂ Ω ∧ Φ(X)],

where ⊂ denotes proper parthood.Reflection theorems were developed formally and independently by Azriel Levy

[Lev60] and Richard Montegue in 1960, and at least since then has provided a fertilesource of active research in set theory; see W.W. Tait in [DO98]. Reflection offersa downward justification of the existence of transfinite numbers. Consider someclaims about the absolutely infinite, which I will here follow Cantor in writing asΩ; let us see how they reflect, following a suggestion in [Ruc82]. The mechanismis: If some property holds of the absolute, it holds of some lesser set.

We can first generate the lowest of the transfinite ordinals, since

Ω is greater than every finite number.

4. THE VIEW FROM ABOVE 69

By reflection some lesser object satisfies this property; and then by wellorder, thereis a least such object. Call ω the least such object greater than any finite; reflectionthus implies the existence of ω. We can restate the upward generalization notionnow in terms of reflection:

The successor operation cannot reach Ω.

This gives the reflected limit ordinals. So the familiar low hanging fruits of settheory obtain quite quickly.

To see how the large cardinals are justified, consider

For all κ < Ω, also 2κ < Ω.

This yields (weakly) inaccessible cardinals, the first large cardinals. Investigatinglarge cardinals was inaugurated in 1908 by Hausdorff, who considered cardinals ofthe form α = ℵα. So similarly, the existence of an inaccessible could be reflectedby a theorem (thm. 5.72) to the effect that Ω = ℵΩ. The reason such cardinalsare called inaccessible is also a reflection issue. We have seen that the absolute isin fact the universal domain of quantification. For an axiomatic system like ZF ,a domain of some least rank θ can serve as a model of ZF . And θ, as a model ofthe classical universe of sets, reflects the absolute. (The details of this are spelledout in my chapter 6.) Large cardinal enthusiasts, in particular Akihiro Kanamoriin his wonderful The Higher Infinite, bring reflection to bear heavily on helping usunderstand the structure of the universe. Reflection is evidently very powerful.

Here then we have a characterization of the Cantorian program as picking outthose aspects of infinity amenable to rigorous theorizing, reflections of the absolute.If this characterization is correct, then there is little wonder that some set theoreticparadoxes are irremediable. The edifice is extracted from an explicitly inconsistentobject.

As with the axiom of choice, naive theory can make sense of what is otherwisea dark doctrine. The view from below is just counting up from the finite, and hasan uncontentious grounding, even if an overzealous extension. Descending fromabove, on the other hand, requires heavier metaphysics if it is to make any senseat all. A great advantage of the naive view will be to prove the reflection theorem(thm 6.2), from nothing but the basic notion of set. In turn, this will prove all thelarge cardinal axioms, and so divest higher set theory of some dubious conceptualabysses.

As with the view from below, so is the view from above not without paradoxes.Indeed, notions like ‘incomprehensible’ or ‘ineffable’ have been recognized at leastsince the Hindu’s Brahman as sources of inconsistency [Pri02a]. These sorts ofparadoxes are indigenous to inclosure. With respect to reflection, the proof issimple. Consider the second order property Ω(X), “Can only be predicated of

70 2. THE ABSOLUTE

Ω.” When predicated of Ω, the predicate is satisfied. But, analytically, no otherobject will do—violating the reflection requirement that some lesser set also satisfy.Reinhardt, in [Jec74], gives an honest and informal discussion of the absolute andreflection principles. After stating the “naive reflection principle,” he notes that theproperty “x = Ω” can only be satisfied by Ω itself; we might call such a propertyirreflective. Thus “something more subtle is required.” Rather than give a tightenedparsing, though, Reinhardt goes on to say

According to Cantor, Ω is unlimited. (Cantor shared with classi-cal Greece a certain distaste for the απειρoν (unbounded, inde-terminate), under which he classed the potential infinite.) To theextent that our thinking is limited, then, it should be compat-ible with what we think or understand (about Ω, for example)that the same could be thought or understood of some κ < Ω.Thus we do not understand (in the requisite sense) the propertyP (x)↔ x = Ω. [Jec74](191)

The argument here reasserts that reflection is meant to capture an aspect of Can-tor’s doctrine concerning incomprehensibility. Because we only understand theabsolute in a limited way, what can be said precisely of it can be said of some lesserset; and what can only be said of the absolute, cannot be said at all.

Plainly, though, we have just said it. And we can all understand exactly whatit means for Ω to be Ω. If not, then the whole reflection enterprise is unintelligible.The statement of the reflection principle—‘anything that can be said of Ω can besaid of something less’—itself would be nonsense. In part two we will see a proof ofKunen’s, from [Kun71], that Reinhardt’s investigations are classically inconsistent.

Rather than stage a retreat, though, consider the potential for self-referencehere:

Ω reflects.

Then there are lesser reflecting sets, too:

If Ω reflects, then (∃X)(X ⊂ Ω ∧X reflects);if X reflects, then (∃X0)(X0 ⊂ X ∧X0 reflects);if X0 reflects, then (∃X1)(X1 ⊂ X0 ∧X1 reflects)...

This provides, again, a fertile source of large cardinal theorems for the paraconsis-tentist. And it explains the non-self-identity of Ω we will see in chapter five; forthough there are lower reflecting sets, still they may all still be just Ω. Like theRussell set in the opening incursion, since Ω properly reflects itself,

...Ω ⊂ Ω ⊂ ... ⊂ Ω...

5. CONCLUSION 71

This is the key mechanism to the wellordering theorem. The transfinite, whenviewed from above, is a part of the absolute.

5. Conclusion

Formerly there were only two taxonomical categories, finite and infinite. Cantoropened a space between the two. The transfinites, as far as possible, are treatedlike finite numbers; though they are without end, they are still graspable objects ofintelligible and precise manipulation. Such is the view from below, which inevitablyinvites a horizon of its gaze, and this leads to the absolute as iterative limit. Viewedfrom above, the absolute has all the incomprehensible aspects of infinity, which canbe understood through reflection.

Laying down criteria to judge whether an infinity is transfinite or absolute, andmutatis mutandis consistent or inconsistent, almost by necessity fails, because as amatter of paraconsistent logic, ¬(Φ∧¬Φ) is a theorem—so no sets are inconsistent.And similarly, the naive approach is exactly to treat the universal set or the set ofall ordinals as sets like any other, so absolutely infinite sets are also transfinite; nosets are absolute. In a sense, this just is the paradox—that it must be somehowapart from the set theoretic inclosure, and it is not.

Totalities necessarily involve limits; without a boundary there is no totality.When true totalities—wholes which are not contained in any greater collection—are considered, paradox results. The absolute is a fixed point where there can benone. In 1930, Zermelo presents a summary of the situation [Zer30]:

The “ultrafinite antinomies of set theory” that scientific reac-tionaries and anti-mathematicians refer to so assiduously andlovingly in their campaign against set theory, these seeming“contradictions,” are due to a confusion of set theory itself, whichis non-categorically determined by its axioms, with particularrepresenting models: What appears in one model as an “ultra-finite non or meta set” is in the next higher one already a fullyvalid set with cardinal number and order type, and is itself thefoundation stone for the construction of the new domain. Theunlimited series of Cantor’s ordinal numbers is matched by justas infinite a double series of essentially different set-theoreticmodels, the whole classical theory being manifested in each ofthem. The two diametrically opposed tendencies of the thinkingspirit, the idea of creative progress and comprehensive comple-tion, which also lie at the root of the Kantian “antinomies,”find their symbolic representation and symbolic reconciliation in

72 2. THE ABSOLUTE

the transfinite series of numbers based on the concept of well-ordering. This series in its boundless progression does not have atrue conclusion, only relative stopping points, namely those limitnumbers which separate the higher from the lower model types.And thus also, the set theoretic “antinomies” lead, if properlyunderstood, not to a restriction or mutilation but rather to apresently unsurveyable unfolding and enrichment, of mathemat-ical science.

What remains is to take this knowledge, which is everywhere expressed by settheorists, and to admit that it is an integral part of set theory itself.

The finite is pressed directly against the limit, against the absolute. The trans-finite is fractal structure in the cracks. And structure demands study. At first,explanations of the gap between finite and infinite enjoined contradictions. Thenthe contradictions became definitions: Infinite sets are equipollent with proper partsof themselves; finite sets are not. Now there is a need for explanation of the gapbetween transfinite and absolute, and there are contradictions, too. The confusioncan be handled in much the same way. As our dialethic mathematics will reveal inpart two, the transfinite is little more than what can be coherently discerned of theabsolute. The task of this chapter has been to till the ground in preparation.

CHAPTER 3

Foundations in the Twentieth Century

Nothing is easier than to devise expressions or notations.... Butif we remove the veil and look underneath, if, laying aside theexpressions, we set ourselves attentively to consider the thingsthemselves which are supposed to be expressed or marked thereby,we shall discover much emptiness, darkness, and confusion; nay,if I mistake not, direct impossibilities and contradictions.- Berkeley, The Analyst, 1734

In the last two chapters we have positioned some conceptual elements of nine-teenth century set theory. That original theory was paradoxical in both the formaland informal sense—for it was clearly correct mathematics, and just as clearly, itwas logically impossible mathematics. In this chapter we will consider in full thenext generations’ reactions, the twentieth century doctrines encasing the mathe-matics of infinite, indicating how the early notions have been transferred into themodern apparatus, contradictions and all.

1. Introduction

A natural thought in the contemporary climate: Isn’t the iterative concept ofset a consistent and adequate set concept?

In their authoritative and influential 1958 book on the foundations of set the-ory, Fraenkel, Bar-Hillel and Levy state, as uncontroversial fact, that set theoryand therefore all of mathematics is in an ongoing crisis [AF58](14). What haschanged in the intervening years? The sections below are my partial answer to thisquestion—that because the crisis has not been resolved, instead the statements of ithave been discredited. The assumption that the iterative notion of set is successfulhas been widely challenged in many places for many years—by Routley and Priest[PRN89], but also questioned by Boolos [Boo98], rebuked at length by Lavine[Lav94] and by Weir [Wei98], as we will be seeing. Some of these charges aregathered together in [Woo03](155-163) as “the dialethic mini-history of set the-ory,” and the lot is synthesized in chapter three of Berto’s [Ber07]. Since I begandrafting this chapter, even, a swath of wider criticisms have been collected in es-says in [RU06], there referring to doubts already aired in [ST00], on the needs of

73

74 3. FOUNDATIONS IN THE TWENTIETH CENTURY

quantification and the concomitant shortcomings of the iterative concept. So thethoughts here are not unusual. Nevertheless, defaulting to the iterative conceptis still basic for most philosophers. The purpose of the chapter, then, is to castfurther doubt on this assumption.

Cries of a foundational crisis arise when the most basic elements of a rigoroussubject are not clearly defined, and the behavior and properties of these elements arenot adequately understood. In mathematics, the first crisis arose when Pythagorasdiscovered irrational numbers but deemed them incommensurable; another crisiscame when Newton and Leibniz discovered the calculus but deemed it infinitesi-mal. The crisis that occupies us here is in the foundations of mathematics itself,in set theory—the first self-proclaimed foundation. Again the crisis concerns theincommensurable, now returned as absolute infinity.

In some circles, it is now considered unsophisticated to talk of paradox havingever caused a crisis in the foundations of mathematics. Kanamori [Kan94](481)recounts a parable of spiders building epic cobwebs in the newly excavated vaultsbeneath a cathedral, foolishly believing their intricate threaded structure to sup-port the stone spires and buttresses above. When one day a wind tears a gap inthe webbing, the spiders are thrown into apocalyptic panic, while “of course, thecraftsmen above hardly raised an eyebrow.” One response to the suggestion thatthere is a problem here at all, let alone a crisis, is dismissal. Jones registers thissentiment in his “A credo of sorts”:

The physicist’s reaction on being told that his proof was riddledwith holes would be essentially to say, “Oh that’s just somethingfor the mathematicians to worry about.” So it is for the math-ematician with respect to the logician. If ever the day comeswhen the logicians find some inconsistency in arithmetic, our re-action will surely be “Oh that’s just a trick of the logicians; letthem worry about it.” And one can almost hear the inconsis-tency coming—perhaps there will be a proof of the existence ofa contradiction, but that contradiction is too long to even con-template, so we may quite happily behave as if it did not exist.[DO98](204)

Shapiro [Sha97] tracks a related notion with the name “philosophy-last-if-at-all.”Has the iterative set concept solved the crisis? No. The iterative set conception

is either inconsistent or inadequate. Routley paints the scene in dystopian hues.

1. INTRODUCTION 75

There are whole mathematical cities that have been closed offand partially abandoned because of the outbreak of isolated con-tradictions. They have become like modern restorations of an-cient cities, mostly just patched up ruins visited by tourists.[Rou80](927)

The problem inherent in the work of Frege, Dedekind, and Cantor—the problem ofdescribing an intensional universe by an extensional theory—remains. The problemof absolute infinity, which expresses and engenders this problem, remains. The basicfact is contradiction, and has been clearly identified. I will show how and why theproblem persists.

Following Cohen’s independence proof in 1963 and Kunen’s inconsistency resultin 1971 (see chapter 6), the most interesting of the subject’s questions, the cardi-nality of the continuum and the axiom of choice, have been left unanswered—andpermanently unanswered in the framework of ZFC. This is at least mathematicallyunsatisfactory, and there is growing dissatisfaction among mathematicians aboutZFC, “the generally accepted axioms for set theory,” writes Woodin—“but I wouldcall these the twentieth-century choice.”[Woo01](567) The problem, I will argue,is rooted in the tension between the descriptive fact that the naive set concept isstill in use, and the proscription that it cannot be, given classical logic.

First I will enter into a reprisal of recent history, and identify a methodologicalissue which is quite general: a myopia—a near-sightedness about history. Themethodological idea arises out of a sociological observation made most prominentlyby Lakatos, that science aspires to be a pinnace, always the last best hope, of aunified human project. As Shapiro puts it, “The historical assertions that conformto one’s views are to be praised as clear, while those that are incompatible aredubbed ‘confused’.” [Sha91](184) One can observe in the literature the foundingand maintenance of a mythology, a revision of history that reifies current trendsand drives out alternatives.

Then I will show, largely through remarks of working set theorists, that theconcepts underlying contemporary set theory are in need of clarification. The diag-nosis is simple. Because the situation is inherently inconsistent, but the theory isembedded in classical consistency, the result is incoherent. A main claim is just thistautology: If we insist a priori upon consistency, then it is a priori ruled out thatset theory is inconsistent. By the same token, if genuine set theory is intractablyparadoxical in places, then it is equally a priori impossible that a purely consistenttheory will suffice.


2. On Method

The abstract sciences differ from others in that they “cannot, and have no needto, negotiate the empirical check.” [Woo03](xi) Pure mathematics and logic areparadigm examples of abstract science, and arbitrating conflict in these—or evencharacterizing the conflict by fixing on what the conflict is about (method? content?opinion? aesthetics?)—is an imposing challenge. All the more so for any philosophyin the vicinity of an abstract science. “How,” Frank Ramsey asked 1931, “can aphilosophical enquiry be conducted, without a perpetual petitio principii?”

While abstract sciences and the philosophy thereof may bypass the empiricalcheck, there is still concrete data: the state and practice of mathematics. Shapiroquite rightly makes a programmatic identification:

Because mathematics is a dignified and vitally important en-deavor, one ought to try to take mathematical assertions literally,‘at face value.’ This is just to hypothesize that mathematiciansprobably know what they are talking about, at least most of thetime, and that they mean what they say. [Sha97](3)

I mean to extend this principle to include not just mathematical assertions, butalso assertions made by mathematicians about their subject. This can subtendsome straightforward observations about set theory: of outright conflict internaleven to the consensus doctrine, of ideas, motives, and methods that depart froma purely iterative concept of set. Let us clarify method as the business of taking“the full range of contemporary mathematics seriously. It is a basic datum that thebulk of mathematics is legitimate ... something to be interpreted and explained,not explained away.” [Sha97](51) In agreement with this dictum, we will try todisentangle what talk from experts is only after a facon de parler, and which partsreally do not fit the orthodox view of set theory occurring in an iterative, non-constructive, and ultimately uncomprehended wellorder.

Burgess [Bur05] invokes the difference between descriptive and prescriptivefoundational mathematics. Frege saw his own work as descriptive, Burgess pointsout, capturing the inferences and assumptions used in informal proofs and therebyshowing that the standing edifice of mathematics, “dignified and vital,” is secure.“I did not wish to represent an abstract logic by formulas,” writes Frege in 1883[Sha91](175) “but express a content ... in more exact and clear fashion....” Theavowedly descriptive character of Frege’s program is what makes logicism a legit-imate foundational, rather than interpretive, program. One can go so far as tomake description an adequacy condition for any foundation of mathematics. Thetool for disentangling the expert’s talk is to figure which expressions are descrip-tions of mathematical structure, and which are expression of prescription about

3. PARADISE LOST, PARADISE REGAINED 77

how that structure should be studied. It is with the goal isolating the descriptivefragment, albeit not a description that would win immediate universal endorse-ment, that we proceed. In following Shapiro and taking what mathematicians sayseriously, the most difficult interpretive and explanatory task is how to respondwhen mathematicians speak in contradictions, all the while claiming consistency.

3. Paradise Lost, Paradise Regained

One received mythology tells us

3.1. A Story. On 7 December 1873, Cantor discovered that the reals areuncountable and thereby invented set theory; and that for a time, for the romanticperiod, all was halcyon in the garden of infinity, as we nibbled cherubic on theordinal fruits of the transfinite, revelling in the place Hilbert called a paradise.

In those days, we thought that any collection at all could be a set. After all,set theory arose out of concerns with arbitrary real-valued functions (a paradigmexample: Dirichlet’s “shotgun function,”

D(x) =

1 if x is rational;

0 if x is irrational,

everywhere discontinuous), functions that do not arise naturally but can neverthe-less be defined and so needed to be understood, to keep analysis above alchemy.From this motivation, forming sets from arbitrary collecting predicates is appropri-ate, intuitive, even tautological. Against this background, set theory was intendedto clarify the foundations.

In those early days, a small band worked to anchor basic conceptual processes inprecise formal methods. There was no marked boundary between first and second-order logic; there were no inhibitions about including hefty metaphysics in themachinery.

Yet, alas, there was a serpent in the garden. Frege was caught unaware, butby 1895 Cantor certainly knew of the inconsistencies in his system. Russell tooknew of Cantor’s paradox and from it abstracted his own paradox at the dawn ofthe 20th century (as Moore describes in [Hin95]). The ship was rocked amidst asquall of related paradoxes, and we realized that we cannot trust our most basic ofintuitions. “It is not for nothing, after all, that set theorists resort to the axiomaticmethod,” writes Quine. “Intuition here is bankrupt...”[Qui69](x) In time, Zermeloand von Neumann salvaged what they could from the wreck of Cantor’s system.Frege’s work is now archaic. The gates of Cantor’s paradise, of which we may stillspeak, but which can never regain its glory, closed.


Or so the old story goes. The thought has emerged and grown up in theliterature that the old story is inaccurate. This culminates in a recent and influentialbook by Lavine [Lav94], who goes so far as to say of the old story that “not oneword of it is true,” and mocks the Russell/Quine diagnosis that “we can never relyon our intuitions again.” Instead there is

3.2. A New Story. “Cantor’s original theory was neither naive nor subject toparadoxes.”[Lav94](3) His paradise is today as it always was, an unshaken ladderfrom zero to beyond. Cantor’s theory had no problems, and still does not. “Theparadoxes posed no problem for Cantor’s theory of sets—transfinite objects thatcan be counted.”[Lav94](76) Cantor knew all about the so-called contradictions,having written to Dedekind well before Russell wrote to Frege, along with anothershort note to Philip Jourdain asserting that the collection of all sets cannot be a set[Hal84](286). Cantor’s followers, notably Hausdorff and, on this account, Zermelo,were unruffled by the so-called crisis in the foundations of mathematics, and withgood reason: There was no crisis. Cantorian set theory was, is, and will remaingood, classical mathematics, and therefore consistent and true.

Lavine’s [Lav94] story of the set concept in precis is as follows. Cantor’s theory,until the 1890s, was of transfinite objects that can be counted (i.e. wellordered),called sets. Cantor did not assume the domain principle, which is a naive compre-hension principle, but instead expected to be able to prove as an instance that thereal numbers are a set (92). He expected to prove that the reals are a transfiniteobject that can be counted. Only late in his career did Cantor begin to see that thepower set axiom might lead to the desired proof that the continuum is 2ℵ0 . Onlyhere did the troubles begin. Because “Cantor needed to allow for the existence ofa set that he did not know how to introduce explicitly via a counting,”(95) nowhis “theory was in trouble, but it was not trouble caused by the paradoxes. Itwas trouble caused by trying to fit the power set axiom into a theory that tookwell-orderings to be primary.”(97)

Ergo the Cantorians, with their consistent notion of wellorder, charged ahead:Zermelo’s 1908 axioms, often assumed to be in part a response to paradoxes, arejust the necessary ingredients for proving the wellordering theorem, and a fortioria demonstration that the wellordering principle is free of soft notions or theolog-ical metaphysics. Similarly, Felix Hausdorff, with wholesale disdain for founda-tional arguments, “the only productive set theorist who is so little irritated by theparadoxes” (according to Hessenberg) in the immediate wake of what he deemedan attack with “medieval weapons” [Hau05](182) published work on ordertypesand showed the first glimmers of large cardinal research. Though under the namePaul Mongre he wrote metaphysical poetry and politically critical novels and plays


[Czy94], (and in another sad bit of biography, discovered his high academic po-sition would not save him and committed suicide along with his family when theNazis conquered Germany), when writing as Hausdorff he approached the credo ofJones: In 1908,

An investigation such as this, which endeavors to add ... to thepositive inventory of the still so new theory of sets in the spiritof its creator, cannot prae limine spend any time to enter intoa discussion ... where presently a somewhat misplaced degree ofingenuity is squandered....1

And unchanged decades later: “In our opinion,” Hausdroff wrote again in 1937, “itdoes not detract from the merit of Cantor’s ideas that some antinomies that arisefrom allowing excessively limitless constructions of sets still await complete eluci-dation and removal.” [Hau57](11) A first tiller over the topology of Cantor’s earth,Hausdorff decried the intellectual efforts wasted on hand-wringing about paradoxwhen these same energies could be put toward fruitful research—“the pursuit ofstructure for its own sake.”[Kan94](xxi)

Wherefore, then, all the mythologizing about paradox? That, the new storyproposes, was all part of a different narrative. Here we have not Cantor, but Frege,toiling away at an altogether different, baroque task of deducing arithmetic fromlogic. Frege’s logic was higher-order and his Basic Law V an unrestricted com-prehension principle, so the confusion with set theory is forgivable, if regrettable.Russell’s discovery of the set x : x 6∈ x, which after all he mailed not to Cantorbut Frege, was devastating only to this latter logicist’s program. It was Russellwho saw a contradiction in Burali-Forti’s paper; it was Russell who went beforethe London Mathematical Society to speak about the dire threat of the paradoxes[Rus05], and it was Russell again who’s reconstruction in the Principia ultimatelyimplicated anything Cantorian. In short, the paradoxes belong to the logicians—we might as well be honest and call them philosophers. Basic Law V conflatesproperties with sets. Isn’t this like conflating astrology with astronomy? Contra-diction is inevitable. Thus, in a distinction running from Godel through Wang, toMaddy and Shapiro and now Lavine, good collections are mathematical sets, andbad collections are metaphysical, ‘logical’ sets.

Or so goes the new story.

1Hausdorff’s comments continue: “To an observer, who in the face of skepticism is not wantingin skepticism, the ‘finitistic’ objections against set theory might roughly fall into three categories:

those that reveal a serious need for, perhaps an axiomatic, sharpening of the set concept; intoanother fall those that would affect the whole of mathematics together with set theory; finally,

there are those that fall into the simple absurdities of a scholasticism that clings onto words and

letters. One shall be able to come to an understanding with the first group sooner or later; one maysafely let the second rest; the third deserves the sharpest and clearest disapproval.” [Hau05](198)


3.3. Which story to believe? Like all stories, both contain elements ofreality and fancy. Michael Hallett leaves the issue complicated and unresolved.Dauben largely supports the old story; Moore does, too. Potter is skeptical aboutimporting anything too modern into the past, observing that while some try “tomake the history of the subject read more like an inevitable convergence on onetrue religion,” really the iterative idea was not clearly articulated until Godel in1944 or [G64], or even Boolos in 1971, and warns us to be wary of the Panglossianview that “the paradoxes are not really so paradoxical if we only think about themin the right way,” since this idea “is hard to find in print before Godel.”[Pot04](26,37). (Potter has probably overlooked Zermelo’s 1930 paper [Zer30], but this wouldstill not impinge on the old story.) In the next sections we can go into some detailabout the origins of the reconstructed set theory and its history. But I think thatthe old story and the new story both have something to teach, not least because oftheir telling; to show this I’ll utilize the familiar, helpful description/prescriptiondistinction.

The old story, for all its shortcomings as apocrypha, is roughly descriptive, andends with a prescription: Here are paradoxes, and here is how we should avoidthem. It states a problematic is and offers a programmatic ought. The new story,on the other hand, is prescriptive. It boasts finer distinctions within the reactionsof the mathematical community, and suggests important insights about the causesof paradox. But the new story also claims to be descriptive: This is what we do,or else there would be paradoxes—but, of course, there are no paradoxes. The newstory makes an is of an ought.

The old story is nostalgic. The new story is myopic. At least, it betrays a char-acteristic of mathematical practise: a preference for perfection, for clearing awayrough edges, leaving only the beautifully syncopated call-and-response of theorem,proof, theorem, proof. Example: The luminary C.F. Gauss published little, terselyand without supporting motivation; of him Abel said approvingly, “He is like thefox, who effaces his tracks in the sand with his tail.” Gauss himself said that “noself-respecting architect leaves the scaffolding in place after completing the build-ing,” or again, “a cathedral is not a cathedral until all the scaffolding is removed.”Mathematics is the science of certainty; and I merely point out that what Lakatoshas called the Euclidian tendency [Lak76](preface), and what Nietzsche in allusionto mummification called ‘Egyptianism’,2 is pervasive not only in mathematics, butmathematical history.

2“You ask me about the idiosyncrasies of philosophers? ... There is their lack of historical sense,

their hatred of even the idea of becoming, their Egyptianism. They think they are doing a thing

honor when they dehistoricize it, sub specie aeterni—when they make a mummy of it.” Twilightof the Idols, “‘Reason’ in Philosophy” I.


Who makes claim to history, it must be said, has little bearing on truth. Thehigh tradition of medieval medicine ought attract no practitioners now. It is prob-ably distracting and specious to trade blows over precedent. Nevertheless, for whatit is worth, the old story is the one I have selectively and interpretively told, be-cause I believe that some version of the old story is true—at least by the principlethat the closer witness is more likely to be correct. The old story is found in Rus-sell’s pre-Principia writings, Hilbert’s epochal 1925 speech “On the Infinite,” and isthen largely propagated by a first hand participant, Abraham Fraenkel, in own hispopular texts [Fra53], [AF58], and historical introduction to Bernay’s [Ber68].There was a romantic period, there were surprising paradoxes, and in time, therewas some consensus reached on a solution. If this is right then paraconsistency canoffer a natural continuation of the process. But, as the remainder of this chap-ter must show, the current paradox-avoiding strategy was built into the rhetoric,canonized, dogmatised; the messy, organic formation process was swept away, hid-den like scaffolding to glorify the cathedral. What was born as ought is promotedto is, and the hermeneutic circle tightens with perfect and timeless mathematicalprecision.

The iterative hierarchy makes claim to being descriptive, of illuminating thebasic ontology of practice; it makes this claim because the claim is needed forlegitimacy, or at least to stave off suspicions of ad hoc-ery. But this claim is false.The iterative hierarchy represents only a fragment of the mathematical universe.Current problems in our understanding of set theory are explained by seeing thatthe iterative reconstruction of set theory always was prescriptive. The point ismade explicitly and overtly not least by Zermelo himself, who leaves no room forhistorical revision (although c.f. [Moo82] at the outset of his 1908 axiomatization:

At present, the very existence of the discipline [of set theory]seems to be threatened by the existence of certain contradic-tions or ‘antinomies’ that can be derived from its principles—principles necessarily governing our thinking, it seems—and towhich no entirely satisfactory solution has yet been found. ...Cantor’s original definition of a set (1895) ... requires some re-striction; it has not however successfully been replaced by onethat is just as simple and does not give rise to such reservations.[Zer67](200)

And von Neumann, at the close of his own sophisticated axiomatization in 1925,sighs that despite much work, still he must “entertain certain reservations aboutset theory” because “for the time being no way of rehabilitating this theory isknown.”[vH67](413)


Now to support the claim that set theory is in ideological confusion, using ourliteralist principle to guide a closer look at the twentieth century technical responsesto contradiction—the iterative concept of set, proper classes and limit on size.

4. Hierarchy

What follows are discussions of the three main turrets of classical set theoreticideology, iteration, proper classes, and limit on size, and some statements mathe-maticians make about them. A literal reading of these statements together betraysconfusion—a collision of intuition and iteration.

4.1. Iteration. Mathematics displays remarkable invariance. In the first halfof the twentieth century, disparate research independently converged on the imageof an infinite hierarchy. This is a means to escape the paradoxes of naive theory. Onemight even say that a certain attitude toward paradox, as a trigger for Dummetian‘indefinite extensibility’, naturally leads to hierarchical structures. Priest identifiessuch structures as “latterday Kantian attempts to retain a certain control overconceptual production.”[Pri06b](48) Cast in terms of the inclosure schema, thisview of paradox emphasizes the transcendence conditions, and denies closure. Andthis is the shortcoming of the iterative sets—not that they are not sets, or bad sets,but only that they are not all the sets. By denying closure, the iterative sets areneither enough sets, nor are they adequately explained.

Russell and Whitehead in Principia, Zermelo in 1930 (though emphatically notin 1908), and then Godel in a popular lecture in 1947, all present what in retrospectmay be called variants of iterative hierarchies. The idea came to the full popularattention of philosophers only in 1971, with George Boolos’ “The iterative concep-tion of set” article, reprinted in [Boo98]. This immediately follows a main momentfor set theory in the second half of the twentieth century, the axiomatic set theoryconference of 1967 [Sco68], [Jec74]. Just a few years after Cohen’s forcing resultsput the continuum hypothesis out of immediate reach, at that conference the com-munity rallied and reached de facto consensus on the foundations of the discipline,fixing the tenets one now finds repeated in the literature. “By all accounts this wasone of those rare, highly exhilarating conference that both summarized the progressand focused the energy of a new field opening up.”[Kan94](115) A leading voice inthe area is Dana Scott, who in his seminal address there proved that a prior notionof ‘stages’ can be employed to derive the iterative notion of set. He declared:

Our original intuition of a set is based on the idea of having al-ready fixed objects. The suggestion of considering all-inclusivecollections only came in later....The suggestion proved to be

4. HIERARCHY 83

unfortunate, and so we must return to the primary intuitions.[Jec74](207)

These intuitions are summed in a standard recursive scheme,

V0 = ∅,

Vα+1 = x : x ⊆ Vα,

Vγ =⋃β<γ

Vβ if γ is a limit.

Often, the closure condition is also included, with On the set of all ordinals:

V =⋃

α∈OnVα

This is an anomaly we return to in discussion of overlarge sets.The iterative concept draws on finite intuitions and deeply implicates a con-

structive metaphor: An iterative set y can only be formed out of pre-existing ob-jects. Therefore it cannot be that y ∈ y, and most of the antinomies stall. If theiterative hierarchy is born as an inductive structure, then the axiom of foundationacts as a maximality condition on its universe—asserting that the induction is ex-haustive, asserting ‘and that’s all.’ It is worth noting that Zermelo in 1930 wasambivalent about ruling out non-well-founded sets, ceding only that the founda-tion axiom “introduces for the moment no essential restriction of the theory.”(in[vA86], emphasis added)

In his highly influential paper, Boolos, having first admitted that the naive ideaof sets “might occur to us quite naturally,” and indeed that it has “great force,”nevertheless modulates his rhetoric a few paragraphs later:

It is important to realize how odd the idea of something’s con-taining itself is. Of course a set can and must include itself (as asubset). But contain itself? Whatever tenuous hold on the con-cepts of set and member were given one by Cantor’s definitionsis altogether lost if one is to suppose some sets are membersof themselves. The idea is paradoxical not in the sense thatit is contradictory, but that if one understands ‘∈’ as meaning‘is a member of’ it is very, very peculiar to suppose it true.[Boo98](18)

Boolos is aware that this is a bald assertion. “There does not seem to be anyargument,” he admits, “to persuade someone who really does not see the peculiarityof a set’s belonging to itself.” Assuming nevertheless that friends of self-membershipare sufficiently shamed, Boolos carries on to explain that a set is something that


lives on the cumulative hierarchy, and he proves it with a reworking of Scott’s stageconstruction (discussed below).

At that 1967 conference, a perspicuous model of ZF out for viewing signalledan end to the embarrassment of antinomies. Reinhardt:

It has been pointed out that the paradoxes of Russell, Burali-Forti etc, never really caused a crisis in mathematics (whereone deals only with unproblematic examples of sets) but ratherin logic (and general set theory), where one tries to provide ageneral and universal frame for mathematics and in particularfor arbitrary sets. We now consider such a frame to have beenprovided for set theory by the clarification of the intuitive ideaof the cumulative hierarchy.... [Jec74](190)

Reinhardt distinguishes between mathematics and set theory, accepting the sacrificeof a foundation for mathematics and ‘general’ set theory if a foundation can beprovided for set theory simpliciter. This is a more subtle analysis, with modestaspiration. Indeed, part of the proposed repair of set theory involves recasting itas one branch of mathematics among many, and not a foundation at all. This is atheme to which we shall return, but it is worth noticing immediately then that sucha retreat makes any explanatory work on the part of set theory impossible. Theexpected payoff is that all mathematical paradoxes evaporate if we only considerthe iterative sets:

It should not be forgotten that the paradoxes never applied toany type structure, and in this sense they are not paradoxes ofset theory.[Dra74](14)

Godel had already intimated such forceful thoughts in 1947: that the iterativeconception, unlike “something obtained by dividing the totality of existing thingsinto two categories, has never led to any antinomy whatsoever.”[G64]

The humble distinction is easily elided over for bolder claims. The hierarchicalclaim is not just that iterative sets are the only sets, but that iteration explains whatsets are. They are offered as a foundation, because not only are all mathematicalobjects reckoned to be iterative sets; also this supplies a raw and primal set concept,and so the reduction to sets is illuminating, worthwhile and sound. This is Boolos’explicit purpose, to show that

without prior knowledge or experience of sets, we can or do read-ily acquire the conception, easily understand it when it is ex-plained to us, and find it plausible or at least conceivably true.[Boo98](89)

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Given these are the aims, many problems emerge for the hierarchy, most of themfatal.

The first is the foremost: The reduction fails because it presupposes naive settheory, including the notion of ordinality. Therefore the iterative concept is notthe foundational concept and is not explanatory. It is viciously circular, as is nowexplained.

There is an obvious problem in explaining set theory in terms of iteration. Asdemonstrated by Scott, Boolos, van Aken [vA86], and most recently Potter in hisinformative text book [Pot04], one can deduce the axioms of ZF from some ‘stageaxioms’, postulates about levels of the hierarchy, or some equivalent ordering notionlike ‘presupposes’. The level of technical detail in presentation varies, but the ideais simple enough. We imagine a first level, or stage, comprising all the previousstages, which is meant to be the empty stage. (Note that the inference if there areno sets then all sets are in ∅, is just a form of explosion: p ` ¬p ⊃ q. So the exnihilo construction in this form is very classical.) At the first and each successivelevel we collect up and take the accumulation to be the next stage. Some postulatesare stated to formalize notions like ‘earlier than’ and ‘is a stage,’ and the processis indexed to keep order. At the very least, then, stage axioms require:

- the existence of stages;- one or more order relations;- and an index set.

Usually, these are exactly the sorts of things developed from within set theory.Especially the last, an index set, is the most egregious. As this is laid out in acontemporary text, Devlin notes that

the description of the set theory depends on the constructionof the cumulative hierarchy of sets, the Vα hierarchy, and this,in turn, depends upon the ordinal number system. We are thusassuming a considerable amount of “set theory” in order to defineour set theory. [Dev79](49)

This is an instance where taking mathematicians at their word is challenging; what-ever the difference between “set theory” and set theory is, it is good indication thatsome foundational conceptual matters are in need of clarification. The unlikelyinsertion that “there is, of course, no real dilemma here” notwithstanding, a stagetheory involving the first, second, nth stages, has overtly and persistently presup-posed the ordinals. Priest and Routley:

The construction of sets presupposes a prior construction of or-dinals. However this raises all sorts of problems about ‘howfar’ the construction can be continued, about sizes of infinities,


etc. Indeed it is just these kinds of problems that the theoryof sets was supposed to solve. We do not deny that once onehas a notion of set one can non-circularly produce ... the cumu-lative hierarchy. But to suppose one can use the notion of anordinal to produce a non-question-begging definition of ‘set’ ismoonshine.[PRN89](500)

When in Devlin we meet the iterative hierarchy, we are told

Before we can build sets of objects, we must have the set ofobjects out of which to build these sets. The crucial word here,of course, is ‘build’. Naturally we are not thinking of actuallybuilding sets in any sense, but our set theory should reflect thisidea. [Dev79](43)

From passages like this, there is some question of what the iterative concept is sup-posed to mean. If nothing else, this is not at the level of clarity or rigor one expects.The scare quotes scattered through Devlin’s discussion indicate that the construc-tive metaphor can be eliminated. Rather than forming or building sets, whichwould become unfeasible not even at the infinite stage, we can paraphrase with thestatements: there is a first stage, there is a ω stage, ... (See Potter [Pot04](36 -40) for discussion of construction metaphors, idealized constructors, etc.) This isnow mathematically cogent, but it eliminates the naturalness or intuitiveness of theiterative view, in Boolos’ sense, and makes all the tortured building or presuppo-sition talk pointless. Whatever naturalness the iterative concept has is lost whenwe fall back on bare existence claims. If the proper way of speaking, the phrasingwithout scare-quotes, is not constructive at all, why not a set containing itself? AsWeyl aptly observes,

No one can describe an infinite set other than by indicating prop-erties which are characteristic of the elements of the set. ... Thenotion that an infinite set is a ‘gathering’ brought together byinfinitely many individual arbitrary acts of selection...is nonsen-sical. [Wey19](23)

Finally, the iterative story does not anyway justify all the ZF axioms. Boolosis dubious about power set; van Aken cannot justify replacement; and Lavine findsthese and more wanting. Zermelo’s axioms are not, and were never meant to be,an axiomatic description of the iterative hierarchy. And so the iterative story isno replacement for, and is indeed parasitic on, the naive set concept. I take theseobjections against the iterative concept itself to be, in Routley’s idiom, “central,deep, and disabling.” Let us turn to some of the other features of the classicalsolution, some already familiar from the last chapter.

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4.2. Proper Classes.

Sooner or later, in any course on set theory, the instructor findsit necessary to introduce the notion of a proper class. And thismoment he usually dreads. For with virtual certainty, one oftwo things will occur. Either his students will suddenly gain theuneasy feeling that a sleight-of-hand method of introducing somevery dubious notion is being employed. Or else the student willaccept the new notion but fail totally to understand what exactlyis being done. For us, this moment has come. [Dev79](57)

One way of introducing proper classes is as the dual of an urelement. For anyy, if ¬(∃x)x ∈ y, then y is an individual ; if ¬(∃x)y ∈ x then y is a proper class.Every set is a class, but proper classes are not sets.

The key behavioral property of proper classes is not to be a member of anything.If proper classes can’t be members, then R 6∈ R but no more, and no Russell’sparadox. What to say about Cantor’s paradox is less obvious—V is not a member ofP (V ), perhaps?—but the general issue is meant to be resolved because his theoremonly applies to sets. One tradition in set theory, e.g. in Kelley’s appendix to[Kel55] or Takeuti and Zaring’s extremely precise [TZ71], is to have a predicatefor ‘is a set,’ which does the work of tagging objects that do not seem to triggercontradictions. Proper classes are the repository for blame when an inconsistencyarises. One could forgive a non-initiate for thinking that to call a class ‘proper’ isjust to say that it is an inconsistent set.

The core argument I will make is this. Either proper classes are not realmathematical objects, in which case talk of them is problematic; or else properclasses are real, in which case they are really problematic. Weir’s “Proper classesare proper Charlies,” in his [Wei98](773), makes the same point rather bluntly.

Presenting proper classes as the dual of an atom at least has some symme-try and aesthetic sense. It is not the usual way. Proper classes are more oftenintroduced via limit on size, that y is a proper class iff y is as big as V . Themost common presentation is as in Jech’s standard [Jec78], which blankly insertsproper classes as devices for maintaining the comprehension principle: “We do thisfor practical reasons: It is easier to manipulate classes than formulas.” The sameholds of Levy’s, Drake’s, Devlin’s, and Kunen’s books, where classes are taken as anotational substitute for predicates; that is, where for every open sentence there isa class such that the comprehension principle applies. On this line, then, it is notexactly the comprehension principle that is to blame for paradoxes; rather it is usmistaking some comprehended classes for sets.


Proper classes are, in a word, intensional. “. . . Sets are sharply delimited anddetermined by their extensions, whereas classes in the sense of properties have anessentially intensional quality.” [Kan94](313) They retain the intensional characterof the naive set concept. And since they are taken as a paradox solvent, freeingsets to be perfectly extensional and consistent, that intensionality is consideredsomething of a stain from the old, romantic psychology:

If we call such things properties then it is clear that the paradoxesare a real problem to be dealt with before a thoroughgoing theoryof properties can be developed. (The consensus of opinion seemsto be that in fact it just cannot be developed, but some system oflevels must be resorted to—that we cannot regard properties asbeing members of the universe to which they apply.)[Dra74](14)

The dangerous operation of “dividing the universe in half”, alluded to by Godel,is the province of property theory, attribute theory—proper classes. Martin (in[Mad83]) gives the standard gloss:

...sets are generated by an iterative construction process. Classesare given all at once, by the properties that determine whichobjects are members of them. ...‘set’ is a mathematical conceptand ‘class’ is a logical concept.

In this attitude, proper classes are volatile, and metaphysically heavy.Given the insistence on iteration as the only legitimate picture of sets, it is

important to recognize why proper classes are still a part of the classical appara-tus. Despite persistent insistence that “the universe does not exist” (as Halmos’introductory text [Hal74](7) announces for maximum shock value), to work in thediscipline of set theory does require the cumulative hierarchy to be a finished do-main. Proper classes offer closure. Proper classes are appealed to often, for bothcombinatorial and conceptual purposes.

This goes to the main problem: Sets were supposed to be collections. Somecollections are inconsistent. Invoking proper classes is an attempt to save the sets,but only because sets are no longer the last word on collections. Nevertheless,we must have some workable notion of collection. To cite a telling assortmentof examples, key results like Scott’s proof that the existence of measurable largecardinals contradicts Godel’s axiom that the universe is constructable, V = L, makeapparently indispensable use of V itself [Kan94]. More generally, the big class Onenjoys constant publicity, since any definition by recursion on the ordinals involvesa function that explicitly takes On as its domain; and even if it is recursion only upto some rank α, where is α? Most tellingly, the set V is the gauge of truth; Shelah[She00] asserts that ‘true’ means ‘true in V ’. To say that the structure 〈V,∈〉

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satisfies a set theoretic formula ϕ is to say that ϕ holds absolutely. Absolutenessis an important property for studying models of set theory, as in [Kun80]; butwithout V , there is no sensible way to articulate it.

Now, if proper classes do properly exist, as they do in von Neumann-Godel-Bernays set theory NGB, then at least some of the psychological discomfort anddisingenuousness is alleviated. But still a proper class y is a member of no z.Thus although we can talk in NGB of the whole iterative hierarchy VOn, this classcannot itself have a model or belong to any interpretation function [Wei98]. Sothese classes are about as ineffable as any facon de parler.

Maddy identifies and investigates this tension in her 1988 paper, which developsa putatively consistent theory of proper classes based on Kripke’s construction in AnOutline of a Theory of Truth. Her core observation is that proper classes are integralto mathematical practise, yet seem neither to be mathematically nor metaphysicallyunderstood.

Much of the talk [of proper classes] is casual, in the sense thatit can be translated away, but in some cases it seems that thetranslated version would never have been reached without theheuristic detour through proper classes... On the other hand,some of the talk is serious and much in need of foundationalclarification. [Mad83](117)

She goes on to suggest that the notion of set needs explaining—and that by improv-ing our understanding here, even problems like Benacerraf’s (which characterizationof ordinals, Zermelo’s or von Neumann’s, is correct?) may be solved. This is true;if stronger theory could pin down an intended model, we would have a firmer ideaof the invariant properties of mathematical objects.

First Maddy suggests that the set/class distinction is like Aristotle’s separationof potential and actual infinity: The boundaries of the realm of sets are nevercompleted, while proper classes make claim to being a finished, yet indefinite, space.Reinhardt takes up the same thought, suggesting that “Cantor shared with theancient Greeks a disdain for the apeiron,” and that this is modelled in the set/classdistinction.

Maddy makes the distinction we noted in the comparison of histories, betweensets that are essentially combinatorial, and classes as given by properties. It is thisdistinction we have been considering all along. There is room for terminologicalconfusion here; for “it is not easy to say what makes a result ‘combinatorial’...It seems to be used for results which are more intrinsic to the sets themselvesrather than relying on additional structure, and which in some way involve cardi-nality.” [Dra74](70) To these remarks, Kunen adds, “By infinitary combinatorics


we mean the field that used to be called set theory before there were independenceproofs.” [Kun80](47). Rather than fiddle with these terms, which are not terriblyrigorous, we settle on the basic dichotomy of extensional and intensional collec-tions. Maddy’s suggestion keeps with her Quinean naturalism, echoing the policyof keeping the universe-dividing properties cloistered into proper classes, while theinnocent extensions of sets are undisturbed.

The picture piques an immediate and irremediable problem. Suppose the it-erative hierarchy is the last word on sets. Where are the proper classes? Whilequalitatively distinct from sets, the proper classes still seem to be members of themathematical universe—“much of the talk is serious”—so the question is not mis-placed. As Reinhardt notes, “Our idea of sets comes from the cumulative hierarchy,so if you are going to add a layer at the top, it looks like you just forgot to finishthe hierarchy....” In trying to answer his own worry Reinhardt offers:

A proper class P may however be distinguished from a set xin the following way (if the reader will indulge another counter-factual conditional): If there were more ordinals, x would haveexactly the same members, whereas P has more to it than that.(We could say that P contains the famous three dots or ‘etc.’ ofmathematics in an essential way.) [Rei74](196)

Whatever sense one can make of this, it goes no distance in answering our query oflocation, save perhaps an oblique suggestion that proper classes are ‘at the top’—“in class theory the ‘universe’ of sets is a completed whole (so one looks downwardat the universe of sets from a high vantage point).”[Dev79](59) In a followuparticle developing a Kripke-construction of sets and classes further, Maddy sumsthe issue up completely. “‘Proper classes’ are either indistinguishable from ex-tra layers of sets or mysterious entities in some perpetual, atemporal process ofbecoming.”[ST00](299)

Approaching from above, it becomes clear that there is no way to locate properclasses in the official mathematical domain—at least not in a way consonant withclassicality. This is the reason that, despite the irresistibility of talking aboutcollections as sets, most wish to continue regarding proper classes as meaninglessabbreviations. But the purpose of proper classes is to provide closure, an inelim-inable conceptual resource; and this brings us to a very serious problem for theprevailing theory.

The main theme of [Pri02a] is the sheer hopelessness of both outrunning para-dox and reaching any kind of broad intelligibility; one can see these two demands

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as transcendence and closure respectively, of unstoppable forces colliding with im-movable objects. Putnam, at one phase (in 1990, quoted in [Wei98](768)), makesthe point succinctly:

The paradoxical aspect of Tarski’s theory, indeed of any hierar-chical theory, is that one has to stand outside the whole hierar-chy even to formulate the statement that the hierarchy exists.... The paradoxes themselves are hardly more paradoxical thanthe solutions to which the logical community has been driven.

To support a hierarchy either heuristically or formally, one must violate the stric-tures of the hierarchy.

The phenomenon is witnessed by Russell’s self-refuting dictum: ‘No propo-sition may quantify over all propositons.’ In effect, the iterative conception of amathematical foundation recapitulates Russell’s vicious circle principle, taking self-inclusion to be the culprit in contradiction. This is demonstrably a misdiagnosis, asis aptly shown by Aczel’s non-well-founded set theory, which is consistent but allowsself-membered solutions to equations. And this is anyway a diversion. Assertingthat closure is ultimately impossible is evidently self-defeating. Priest’s [Pri02a]is a book-length argument on the persistence of paradox over hierarchies.

Iteration fixates on transcendence and denies closure; the iterative view is con-genitally unfit to take ultimate stock of itself, and so is incapable of providing asatisfactory basis for set theory. This is what proper classes are for: Iteratively,there is no way for set theory to express the existence of its own domain. Weirwrites, “An infinite regress at which there is no explanatory power at all at level αunless the explanation at α + 1 works is indeed vicious, for the usual reason: ex-planation must come to an end.”[Wei98](780) To bring about conceptual closurewould entreat paradox all over again. Any completing stage will feature the incon-sistency found at the start. Ergo we have an incompleteness, and not a foundation.

Since inclosure is dilemmatic, a remaining possibility is to opt for outrunningcontradiction and to abandon hopes of intelligibility—of our practice, of mathe-matics, of truth. This is to give up foundations as ineffable, as seems to be theoutcome of the appendix to Kanamori [Kan94]. Claims about ineffability may beanswered in kind.

Writing on the current state of foundational set theory, and reflecting on theirown past work, Shapiro and Wright sum up neatly: “Invoking proper classes isan attempt to do the very thing we are intuitively barred from doing. ... Set issupposed to encompass the maximally general category of entities of the relevantkind.”[SW06](272) As Boolos says, “you can’t get out of this paradox merely bysubstituting one word for another.” Proper classes and the essential conceptual


resources they provide show that the cumulative hierarchy is not a descriptivefoundation for mathematics. And if, as Maddy thinks, the proper classes reallyare required, and if calling a set ‘proper’ is to label it with a warning of nascentcontradiction, I again ask whether or not the proper classes do anything more thatsignal inconsistency.

4.3. Overlarge Sets. In the last chapter we met the doctrine of limit on sizeas a natural enough formalization of Cantor’s absolute: A collection is overlarge ifit is the same size as the universe, V . As early as 1904, Russell had proposed alimitation on size as a solution to the antinomies. This idea, always presented withscare-quotes, states that some collections are ‘too big’ to be sets. Von Neumannagreed and offered the more precise criteria: Any collection containing as manyobjects as the universe is not a set.

The original and still forceful objection to a limit on size is: Any limit, withoutfurther explanation, is capricious and arbitrary. To paraphrase an insight of DavidLewis, if there are constraints on a mathematical object, the mathematics must tellus what those constraints are. Upon proposing it, in fact, Russell admits: “A greatdifficulty of the theory is that it does not tell us how far up the series of ordinalsit is legitimate to go.”[Rus05] The large cardinal work studied in my chapter 6shows indeed that the extent of the ordinals is still an open classical question, thatthere is still a lot to learn about when and where inconsistencies in the ordinal linearise. The limit doctrine isn’t doing any work.

Even supposing the limit on size is clear, we encounter the same demand forresources that plagued the doctrine of proper classes. Regarding the sheer size ofCantor’s universe, we find direct ascriptions of plenitude that simply outstrip theiterative construction and show that comprehension, and with it overlarge classes,remains the unspoken assumption. The general Cantorian initiative is: Whateverfurther ordinals there could be, there are [Pot04](210). The mathematical universeis utterly maximal; the principle of plenitude is a closure condition. Reinhardt:“Cantor through his doctrine of the Absolute had intended the universe of sets tocomprehend all possibilities.” And this is correct: The domain principle is a clearrestatement: Potentiality presupposes actuality. Such plenitude is appealed to inmathematical practise, as a justification for assumptions. For example, Gaifmanexplains that “All embeddings of the form j : V −→ M are assumed here tobe definable in V . This is a natural assumption; since V is Cantor’s universe,everything should be there already.” [Jec74](39)

Here is a stark commitment to the largeness of the universe, in particular themaxim that all the ordinals are already there. Tarski would observe that “thosewho share this attitude are always ready to accept new ‘construction principles’,

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new axioms securing the existence of new classes of ‘large’ cardinals ... but arenot prepared to accept any axioms precluding the existence of such cardinals ....”Perhaps the maxim is meant only heuristically, or as some kind of figure of speech.How might such overlarge thoughts be defined away, especially if the paraphrasingmust be in terms of iteration? When offered statements of this kind by expertslike Gaifman, the clearest course is to take him at his word, rather than attemptsome radical translation. A literal reading of set theoretic intuitions requires amathematical universe that outstrips the conceptual resources available from thecumulative hierarchy.

The discussion of proper classes has already touched on this issue. Two furthercomments are in order.

First, from the naive viewpoint, limit on size makes V look, ironically, like aset. If there is no universe, then what, exactly, is it’s size? On the other hand,if V is a real object in the range of a one-one correspondence with some M , weneed an explanation as to why V and M are not sets. Otherwise these would becollections deemed largely unintelligible or somehow conceptually defective, becausethey are wellordered and in one-one correspondence. If Lavine is right that setsare transfinite objects that can be counted, then they are sets. As I’ve alludedto, I think the cause of this strange doctrine is an attempt to articulate a limit ofiteration or expression qua Priest. But without resituating into a dialethic context,this attempts to say something where, by consistent lights, nothing can be said. Itis indicative of the is-ought, description/prescription confusion.

This makes way for the second comment germane to the doctrine. Consistencycannot be guaranteed by assertion of consistency; “one thing that is clear is thatadding premises cannot possibly reduce threat.” [ABD92](503) To take an analogy,murder is a crime. It is illegal to commit murder. This does not, however, ensurethat murder never takes place. The very fact of murder is what makes the lawsurgent. Now, inconsistency and size are closely linked, as we will prove in parttwo. If limit on size is recommended as mere legislation, then so be it; one maytry to regard it as a protective barrier, as something imposed by fiat. This is notthe current attitude towards limit on size, which reverses the roles of prescriptionand description. The standing argument, rather, accepts the premises but drawswayward conclusions. With the murder analogy, consider: If the President murderspeople, then he has committed a crime; but the President is the President, andcannot commit crimes; therefore, by a limit on size style reductio, he cannot havemurdered anyone—or even more baroquely, if he did, then the President does notexist. With respect to set theory, the ordering of premises is similarly tortured. Weprove some set M to be V -sized. Then we are to reject that M has an intelligiblesize, or even that there is such an M .


Now, if this were a cogent method, then there is of course no worry that settheory is, was, or ever could be inconsistent. Any inconsistent element of the theoryis rejected from the theory, ruled out a priori. But plainly, as with murder, onecannot simply eliminate a problem by defining it away. Having found a collectionto be overlarge, there is no time to invoke limit on size; it is already too late. Oncea class is recognized to be a proper class, this means that we have proved a set isinconsistent. Faith, and faith alone, intimates that it was not a set to begin with.Faith alone is propping up limit on size, which on a literal reading is the simplestatement that set theory (albeit not necessarily ZF ) is inconsistent.

The same problem was incipient in Cantor’s theory. A collection is a set whenintensionally bound, and in this breath, as Russell recognized, all collections shouldbe sets. In particular the collection of all sets is intensionally bound, by this verysentence, and so should itself be a set. Cantor flatly denies this with the name ‘ab-solute’. As we’ve seen, the absolute simultaneously is just the universe of sets anda prohibition against saying so. Cantor did not provide sufficient criteria for de-marcating the difference between transfinite and absolute, and a fortiori providedmotivating arguments that immolate the difference altogether. Exactly as muchwork in the service of consistency is achieved by the words ‘proper’ and ‘absolute’;when viewed without a classical bias they are names of dialethias. Cantor’s dis-tinction between consistent and absolutely infinite multiplicities is not a solutionto the problem of the quantitative boundary of consistency. The stigmatizing ofoverlarge sets is similarly but an illustration of the basic paradox of set theory.

5. Of Comprehension

The naive comprehension principle is an axiom in the old sense: simple andself-evident. My case comes down to the inalienable truth of comprehension, andthe fact that an honest assessment of mathematical statements shows that it has notbeen alienated. In this last phase of the argument, we observe that in fact classicalmathematics has not, because, as the discussion so far has already suggested, itcannot abandon the naive view.

Zermelo’s 1908 axioms met with “intense criticism,” not only over the axiom ofchoice but because of his decision to molest the comprehension principle. Schoen-flies, Bernstein, and Poincare all rejected this possibility out of hand [Moo82](111,117). In his famous 1914 textbook, Hausdorff too expressed doubt about Zermelo’ssystem—“At present, these extremely ingenious investigations cannot be regardedas completed, and introducing a beginner by this [axiomatic] approach would causegreat difficulties. Thus we wish to permit the use of naive set theory here...”[Hau57]

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George Boolos, as we’ve seen, was an important spruiker for the iterative setconcept as a philosophically satisfying replacement for the naive view. The work,if such work could be done, is necessary, because the naive view is otherwise thedefault position. Boolos’ discussion is very honest. On Frege’s fifth law (“it soundsutterly obvious, doesn’t it?”) and the Russell paradox, he writes

It should be noted that, although logic forces us to accept thatthere isn’t any such set, it’s highly paradoxical that there isn’t.... Let me invite you to fix your attention on the sets that don’tcontain themselves, the set of evens, the set of human beings,etc. Doesn’t there HAVE to be such a thing as the collection ortotality of things you are thinking about, the sets that don’t con-tain themselves? And isn’t a collection or totality just the samething as a set? How COULD there NOT be a set containing alland only the sets that don’t contain themselves? [Boo98](148)

And he goes on to chide the renaming certain collections as sets and others asclasses as a hopeless diversion.

The aporic situation is apparent in the opening pages of most set theory books.Devlin opens as many do with exposition of basic Cantorian theory. Endorsing(and crediting to Paul Halmos3) the epithet of ‘naive’ set theory, Devlin explainsthe discipline to be

a rigorous theory, based on a precise set of axioms. ... [But] theaxioms can only be fully understood after the theory has beeninvestigated sufficiently. This state of affairs is to be expected.The concept of ‘set of objects’ is a very intuitive one, and, withcare, considerable, sound progress may be made on the basis ofthis intuition alone. Then, by analysing the nature of the ‘set’concept, the axioms may be ‘discovered’ in a perfectly naturalmanner. [Dev79](1)

The set concept Devlin intends to use initially is naive comprehension. “In settheory, there is really only one fundamental notion: The ability to regard any col-lection of objects as a single entity (i.e. a set).” On the basis of our method oftaking people at their word, comprehension is still the guiding notion of set; theZF axioms are supposed not even intelligible without naive acquaintance with sets.This is not a controversial claim. “Since the concept of set is so intuitive (is it?),one might expect the axioms to form a neat collection of about three statements,”

3Who at the close of his charming autobiography I Want to be a Mathematician names creditsto himself the abbreviation ‘iff’ and the little square ‘tombstone’ marker at the end of proofs.


Devlin writes. “In fact we shall obtain what can only be called a motley collec-tion of some nine statements.” [Dev79](49) C.f. van Aken: “A survey of the [ZF]axioms does not suffice to reveal the source of their attraction.”[vA86](992)

The explanation from Takeuti and Zaring, in their dense and elegant text[TZ71], is paradigmatic of the way heightened precision also exposes the gapeof a conceptual gap. They make respect for comprehension an adequacy condition.Russell’s paradox notwithstanding, “classes, or arbitrary collections, are howeverso useful and our intuitive feelings about classes are so strong that we dare notabandon them. A satisfactory theory of sets must provide a way of speaking safelyabout classes.” And again in the midst of technical development, “The idea of thecollection of all objects having a specified property is so basic that we could hardlyabandon it.” [TZ71](3, 9)

Scott, the principle architect of stage axioms, at some points in his influential1967 symposium talk belittles the naive view, insisting on the myopic idea thatintensional set collection was never a part of Cantorian practise. Such rhetoric ishighly local, however. Speaking of the Zermelodic separation scheme, Scott admitsthat “it is a great temptation to erase the condition x ∈ z [from (∀z)(∃y)(∀x)(x ∈y ↔ x ∈ z ∧ Φ(x))], thus simplifying the axiom schema; but we all know whathappens.” The localization of the rhetoric against the ‘logical’ notion of set suggestsit is not entirely sincere.

In general, and in parallel to the historical revisions, a ritual has evolved ofbringing out the formal comprehension axiom and discrediting it. This officiallydone in public, set theory then reverts to what it always was, a study of compre-hended collections. As Church bluntly states in [Jec74], “We assume all of naiveset theory except the comprehension axiom, and then assume as many special casesof the comprehension axiom as we dare.” De jure, comprehension is in disrepute.De facto, the full comprehension axiom is still in force. After all, “simply sayingthat we ought never to have expected any property whatever to be collectivizing,even if true, leaves us well short of an account which will settle which propertiesare...”[Pot04](27)

In light of what has been argued, the revisionism in the new history of set theory§3.2, with Cantor a forerunner of modern researchers, is telling. Lavine’s story isnuanced, detailed, thoughtful, and ends in exactly the contradiction it meant toavoid: the existence of collections that are obviously sets, but that outstrip theresources of any set concept save the full naive one. Lavine himself rebukes theiterative concept for his own reconstruction of Cantor’s theory; but the differencefor us is immaterial. A theory taking e.g. wellorders as primary will be even lessindependent, presupposing even more naive set theory, than the iterative account.Whatever point an inclosure paradox is admitted, it is the same, inevitable paradox.

6. CONCLUSION 97

6. Conclusion

The paradoxes show that the naive view is inconsistent. The naive view isanalytic. Persisting with classical logic, then, is an act of outright incoherence, inthe formal sense of being trivialized by explosion. The twentieth century crisis wasa conceptual battle of the need for and admission of explanatory closure, againstthe assumption of consistency. The broad situation is captured by Dummett.

The beginner can be persuaded to that it makes sense, afterall, to talk of the number of natural numbers. Once his initialprejudice is overcome, the next stage is to convince the beginnerthat there are distinct transfinite cardinal numbers.... When hehas become accustomed to this idea, he is extremely likely toask: ‘How many transfinite cardinals are there?’ How should hebe answered? He is very likely to be answered by being told,‘You must not ask that question.’ But why should he not? If itwas, after all, all right to ask ... in the sense in which ‘number’meant ‘finite cardinal’, how can it be wrong to ask the samequestion when ‘number’ means ‘finite or transfinite cardinal’? Amere prohibition leaves the matter a mystery. ... And merely tosay, ‘If you persist in talking about the number of all cardinalnumbers, you will run into a contradiction’, is to wield the bigstick, but not to offer an explanation.[Dum91](315)

The bottom line of the explanation we get is this. There is of course no worryabout paradox. Paradoxical objects don’t exist, or else they would be paradoxical.

Now, a friend of the mainstream theory could still think that all the apparently‘bad’ talk, the appeals to various notions in scare quotes, can be paraphrased ordefined or explained away, leaving a consistent concept in place. Enough care andthought have gone into isolating the consistent fragment of set theory, the thoughtgoes, that it is now safe to show a bit of naıvete; the full consistent machinery canbe invoked if any trouble arises. Kunen and Drake, for example, devote consider-able energies to showing how class terms can always be reduced to basic, legitimatesets; Levy has a full appendix to prove what he calls the “conservation and elim-inability theorems.” The starkest and most pervasive example of this is to have thewellordered class V on hand, but always to work at a suitably high rank Vα and indoing so avoid risk of antinomies. ‘Scott’s trick’ in his ultrapower construction is agood example of this idea being stretched to the limit; see [Kan94](47).

Thus, my literal reading of mathematical talk could be accused of being simpleminded. When we use heuristics, metaphors, aren’t these just means of gainingtraction on difficult and abstract ideas? In principle, this can all be replaced with


utterly precise and clear propositions, in the same way that in principle a proofin mathematical English can be displayed recast as a fragment of Principia Math-ematica. Russell himself said in 1901 that “A book should have intelligibility orcorrectness; to combine the two is impossible.” In selecting statements from settheory texts, perhaps I am being unfair.

This is a serious concern. The reply is that mathematics is serious, and weshould not mistake appeasement for actual ideas. To take an analogy, Kant’s Cri-tique of Pure Reason states explicitly that noumena are that about which nothingcan be said. He then writes in excess of 500 pages about the noumena. If one wishedto defend Kant against charges of inconsistency, something like a use-mention dis-tinction could be introduced and an algorithm offered to reduce any apparent talkabout noumena to simple mention of ‘noumena’. Perhaps this could be done. Sim-ilarly for set theoretic foundations and the paradoxes of infinity, perhaps mathe-maticians, being extraordinarily clever people, can and have found technical devicesto silence complaints from logicans and philosophers.

The question is whether this in-principle solution makes contact with the con-tent of mathematical propositions. Is it really descriptive? The thought that largeportions of set theory books are written in a kind of eliminable slang runs againstthe idea of taking mathematics seriously; no vital and dignified discipline is con-ducted in scare-quotes. Before Cantor, but after Newton and Leibniz, it was clearthat talk about infinity was required, yet terribly murky. Gauss famously excusedthis cloudy exposition as a mere facon de parler. The transfinite shows Gauss’evasion is unnecessary, and frees up analysis for honest practice. There may beworkable translation algorithms, but they are in the service of prescribing safebehavior rather than describing mathematical structure. We now have an oppor-tunity to trade in prescriptive dogma that has overstayed its need, for explanationspointing to a way ahead.

To conclude, let’s brings together the main themes of part one, the paradox ofabsolute infinity at the meeting of intension and extension.

Cantor overtly and repeatedly explained that his extensional set theory is ulti-mately intensionally bound. Similarly for Frege’s system, it is an extensional theorymortared by intensionality—a notion that, if not for contradiction, was never inquestion and even now remains. Zermelo and von Neumann found a neutral wayto phrase Cantor’s doctrine, but, as we’ve seen, did so by neutralizing the mean-ing of their axioms, or more charitably, by allowing that there is no explanationfor transfinite set theory. Plainly unhappy with making Russell’s quip about thevacuity of pure mathematics (as the “the subject in which we never know what weare talking about, nor whether what we are saying is true”) a reality, a programwas launched to replace the intensional universe with an extensional story about

6. CONCLUSION 99

iterative set formation. We have seen that the outcome of this program is inade-quate. All that I’ve said can and has been repeated mutates mutandis with respectto naive semantics and Tarksi’s hierarchy, and there the data is not sequestered toexpert mathematics but to the working praxis of language.

The approach we will now take up, that of dialethic paraconsistent set theory,is designed to overcome as many of these difficulties as possible.

Part Two—Foundation

“One of the most compelling motivations for the construction of paraconsistentlogic,” said Newton da Costa in 1998, is “the possibilities it opens up in the founda-tions of set theory.” The ratio of programmatic statements about paraconsistent settheory, though, to the proven theorems of paraconsistent set theory, is imbalanced.A developed set theory based on full comprehension is long overdue, and is begunin this part.

First I fix the logic for the theory, DLQ, and discuss some of its properties.Then, in the most important chapter, I work up from the axioms through thetransfinite numbers, recapturing classical results such as Cantor’s theorem, andalso deriving novel theorems, such as the ‘axioms’ of infinity and choice, Burali-Forti’s contradiction, and some indications of distinctive transcendence phenome-non. Then, in chapter 6, the transcendence results prove the existence of all themajor large cardinals, showing serious advances in the proving power over com-petitors. All this shows that, to a large extent, infinitary mathematics is just areflection of the paradox at the limit of ordinality, discussed in chapter 7.

At the least, this part proves that a robust, independent paraconsistent settheory exists, and a fortiori that a lot of set theoretic mathematics can be hadwithout need or assumption of consistency.

∗ ∗ ∗

Logistic is no longer barren.It engenders antinomies.— Poincare

101

CHAPTER 4

On A Logic for Naive Set Theory

This chapter studies a logic appropriate for naive set theory, focusing mainly onthe technical problems that arise for weak relevant logics in inconsistent settings.

1. Introduction

The goal of this work is to embed a viable naive set theory. Paraconsistencyis therefore a requirement; but there are many systems of logic in which explosionfails. Which logic to use? Once committed to a robust abstraction principle like fullcomprehension or Tarski’s schema, the field narrows quickly. In fact, for reasonsshortly to be given, my current suspicion is that there is (basically) only one goingoption, the DLQ of Routley and Meyer [RM76]. I will be calling it dialethic logic,in keeping with the change in nomenclature since 1976. This chapter is not an ex-haustive survey of the many non-classical, non-explosive systems available (for thissee [PRN89], [Pri01], [DR02], [Pri02b]), but rather scene-setting considerationsand solutions to preliminary formal woes.

This chapter is heavily concerned with identity as it occurs in mathematicalpractice. I use a technical result on relevant restricted quantification in §5 to mo-tivate a new approach to thinking about identity and naive set theory in general.The technical result shows that a relevant conditional cannot, despite a recent sug-gestion [BBH+06], be made to mimic the material conditional without lapsinginto absurdity. The main subsequent claim is: Some sets, contrary to first impres-sions, are not coextensive. While it is standing wisdom that set theory can beeither extensional, or have unrestricted set-formation principles, but not both, itis here redressed what is meant by extensional; and this done, the standing wis-dom is undercut. The meaning of extensionality in a relevant naive set theory isdifferent and novel. My point is a straightforward one: Identities are only ‘lost’ ifwe start from the assumption that they are had. We need not assume so; in theset theory to be considered below, they are not. And a fortiori, to develop theinternal mathematics of paraconsistent set theory, as a robustly independent anddignified discipline, we ought to take the paraconsistent view as independent anddignified. Naive set theory in relevant logic is not extensional when evaluated fromthe standpoint of classical set theory. It is perfectly extensional when evaluated on

103

104 4. ON A LOGIC FOR NAIVE SET THEORY

its own terms. The methodological import of this is that metatheory and studyof models of set theory with respect to naive set theory is best conducted in anappropriate non-classical logic. Some preliminary notes on that project are foundat the end of this thesis. The criteria we bring to bear on a theory has radicalimplications for our attitudes toward and the status of that theory; crucially, ifclassical logic continues to be the logic in which we prove meta-theorems aboutnon-classical systems, classicality will be the standard by which competing theoriesare judged. Trivially, if any other set theory diverges from classical set theory butis judged by the latter, then the former will always be a deviant. It is no longer aucourant to regard non-classical logics as ‘deviant’, and it should not be so for settheory, either.

2. Logics of Formal Inconsistency

2.1. A Map of the Paraconsistent. Lukasiewicz opened the way to non-classical logic in its modern form in 1917 in Poland, shortly after the introductionof classical logic in its modern form. Paraconsistent formal logic appeared with thework of Jaskowski, himself a student of Lukasiewicz, on non-adjunctive logics in1947. Not long thereafter, Newton C. A. da Costa in Brazil began a direct inves-tigation of paraconsistent logics [dC74], [dCKB04] and in 1976 F. Miro Quesadaof Peru coined the neologism, paraconsistent. South America is prominent on theparaconsistent landscape, even if da Costa was not its sole creator as is sometimesclaimed there; recently we have [Mar05], from whom the title of this section isborrowed.

The programmatic leader in Brazil, early on da Costa stated four methodolog-ical principles (see [Urb89]):

- From Φ and ¬Φ should not follow Ψ for arbitrary Ψ;- The fragment of classical logic that does not cause explosionshould be maintained;- The schema ¬(Φ ∧ ¬Φ) should not be derivable;- The full system must be at least first order (with equality).

The second principle may sound simple enough, but working out just what fragmentit alludes to is a substantial and controversial topic.

The third principle is a good indication of da Costa’s thinking. Other paracon-sistentists [PR89b] see a difference between holding the law of non-contradictionas a normative principle, and having the formula ¬(Φ ∧ ¬Φ) as a theorem. Theformer needs to be overcome; the latter is, arguably, an analytic feature of nega-tion any formal system should capture. In da Costa’s systems the matter remainsundifferentiated. Ultimately da Costa founds a hierarchy of logics, the Cn systems,0 ≤ n ≤ ω, which includes an operator Φn read that Φ behaves consistently in

2. LOGICS OF FORMAL INCONSISTENCY 105

Cn−1. Most often this is presented in C1, where the classicality operator is Φ,since C0 is just classical logic. The opening set theory exercise showed C=

1 allowsfor some natural arguments, but we cannot develop naive sets in da Costa’s sys-tems, because its conditional is wrong: Arruda found early in her investigationsthat naive comprehension is too strong for the C systems, deriving the “unpleas-ant” result that a = b for every a and b; see [AB82], [Arr89](112). In particularthe conditional in any Cn supports contraction and therefore falls afoul of Curry’sparadoxes—all of which is discussed in detail below.

A paraconsistent school in Europe is thriving. In Ghent, Batens and his schoolstudy dynamic dialectical logics [PRN89](ch6). From Brussels come innovativemodels in the works of Hinnion [Hin94] and Libert [Lib04]. Hinnion’s work grewout of another alternative set theory, Quine’s NF set theory, to which he hascontributed plentifully [For95]. Focusing heavily on topological models for thecomprehension axiom, their approach pays equal attention to paracomplete (truth-gappy) theories, beginning from Gilmore’s work on partial set theory [Gil74].

The Australian school of paraconsistency is closely allied with the relevancelogic program there [RPMB82], and is most attributable to Routley (later Syl-van), Meyer, Brady, and Priest. In 1976 Routley and Meyer questioned the con-sistency of the world [RM76], in a Marxist context. In 1979, Priest published hislandmark [Pri79], the “Logic of Paradox.” This paper brought the burgeoning ideaof dialethism to a wider audience than it had found previously, and paraconsistencyin Australia is now often conflated with dialethism, if not with Priest himself.

2.2. A Point of Principle. As we go, there is a methodological question,which we will return to in the conclusion of this chapter. On the one hand somelogic must be in place to guide steps in proofs. To know what is logically valid onthe other hand is by most accounts to know what is valid in every model; modelsbeing set theoretic constructs, logic relies on knowing what sets there are. Roughlyspeaking, the number of logical truths is inversely proportional to the number ofsets, since a smaller universe makes it more likely that a proposition is ‘universally’valid. (See Jesus Mosterin’s article in [Wei94].) Determining either logic or settheory in complete isolation is an impossible task.

A proposed approach so far has been to bootstrap. Priest uses the collapsinglemma, that a consistent model can be extended by adding truth values to obtain aninconsistent model [Pri97]. The proof is by induction, a non-trivial mathematicaltool. The collapsing lemma can then be applied to show that LP is just an extensionof classical logic—but all this under the umbrella of classicism. And if we areconcerned that the classical umbrella will not hold up against what we suspect areal deluge of contradiction, then this whole line is in serious danger.


My approach is to treat the logic, at least initially, as a formalist—to give aHilbert-style axiom list, comprised of uninterpreted symbols which may be shuf-fled by a few given rules. Talk of truth, meaning, or interpretation will have towait for the set theory. At the end of the chapter we can return to the ques-tion of classicality in the metatheory. Newton da Costa explains that “at leaston philosophical grounds, it is needed to have a paraconsistent set theory alreadyarticulated if one intends to have a reasonable semantics for paraconsistent logic(given that the semantics shall be constructed within set theory).”[Mar05], xlii)The methodological maxim underlying this chapter is that, because semantics arepresented in terms of set theory, there can be “no semantics for a paraconsistentlogic without a paraconsistent set theory.”

2.3. Which Logic to Use? The two concerns in setting up a paraconsistentset theory are: the formulation of the non-logical assumptions and basic definitions;and the choice of the background logic itself. Systems have shown themselves tobe remarkably sensitive to small changes in either. The chapter is not a survey ofother attempts. Rather we focus in on a version that seems to work, by fixing the(sometimes competing) goals of finding a system

- strong enough to deliver real mathematical theorems, both classical and non-classical;

- weak enough to be non-trivial, else the enterprise is absurd;- and as far as possible still respecting our intuitions about the meaning of core

concepts, such as predication, set membership, identity, and order.Achieving the first two goals is more or less a matter of technical balance,

approachable by trial and error. The third is far more difficult, since intuitionsseem to diverge—or, more contentiously, are overdetermined. A rule of thumb isto avoid excessively prolix or abstruse requirements that would lead to the systemsbeing inflexible and unattractive to users.

Priest’s work has enjoyed continuing importance in part because Priest tieshis philosophical argument to a natural enough looking logic, LP . This “logic ofparadox” is Kleene’s strong three valued logic K3, with the middle value designated,and the intended interpretation of that value to be overdeterminate rather thanindeterminate. (A similar change of designation to K4 yields Dunn’s first degreeentailment.) Because LP is truth-functional, the following initially alluring resultfollows:

Every classical tautology is an LP tautology.

The picture presented is one of a straightforward conservative extension of our rea-soning apparatus, likened to the addition of irrational numbers to the rationals forthe completion of a structure. A naive set theory in LP was briefly investigated

2. LOGICS OF FORMAL INCONSISTENCY 107

by Restall in [Res92], with solid model-theoretic results such as non-triviality, asevinced by the failure of the foundation axiom. Non-trivial models for inconsistentarithmetic in LP have been completely characterized in [Pri97] and [Pri00a]. Of-ten dialethic paraconsitency and LP are treated as interchangeable: LP is assumedto be the correct logic for a dialethist.

As a logic for set theory, though, LP has some insuperable difficulties. Pre-eminently, LP has no conditional; Φ ⊃ Ψ is defined as usual as ¬Φ ∨ Ψ, and so,because disjunctive syllogism must fail, is non-detachable:

Φ,Φ ⊃ Ψ 6`LP Ψ.

Indeed, LP is just a de Morgan lattice with excluded middle, so it does recaptureclassical tautologies, e.g.

`LP Φ ∧ (Φ ⊃ Ψ) ⊃ Ψ,

but exactly because it is too much like classical logic, these tautologies sit inert. Nologic, let alone any serious mathematics, can proceed without a conditional thatrespects modus ponens.

There are several more restrictions on the logic, discussed in §3 below. Theseconsidered, then amongst the well-understood paraconsistent logics, this leaves onlythe intensional relevant systems of the Australian school. Again, much of thiswork has been motivated by Anderson and Belnap’s relevance program initiated in[AB75], and carried on by Dunn, Slaney, and more recently, Mares [Mar04]. Thebest investigated relevant logic is R, but this is too strong for naive set theory byseveral orders, and as a home for arithemtic has been investigated into a cul-de-sacby Friedman and Meyer [FM92]. Mathematics couched in both a relevant anddialethic light is Mortensen’s [Mor95].

When I began to explore naive set theory, my principle was pragmatic: Usethe strongest logic possible without triviality. I did not fix on any one logic, butrather guessed and checked, employing many natural inference principles, seeingwhere they led, much in the spirit of the initial excursion of this thesis, chapter 0.They led, as it turned out, to several explosions, and one proof that a 6= b for anya, b. Some real time was required to break from intuitions honed on classical logic.

Aside from an absolute insistence on modus ponens, it came clear that someprinciples, especially negation principles, are indispensable for a study of collections.Particularly, it is important to keep contraposition,

(Φ→ Ψ)→ (¬Ψ→ ¬Φ),

and a counterexample principle, at least in rule form:

Φ ∧ ¬Ψ ` ¬(Φ→ Ψ).


Contraposition is essential because the naive intuition is that set membership andpredication are basically the same. Without contraposition, much of this conceptuallink is severed; if n is not a number, it is not in the set of numbers. Socrates’ schemaasks not only when an action is Φ, but also when it is not. Contraposition, though,demands adherence to a relevant logic in dialethic context; in particular, we cannothave weakening, Φ ` Ψ→ Φ. With weakening and contraposition, we could argueas such, with Λ a true contradiction:

1 Λ premise

2 ¬Ψ→ Λ 1, weakening3 ¬Λ→ Ψ 2, contraposition4 ¬Λ premise

5 Ψ 3, 4,modus ponensBut Ψ is arbitrary. This step to relevant logic does not really increase our

philosophical commitments, though, since in the presence of contrapositon, theinference in question here is just Φ ` ¬Φ→ Ψ, a form of explosion.

Similarly, the counterexample principle is needed to respect the nature of col-lections, specifically the subset relation a ⊆ b. If x ∈ a but x 6∈ b, it must be thata is not a part of b; else there would be no way of distinguishing sets. Both theseprinciples are about negation.

Paraconsistent negation is a nuanced issue; Dunn, Priest, Restall, and manyothers, have written extensively on it. I am only stating my intuitions on thematter, combined with some experience trying to prove theorems.

Negation, while it cannot be entirely boolean, nor intuitionistic (see section5 below), still has most of its other properties: contraposition, double negationintroduction and elimination. With contraposition, the law of excluded middleactually just follows from axiomatic counterexample,

(Φ→ Ψ)→ ¬Φ ∨Ψ,

with Φ for Ψ. The claims about extensionality in this chapter hinge on a ro-bust negation. The axiom of extensionality should tell us not only when sets areidentical, but when sets are not identical. This is as much a part of the set con-cept as comprehension—to tell us, in Socrates’ idiom, which actions are just andalso which actions are unjust. Some would jettison excluded middle—and eo ipsocounterexample—to preserve consistency, e.g. Brady [Bra06]. For without ex-cluding the middle, instances of comprehension like x ∈ R ↔ x 6∈ x, the Russellcondition, go no further.

But a great many (most?) proofs in e.g. the theory of ordinals make appeal toan exhaustive set theoretic universe. I can see no way at all to argue for theoremswithout it, and this leads me to suspect that exhaustion is a deep property of

3. A PARACONSISTENT LOGIC 109

the set concept; my suspicions are borne out since Brady in [Bra06] postulates,rather than proves, all the desired facts about order, ordinals, and cardinals. Froma perspective that does not assume that classical set theory is the antecedentlycorrect set theory, the axiomatic postulation method makes little sense.

Anyway, blocking the derivation of the set theoretic paradoxes, given a settingin which they will not lead to absurdity, deprives us of knowing what sort of novelinformation the paradoxes contain. It constitutes a decision, as it were, not tostudy the full naive theory; and this is not in the paraconsistent spirit. There is awidely noted duality between paraconsistent and paracomplete systems, and therehas been a good deal of investigation into mathematics that is consistent but withgaps; at the very least, it is of some scientific interest to consider dual systemswithout gaps that are inconsistent.

The logic DLQ was first proposed by Routley and Meyer [RM76]. There it wasexplicitly a dialethic (at the time called dialectical) logic, with p∧¬p included as anaxiom, to ensure inconsistency (!). In his highly influential article [Rou77], wherehe argued for and sketched the beginnings of a naive set theoretic recapture, Routleyused the logic DKQ, which is slightly weaker than DL but still has excluded middle,Φ ∨ ¬Φ. Routley called the theory obtained from joining DKQ with the axiomsof comprehension and extensionality DST , dialectical set theory. Brady [Bra89],[Bra06] proves that DST has a model and is non-trivial. In the last decades Bradyweakened the logic further, dropping excluded middle to obtain DJQ, which leadsto a consistent set theory.

A weak member of the relevance family, DLQ nevertheless proves more ap-propriate than Cω or LP . On the one hand, every classical tautology (since theyinvolve only ∧,∨, and ¬) is a theorem of DLQ [Rea88](71), just like LP . Butthe logic is not cast in a classical philosophical paradigm. As it turns out, drop-ping ex falso quodlibet is a tall order; the majority has not been wrong to thinkthat classical logic is deeply inured in, if not based upon, consistency. Abandoningconsistency requires a qualitatively distinctive approach to formal reasoning, andthis is the raıson d’etre of DL. A drawback for DLQ is that there is as yet nonon-triviality proof for it. Brady’s construction is not easily adapted to encompassDLQ [Bra06](244).

3. A Paraconsistent Logic

Let us have the language of first order set theory: with primitives ∧,¬,→,∀, =,∈, and · : −; variables x, y, z, ...; names a, b, c, ...; and formulae Φ,Ψ,Υ, ..., builtup by standard formation rules. The usual shorthand is used: Φ∨Ψ for ¬(¬Φ∧¬Ψ);Φ↔ Ψ for (Φ→ Ψ) ∧ (Ψ→ Φ); ∃ is ¬∀¬. All instances of the following schemataare theorems:


Φ→ Φ,Φ ∧Ψ→ Φ, Φ ∧Ψ→ Ψ,Φ ∧ (Ψ ∨Υ)→ (Φ ∧Ψ) ∨ (Φ ∧Υ),(Φ→ Ψ) ∧ (Ψ→ Υ)→ (Φ→ Υ) (conjunctive syllogism),(Φ→ Ψ) ∧ (Φ→ Υ)→ (Φ→ Ψ ∧Υ),(Φ→ ¬Ψ)→ (Ψ→ ¬Φ),¬¬Ψ→ Ψ,(Φ→ Ψ)→ ¬(Φ ∧ ¬Ψ) (counterexample),(∀x)Φ→ Φ(a/x),(∀x)(Φ→ Ψ)→ (Φ→ (∀x)Ψ),(∀x)(Φ ∨Ψ)→ Φ ∨ (∀x)Ψ.

The last two axioms have the caveat that x does not appear free in Φ. For goodmeasure, there is no danger [Bra06](200) in adding the hypothetical syllogism pair,

(Φ→ Ψ)→ [(Ψ→ Υ)→ (Φ→ Υ)],

(Φ→ Ψ)→ [(Υ→ Φ)→ (Υ→ Ψ)]

suffixing and prefixing, respectively, though this takes us beyond DLQ, nearby towhat is called TLQ in [Bra06](242).

The following rules are valid:

Φ,Ψ ` Φ ∧Ψ (adjunction)

Φ,Φ→ Ψ ` Ψ (modus ponens)

Φ→ Ψ,Υ→ ∆ ` (Ψ→ Υ)→ (Φ→ ∆)

Φ ` (∀x)Φ

Including adjunction is in contrast to the earliest paraconsistent systems of Jaskowski.Contraposed, counterexample show the reason for its name: Φ∧¬Ψ→ ¬(Φ→

Ψ). This preserves essential features about ∈ and =, namely, that sets satisfyingdifferent predicates are members of different sets, and that different with respectto membership are different. This is captured in a simple theorem (ch5, thm5.7),that sets differing with respect to membership are not identical. In particular, theRussell set R = x : x 6∈ x exists and R 6= R. The well-known inconsistency of theRussell set thus has a fairly natural characterization in naive set theory: R is notidentical to itself.

As is suggested by the brevity of the above list, very many common inferencesare not appropriate for the study of inconsistent structures. We review notableabsences and their attendant proofs.


Central in the relevance debate of the last decades is disjunctive syllogism,

Φ,¬Φ ∨Ψ ` Ψ.

This is due to C.I. Lewis’ famous argument in [LL59](250), showing that disjunctivesyllogism, or material detachment, is explosive. Suppose both Φ and ¬Φ. From Φfollows Φ ∨ Ψ. But given also ¬Φ, follows Ψ by material detachment, where Ψ isarbitrary.

The other major casualty is contraction,

Φ→ (Φ→ Ψ) ` Φ→ Ψ

due to Curry [Cur42]. By comprehension, consider the set

x ∈ C ↔ x ∈ x→ Ψ

which we will call Curry’s set. Then1 C ∈ C ↔ C ∈ C → Ψ2 C ∈ C → Ψ contraction

3 C ∈ C 1, 2,modus ponens4 Ψ 2, 3,modus ponens

where Ψ is arbitrary.Closely related to Curry’s problem is axiom (or pseudo) modus ponens

Φ ∧ (Φ→ Ψ)→ Ψ

as found in ([MRD78]. Using Curry’s set, if pseudo-modus ponens were a validscheme then we could instantiate thus:

C ∈ C ∧ (C ∈ C → Ψ)→ Ψ

But C ∈ C is equivalent to C ∈ C → Ψ, so substituting we have

C ∈ C ∧ C ∈ C → Ψ,

which reduces to C ∈ C → Ψ. Again, though, this is equivalent to C ∈ C; so bymodus ponens, Ψ, where Ψ is arbitrary.

Upon this discovery, a venerable vanguard of the relevance/naive set traditionexpressed despair; the loss of modus ponens in axiom form was shocking enough todrive the tough-skinned authors of [MRD78] to write:

Our examination of Curry’s paradox is discouraging in the ex-treme for the hopeful naive set theorist.... We were willing togive up the usual aversion to contradiction. We faced with equa-nimity the sacrifice of the deduction theorem. To continue withthe project, a minimal, decent respect even for modus ponensmust be given up as well. ... A naive set theory is untenable.


However, history showed this to be overhasty. Routley conjectured that the ruleform of modus ponens could be salvaged; and indeed Brady proved that this rule isnot explosive even under unrestricted comprehension. The present approach is of akind with Routley’s: Grim pronouncements about extensionality are unnecessarilypessimistic, because they are unnecessarily steeped in classical thinking. As Priestcounters,

this way of putting it prejudices the issue by claiming that themodus ponens axiom is something we have ‘got’ which we willhave to lose if we want to adopt naive set theory. Possession, aseveryone knows, is nine tenths of the logical law. ... We cannothold both the abstraction scheme and the modus ponens axiom.However, both are candidates for rejection. It is not true to saythat we already ‘have’ one which we have to give up. We haveboth in exactly the same sense, i.e. we appear to start out witha belief in both. [Pri80](430)

Classically, axiom and rule modus ponens are indistinguishable; the finer lens ofparaconsistency detects important variations. What is persistently needed is to seeunderlying connections—to see that there are only a few major surprises, manifest-ing in different guises.

For example, the loss of material detachment when blocking explosion is verysurprising. But it can be explained, by putting the premise (Φ ∨ Ψ) ∧ ¬Φ into anequivalent form, (Φ ∧ ¬Φ) ∨ (¬Φ ∧ Ψ). From this form, one ought to be able toargue by cases to Ψ. Now, from ¬Φ ∧ Ψ follows Ψ immediately; but Φ ∧ ¬Φ onlyyields Ψ by explosion. [Sha05] (742) This shows that disjunctive syllogism is nota proof for but a restatement of explosion. We have rejected explosion; not as afurther consequence but eo ipso we reject disjunctive syllogism. Similarly, we haverejected contraction, and therefore pseudo-modus ponens too.

The extensional picture for naive sets is similar. A small class of novel phe-nomenon trail paraconsistent identity, inconsistency most saliently, but no morethan these. What appears to be a problem with extensionality is in fact a base factabout inconsistency, and if the latter is dealt with carefully, there will be no morecause for concern about the former.

There is also a trouble with permutation,

Φ→ (Ψ→ Υ) ` Ψ→ (Φ→ Υ)

due to Slaney’s argument in [Sla89], concerning the related logic RWX. In fact theargument shows that excluding the middle and permutation are not jointly tenable.The cause is, again, a close relative of curry’s paradox. Let us have a constant f


that obeys the two-way rule¬Φ a` Φ→ f.

Slaney shows that (Φ → ¬Φ) → ¬Φ. Expanding this out, we have an instance ofcontraction:

[Φ→ (Φ→ f)]→ Φ→ f

Then, using permutation, Slaney argues to triviality [Sla89](477). No one hasanyway ever produced a non-triviality result on naive set theories with permutation.Slaney’s argument, aside from seriously weakening an already emaciated apparatus,calls attention again to the difference between negation, or denial, and rejection.

Looking ahead to possibilities in paraconsistent metamathematics, the elimi-nation of contraction also blocks Lob’s theorem, which uses a provability predicateProv and ` as a sign of theoremhood:

If ` Prov(pΨq)→ Ψ, then ` Ψ.

Lob’s is itself a sinister version of Godel’s theorem. Since the diagonal lemma andother topics from recursion theory are involved, this point is not as elementaryas the others being rehearsed in this section, and is for now omitted, but it isimportant to see some of the wider avenues opened up by moving to weaker logics—namely a regaining of expressive power and thwarting of limitive theorems. LetProv be a provability predicate and p·q be a name-forming operator. We assumethat theorems are provable and that provability distributes over implication. Lob’stheorem states that if Prov(pΨq) → Ψ is a theorem, then so is Ψ. Dependingon formulation, the argument uses either contraction or permutation, or both. See[BJ89], chapter 15).

In a paraconsistent logic, ¬-contradictions are tolerated. A recurring villainof the piece, however, is absurdity, which serves much the same purposes that in-consistency serves for classical mathematics—fixing boundaries of what is possible.Take ⊥ to be the formula

(∀x)(∀y)x ∈ y.

This is an absurdity, as Slaney [Sla89](476) puts it, “that than which nothing silliercan be conceived.” To show ⊥ is so aptly described it suffices to observe that

⊥ → (∀y)a ∈ y

→ a ∈ z : Φ

→ Φ.

So ⊥ → Φ for every formula Φ. A ⊥ formula allows a kind of negation, Φ →⊥, what Curry called absurdity negation. As Slaney shows, this negation is notparaconsistent even in relevant logics close to the one we are using, because one


can obtain an instance of contraction. More generally, any contradiction with thisnegation explodes by modus ponens:

Φ,Φ→ ⊥ ` ⊥

To repeat the analogy with contradiction in classical logic, the appearance of ⊥ is aclear signal that something is awry; the key, and the key to preserving extensionality,is isolating just what it is that is to be rejected.

4. Relevant Identity

The language of set theory is distinctive for its connectives ∈ and =. Nextchapter we will study membership. More general considerations go into identity,which is the focus now.

4.1. Extensionality. A good deal of concern has been expressed about if orhow a set theory with full comprehension can be extensional. A fully comprehensiveextensionality means:

- every predicate to determine a set;- identity is completely determined by membership; and- identical objects are intersubstitutable.Put directly, though, there is sometimes joint incompatibility between a = b,

Φ(a), and Φ(b), for some sentences Φ. The motivating cases are, prima facie,problems with the intersubstitution of sets a, b when they seem to have the samemembers. This piques two questions: When is it appropriate to say that a hasthe same members as b, and when can a be intersubstituted with b? To take anexample, here are two sets:

U = x : x = x,

V = x : (∃y)x ∈ y.

(The appeal of a naive theory is that these really are sets, not proper classes or someother contrivance.) Classically, both are universal, in that they have all objects asmembers, (∀x)x ∈ U and (∀x)x ∈ V . But absent a material conditional, there is noway to prove that V is a subset of U . In the logics we will be considering, in fact,the sets are distinct, U 6= V . And further, were it the case that U and V satisfyall the same sentences, we could prove everything—triviality, as below (§ 5). Thisis not paraconsistent.

Thus, as has been widely observed at least since White [Whi79], but seeminglyalso earlier in Chang [Cha63], some extensional properties of sets are ‘lost’ whenone moves to a substructural logic. The point is also independently made by Brady[Bra71] and Gilmore [Gil74]. Working on set theory in a logic without contraction,

4. RELEVANT IDENTITY 115

Uwe Petersen writes that “...the failure of weak extensionality can be sharpenedto the point of destroying all illusions about possibilities of finding a realm ofobjects which is ‘well behaved’...” [Pet00](384). Libert in recent work declaresthat “indeed, it is hopeless to combine extensionality and abstraction [terms] withthe equality relation in formulae defining sets.” [Lib04](31).

As stated in the introduction, I think there is a way to understand naive sets insuch a way as they are still comprehensively extensional. There are certainly issuesto face, such as the co-extensionality of sets V and U , but it is worth rememberingthat classically, neither V nor U are even sets at all. How should we approach iden-tity, both in terms of its definition and the properties we expect of it? Inalienableproperties are clustered around the needs of substitution and equivalence relations;we will consider these in turn. Then we will examine a few options open in theliterature, which advocate sacrificing some expressive resources in order to be asliberal with identity as possible. What forms of identity get enough set theoryright, while not proving everything?

4.2. Equivalence. Mares [Mar92] discusses various formulations of identityin a neutral setting, and then Kremer [Kre99] goes farther, asking not just whichformulations of identity are possible, but “which axioms to accept, or at least... the philosophical consequences of accepting one set of axioms rather than an-other.”(200) Any candidate for identity at least ought to be reflexive, symmetricand transitive. The principle of extensionality, given any relevant logic as strongas B (which DL contains), immediately delivers the properties:

x = x,

x = y → y = x.

Transitivity has two non-equivalent formulations:

x = y ∧ y = z → x = z,

x = y → (y = z → x = z),

which we will follow Kremer in calling conjoined transitivity and nested transitivity,respectively. Logics that include conjunctive syllogism validate the conjunctiveform. Logics that include the hypothetical syllogism pair of prefixing and suffixingvalidate nested transitivity. Either form of transitivity, then, will be validated bythe extensional definition of =.

On the other hand, conjunctive and nested forms cannot be equivalent ; or atleast, both directions of

Φ ∧Ψ→ Υ a` Φ→ (Ψ→ Υ)


need to fail. Otherwise, from left to right, we can have Φ ∧ Ψ → Φ lead to Φ →(Ψ→ Φ), which is weakening; or from right to left, (Φ→ Ψ)→ (Φ→ Ψ), which isan instance of an axiom, yields Φ∧ (Φ→ Ψ)→ Ψ, which is pseudo-modus ponens.

4.3. Substitution. Identical sets should be intersubstitutable in a fairly nat-ural way. If x = y, then x and y satisfy all the same formulae. In the context ofnaive set theory’s comprehension principle, this is equivalent to saying that x andy are members of all the same sets. In [BR89], the sign ‘=’ is simply defined interms of membership coincidence: (∀z)(z ∈ x ↔ z ∈ y); Routley and Brady usethe name ‘extensionality’ for what is more commonly called Leibniz’s law or thesubstitution of identicals. As discussed at least since [Rou77], though, there aredifferent and definitely non-equivalent forms of substitution to consider.

Modeling predication in relevant logic in general is studied by Dunn [Dun87],to distinguish those properties an object has essentially rather than accidentally.In that study identity is integral: Any a is said to be relevantly Φ iff (∀x)(x = a→Φ(x)). Dunn’s analysis makes substitution, therefore, a core part of the meaning ofgenuine predication; and when substitution is governed by→, it is also a non-trivialoperation.

The most obvious formulation of a substitution principle,

(1) x = y → (Φ(x)→ Φ(y)),

is shown [Dun87] to be unpalatable on relevance grounds. For one can quicklyderive the statement x = y → (p→ p) for some (irrelevant) proposition p. Similarly,

(2) Φ(x)→ (x = y → Φ(y))

has as an instance p → (x = y → p) for arbitrary p. Even for those not primarilyconcerned with relevance, both of these schemes are basically untenable in a naiveset theory. Since we expect the theory to be inconsistent, we reason as follows.

Let p,¬p be theorems. Choose 0 for both x, y respectively. Contraposingprinciple 1, we get ¬(p→ p)→ 0 6= 0. But by counterexample, p∧¬p→ ¬(p→ p),so two applications of modus ponens yields 0 6= 0. Similarly, version 2 impliesp → (¬p → 0 6= 0). Two applications of modus ponens again reduce to 0 6= 0.This is an instance of a more general fact, that given any contradiction, either ofprinciples 1 or 2 lead to ∀x(x 6= x).

Now, because we are in a paraconsistent setting, it is not obvious whethern 6= n for all numbers n being provable is grounds for rejecting the substitutionprinciples involved. Models of relevant arithmetic due to Meyer and Mortensenin fact allow just this—that every number be non-self-identical—without triviality

4. RELEVANT IDENTITY 117

[Mor95]. Trivial or no, though, since n = n is a theorem for every n, global non-self-identity does introduce a lot more inconsistency than perhaps even a steely-nerved paraconsistent set theorist was expecting; and there does not seem to be anyphilosophical reason to expect or explain inconsistency in the vicinity of 0 (unlike,say, in inconsistency in some transfinite numbers [Can67](in [vH67]), or evenlarge finite numbers [Pri94]). What counts as unacceptable short of full trivialityis a delicate issue [Pri89], [Pri06a], [Ber08]. At this point, if we can isolatesubstitution principles that do not render all small finite numbers inconsistent,that would be worth knowing. For the duration, principles 1 and 2 are no longerunder consideration.

This leaves a few options. A conjoined formulation of substitution is

(3) x = y ∧ Φ(x)→ Φ(y),

but is very close to an unacceptable form of Curry’s paradox, Φ ∧ (Φ ↔ Ψ) → Ψ.This leaves the principle

(4) x = y ` Φ(x)→ Φ(y),

which has the deductively equivalent formulation

x = y ` (∀w)(x ∈ w → y ∈ w).

Both (4) and its deductive equivalent follow from the related formulation

x = y ∧ w = w → x ∈ w → y ∈ w.

All of these are effectively non-contraposable, since given Φ(x),¬Φ(y) there is noinference to x 6= y. Principle 4 is the one to endorse, as with [Rou77], [Bra89]. Aconsequence of the principle is

x = y,Φ(x) ` Φ(y),

which is like the (acceptable) rule form of modus ponens.

4.4. Expressive Restrictions. A widespread response to issues around ex-tensional identity is to restrict formulae defining sets in various ways. In [BR89]the non-triviality of what is there called extensional dialectical set theory is proved,where ‘extensional’ highlights the restriction on comprehension that no occurrencesof → are allowed in formulas defining sets. We saw Mares make this same sugges-tion §4.3 above. Other suggested restrictions have been, variously: not to involveabstraction terms [Lib04], or negation (‘positive’ set theory) [Gil74], [FH89],[HL08], or any combination of these. An important form of Curry’s paradox runsthrough these considerations, beginning with Gilmore.


When recommendations on expressive restriction are followed out, one obtainsdifferent systems with different properties. Whether or not = is defined or primitive,and what connectives and operators can appear in formulae defining sets, can allowfor example embedding into the logic RM3Q, as in [Lib04], which is otherwise toostrong because it includes contraction. An interesting mathematical item is to knowin which resulting models the axiom of choice holds, which Esser has investigated.Understanding these differences goes back to the core decisions in mounting a settheory: the definitions, formulations, and assumptions; and the chosen underlyinglogic.

The original motive for maintaining naive comprehension is that every opensentence of the language determines a set. If possible, an arrangement that doesnot necessitate restrictions on the formulae themselves is desirable. We are alreadypaying a hefty sum of inference rules for the right to practice naive set theory; if wemust also give up some predicates as set defining then this is becoming unaffordable.Set theory without restriction on the comprehension formulae is proven non-trivialin [Bra89], indeed, without even the restriction that the set itself not appear in itsdefining condition. The goal is to maintain expressive power, and set theories inwhich this is done are the main object of study here.

5. Restricted Quantification

This section looks forward; some of the theorems claimed here are officiallyproved next chapter.

Perhaps the most difficult challenge to set theory in relevant logic is formalizingrestricted quantification; for there do seem to be many cases in which we want to saythat all Φs are Ψs, without meaning or being able to prove that ∀x(Φ(x)→ Ψ(x)).An entailment of this form may well be stronger than what is meant by, e.g. “allthe beer is warm.” See [Mar04](198 - 200).

The most recent proposal for restricted quantification is by Beall, Brady, Hazen,Priest and Restall [BBH+06]. Using a semantic characterization, they produce aconditional, the “restricted arrow,” with a number of desirable properties. Let Rbe a three-place relation on setups x, y, z.

Φ→ Ψ is true at x iff, for all y, z such that R(xyz),Ψ is true at z if Φ is true at y.

Various constraints on the relation R produce different conditionals, and so differ-ent logics [Pri08]. The innovation of [BBH+06] is to control implication by theaddition of a heredity constraint. Take x ≺ z to mean that everything true at x isalso true at z. Define a new three place relation,

R′(xyz) := R(xyz) ∧ x ≺ z.

5. RESTRICTED QUANTIFICATION 119

This gives the clause for restricted arrow,

Φ 7→ Ψ is true at x iff for all y, z such that R′(xyz),Ψ is true at z if Φ is true at y.

This seems like a promising way to engineer a subset relation.1 For us the importantfact about this new conditional is

∀x(x ∈ X) ` ∀x(x ∈ Y 7→ x ∈ X).

This is a weakening that, roughly, suggests an indifference to context. The indif-ference is what will allow recapture of some material implications and extensions,presuming that extensions are meant to be indifferent to context.

At least since Routley’s [Rou77], the basic idea has been to outfit → witha similar merging device; in a more sophisticated version, this is still the idea of[BBH+06]. To show the generality of our main result, further detours throughsemantics may be avoided using a syntactic constant. The constant t is addedconservatively [DR02](11) through a two way rule:

Φ a` t→ Φ

Then the enthymematic conditional is defined:

Φ 7→ Ψ := Φ ∧ t→ Ψ.

Whether or not using t agrees in other points of detail with the semantic definitionof the arrow, for what follows, makes no difference (as the authors of [BBH+06](footnote 16) point out; c.f. [Pri06b](ch18)).

The main facts about 7→ is that it obtains if an →-entailment does, but, im-portantly, 7→ does not contrapose. Since Φ ` Ψ 7→ Φ, contraposition would furthergive us Φ ` ¬Φ 7→ ¬Ψ. If, though, both Φ and ¬Φ hold, then arbitrary ¬Ψ follows,which is absurd. The authors of [BBH+06](589) note that contraposition mustfail for restricted arrow given their desiderata.2

Like DLQ’s counterexample axiom, 7→ takes extensional information from in-tensional information. The arrow is intended for use in set theory. For example, ifU is the universe of our example in §4, then it seems that every set is also a subsetof U , just because everything whatsoever is self-identical. And so it is tempting todefine subsets as

a v b := ∀x(x ∈ a 7→ x ∈ b).

We can go back to the formulation of the extensionality axiom and write:

[Ext7→] x v y ∧ y v x↔ x ≡ y.

1Asmus has already worked out proofs systems and a method for obtaining completeness resultsfor this conditional [Asm09].2Brady has also recently pointed out (in communication) that a Curry paradox is in the vicinity:x ∈ C ↔ t ∧ (x ∈ x→ Ψ) leads to Ψ.


Ext7→ seems useful. We have already met the doppleganger problem, which ingenerality involves at least two sets,

x : Φ(x),

x : Φ(x) ∧ p,

where p is a tautology. Now the dopplegangers collapse: Φ(x) 7→ Φ(x) ∧ p holdswhen p does. By Ext7→, x : Φ(x) ≡ x : Φ(x) ∧ p, after all.

Recalling U = x : x = x and V = x : ∃y(x ∈ y), we put the idea to work.From the logic of = and ∃, and the stronger fact that ∀x∀y(y ∈ x → y ∈ V ),

we have ∀x(x ∈ U), ∀x(x ∈ V ), and ∀x(x v V ).

Theorem 4.1. ∀y(y v U).

Proof. Consider a ∈ U .

a = a ` t→ a = a.

Then a = a ` a ∈ y 7→ a = a. Then ∀x(x = x) ` ∀x(x ∈ y 7→ x = x). By modusponens, ∀x(x ∈ y 7→ x = x). Ergo ∀y(y v U) by abstraction, as desired.

Theorem 4.2. V ≡ U .

Proof. By Ext 7→ and the last two theorems.

Now define P (x) := y : y v x.

Theorem 4.3. V ≡ P (V ).

Proof. Because ∀x(x v V ), also ∀x(x ∈ P (V )). Since

∀x(x ∈ P (V )) ` ∀x(x ∈ a 7→ x ∈ P (V ))

it follows just as before that ∀y(y v P (V )), so in particular, V v P (V ). Therefore

V v P (V ) ∧ P (V ) v V,

showing, by Ext7→, that V ≡ P (V ).

While this is the anomaly that undid Frege’s system, it should be expected,possibly even welcome, that such prestigious paradoxes are theorems here.

Obviously x = y ` x ≡ y. The question, roughly, is the converse. Less roughly,we know that restricted arrow does not contrapose, as the arrow of DLQ does.Perhaps, though, there is a way to integrate the two identity strengths, so thatextensional behaviors of identity and subset are preserved, but where the importantnegation facts about sets—facts needed to know what sets are, and for proving

5. RESTRICTED QUANTIFICATION 121

theorems about sets—are preserved too. Clearly → cannot be interchangeablewith 7→, but we would like to know how far they can go together. Maybe = and≡ are a special case of implication where we can have it all. This is the hypothesisunder investigation.

Our main result3 for the chapter concerns a substitution principle for Ext 7→,

(5) x ≡ y ` Φ(x) 7→ Φ(y).

If ≡ is serving as identity, and is extensional, it must satisfy this principle.Here are two empty sets,

x : x 6= x,

x : ⊥.

Given contraposition and double negation properties, these are the compliments ofour universes (c.f. chapter 5, prop. 5.15),

z ∈ V ↔ z 6∈ x : ⊥,

z ∈ U ↔ z 6∈ x : x 6= x.

Even with Ext7→, we do not get a unique empty set, since 7→ does not contrapose[Pri06b](255). That is good news, as we are about to see. Notice in particularthat

x 6∈ V ↔ ⊥,

that V is exhaustive, and x : ⊥ is void, on pain of absurdity.

Theorem 4.4. Principle 5 is trivializing.

Proof. Ext7→ implies that V ≡ U . However, R 6= R by theorem 5.7. ThereforeR 6∈ U . Then from principle 5,

U ≡ V ` R 6∈ U 7→ R 6∈ V.

By two applications of modus ponens, R 6∈ V . Therefore ⊥.

To see what is going on from a slightly different angle, consider Russell’s wit-nesses,

ν(R) = x : R ∈ R.

Since R ∈ R from theorem 5.7, it follows that ∀x(x v ν(R)), as in theorem 5.2.Now we’ve established that ν(R) v V since everything is, and V v ν(R) for thesame reason; so

ν(R) ≡ V.

3This initially developed while studying Libert’s limitive extensionality results in [Lib04].


But since R 6∈ R it follows that ∀x(x 6∈ ν(R)). In particular, R 6∈ ν(R). Now bysubstitution principle 5 and modus ponens twice, ⊥.

Similarly and a fortiori, even though ¬∃x(x ∈ ν(R)), any proposals for exten-sionality that yielded ν(R) ≡ x : ⊥ would be disastrous.

The result seems most directly related to Dunn’s finding [Dun88] in [Aus88]that, if a logic meets some basic conditions and extensionality 7→ is enforced thencollapse back to classical logic (triviality, for the naive set theorist) is inevitable.Our result also relates, then, to the ‘slingshot’ argument, as Dunn points out. Ithink this shows that v and ≡ are regressive notions of subset and identity, as Inow urge.

Unrestricted use of restricted quantification is therefore unacceptable. Theextensionality axiom in terms of v cannot be maintained. Any heavy use of t willreintroduce exactly the problems relevant machinery is meant to solve with respectto set theory. This result can be viewed in two ways. On the one hand, we mightsay that certain classical inferences are explosive, regardless of whether we welcomethem explicitly or covertly through devices like t; and this suggests that there isnothing to be gained in being half-hearted about paraconsistency. On the otherhand, we may say that there are simply certain facts about the full set theoreticuniverse, understood as the domain of a non-trivial but inconsistent theory; and anon-trivial but paraconsistent logic will bring out those facts, no matter how we tryand subvert them with classicality-preserving arrangements. Defining subsets witht reproduces nice classical behavior that we traditionally expect from subsets. Butthe argument to explosion shows limits on what can be inferred—namely, that ≡ isnot identity. While there are multiple subset-like relations to work with, some arestronger than others. At the precipice of absurdity, serendipitously, it is difficult toprove that two sets are truly identical. As I will argue in the concluding section,these are not, even extensionally, the same sets.

Relevant logics, and paraconsistent logic in general, are designed to handleparadoxical objects by abstaining from making formerly innocuous logical infer-ences. The introduction of restricted quantification via t is intended to relieveseparation anxiety. But the relations 7→ and ≡ bring back moves that were prob-lematic to begin with. It is to be expected that the same problems resurface whenthese relations are reintroduced.

6. An Iatrogenic Disorder

A medical condition is iatrogenic when it is induced by misdiagnosis or mis-treatment. Some diagnose dopplegangers and all the other attendant problems asa failure of extensionality. In light of the triviality argument in the last section,this perceived failure is chronic. Weakening the constraints of identity using t are

6. AN IATROGENIC DISORDER 123

to no avail. Nor is it possible to alter these facts by honing the criteria for identityfurther, saying that x = y iff they share an extension and an anti-extension, i.e.

x = y ↔ (∀z)[(z ∈ x↔ z ∈ y) ∧ (z 6∈ x↔ z 6∈ y)]

The suggestion is well intentioned, but inert. In effect, since the logic DLQ includesan axiom for contraposition, we already have this.

The time has come to question whether the objects in question really are dopp-legangers after all. Yes, both V and U are universal—but they are not really thesame set, even extensionally. To give the proof requires appeal to the counterexam-ple rule. Some set, e.g. Russell’s set R, is in V but not in U ; that is, R ∈ V ∧R 6∈ U ,which shows that ¬(∀x)(x ∈ V → x ∈ U), and ergo that

V 6= U.

Sets like V and U both have all objects as their members, and yet are not identical.They have different members. Some objects are demonstrably not in U , whereasanything not in V implies triviality. The situation is even more dramatic with ν(R),of which both (∀x)x ∈ ν(R) and (∀x)x 6∈ ν(R) hold. Both x : x 6= x and x : ⊥have no members, whereas e.g. ν(R) is a member of the first but not the second.

This is a signature of inconsistency: Surprises are hidden even where we havealready looked for them. In the general case, Φ(x) and Φ(x)∧p would seem to pickout all the same objects if p is just true—but there remains the possibility that ¬pis also true, and hence that the latter set excludes as many objects as it embraces.

Since Frege we have known that naive set theory is inconsistent. The phenom-enon we are witnessing is explicable through inconsistency. But nothing indicatesthat inconsistent set theory is not extensional; rather, these show what extensional-ity means in an inconsistent setting. The logic does not demand that V and U arethe same set because they are not the same set. They are not the same extension.

Any reported failure of extenstionality, then, amounts to this. What one canprove using an insensitive operator like the material conditional, working undertruncated axioms like those of ZFC, diverges from what can be proven elsewhere.More specifically, sets that appear to be identical when viewed through a classicallens already incapable of perceiving many (most?) of the objects in the wider settheoretic universe are no longer identical when viewed using a relevant paraconsis-tent lens. The reported failure, in short, amounts to observing that paraconsistentlogic is not classical; the report amounts to not very much.

Future discussions of naive set theory can be recast by taking a thoroughgo-ing paraconsistent approach. Much brilliant work notwithstanding, the unstablemismatch of classical metatheory to inconsistent object theory causes misdiagnosisand, subsequently, mismanagement. Even in situations reckoned to be consistent


(notably, the full theory of ordinals and cardinals not being one of these), the ap-paratus of classical set theory and logic is inappropriate, for two reasons. The firstis just as the authors of Entailment warn:

One might think as follows. The point of [paraconsistency] is totake seriously the threat of contradiction. But there is in thisvicinity no such threat. ... That sounds OK; but is it? Afterall, we supposed that ‘here there is no threat of contradiction’is to be construed as an added premise ... bound up with thethreat of contradiction, and one thing that is clear is that addingpremises cannot possibly reduce threat. [ABD92](503)

The second reason goes beyond concern about inconsistency. It is concern with thebasic integrity of a robustly non-classical project. The point is emphasized in anhonest complaint from John P. Burgess:

How far can a logician who professes to hold that [paraconsis-tency] is the correct criterion of a valid argument, but who freelyaccepts and offers standard mathematical proofs, in particularfor theorems about [paraconsistent] logic itself, be regarded assincere or serious in objecting to classical logic? [Sha05](740)

Once we take a coherent naive theory seriously, no longer judging it by intuitionsfed on classicality, the force of results, in particular the status of paraconsistentextensionality, changes.

Brady’s non-triviality result constructs a transfinite model for set theory—using already a considerable amount of classical set theory and classical semanticmodeling assumptions. The fine works of Hinnion and Libert [HL08] rest on highlydeveloped topology. This is not to suggest that their work is any less valuable, onlythat certain perspectives inevitably taint results, and especially the interpretation ofresults. If one is only interested in conventional metatheorems about unconventionalstructures, then paraconsistent information about the structures will be distorted,lost or ignored. Many of the difficulties that an embedding of naive set theory hasfaced are iatrogenic, because logicians of classical schools are simply not trained totreat the altogether different pathology of dialethias.

If one takes the features of the classical universe to be the gold standard againstwhich all else is judged, then something in naive theory (or at least a naive theoryin which every formula is taken to be set-defining) is terribly non-extensional. Onehas also essentially ruled out the possibility of alternative criteria: Differences willalways be deviants, abominations. Here we still know that

- sets which can be proven to have the same members are identical; and- identical sets can be intersubstituted.

6. AN IATROGENIC DISORDER 125

Sets that are not identified by =, on the other hand, are simply not co-extensive.There is no ‘failure’ of extensionality over a relation like ≡ any more than there isa failure of explosion over a paraconsistent consequence relation. Far from being adisorder, it seems very much a part of the intensional set concept that the propertiesΦ and Φ ∧ p should not be even apparently coextensive. Non-classical behavior isnot the same as non-extensional behavior. Under the analytic axioms of naive settheory, this is what it means for sets to behave extensionally.

We now have some sense of the logic DLQ and the initial definitions that willsuit naive pursuits, especially with respect to identity and the universal set.

We are now ready to begin.

CHAPTER 5

Elements

This chapter develops the basic elements of naive set theory—the consequencesof the full comprehension principle—in a paraconsistent logic. Results divide intotwo sorts. There is classical recapture, where the main theorems about ordinalsand cardinals are proved. Then there are major extensions, including proofs of theaxiom of choice and a counterexample to the continuum hypothesis.

1. Introduction

A set is any collection of objects, and is itself an object. The members of a setdetermine the identity of that set. The first clause is the principle of comprehension,and the second is the principle of extensionality. These statements define the setconcept, and on the basis of these principles, the mathematics of set theory can bedeveloped, as this chapter demonstrates.

The resulting theory has several important features. It contains as theoremsall the basic theorems of standard ZFC set theory, initially proving the ZF axiomsand building to the theory of ordinal and cardinal numbers. This has been calledthe classical recapture, and has proved startlingly elusive. The demands placed onthe underlying logic are simply very stringent; speaking to this point, the usuallyoptimistic authors of [MRD78] conclude in dramatic fashion that “naive set theoryis untenable.”(128) Priest summarizes the situation:

Since the early days of paraconsistent logic, it has been clear thatthe rejection of ex contradictione [quodlibet] is not possible with-out the rejection of other things which appear to be much moreintegral to classical reasoning.... Several logicians (includingBrady, Meyer, Mortensen, Priest and Routley) have attemptedto reconstruct various fragments of classical reasoning.... Whilethe results are not definitive, they are not terribly encouraging.Though ... a highly important—indeed, essential—enterprise, itwould now appear that the aim of reconstructing sensible clas-sical reasoning in this way may not be realizable.[Pri06b](222)

This chapter begins to show that the goal is realizable.

127

128 5. ELEMENTS

Recapture of higher theories, most strikingly arithmetic, is not included herebeyond showing that the Peano axioms hold. Not all classical consequences of thesetheorems are paraconsistent consequences—at least not automatically—so the issueof what can be proved from here is open. Generally there is much more to be done;the best days of such a set theory lie ahead.

Paraconsistent set theory can prove more than its classical counterpart, becauseit is using the full set concept. We will be able to prove the axiom of choice andprovide a counterexample to the generalized continuum hypothesis. And the theorybegins to address new mathematical questions that arise in coherent inconsistentsettings. Inconsistent objects are characterized and an approach to counting themis developed, thereby proposing an answer as to how big is e.g. the singleton awhen a differs from itself with respect to membership. The work here, then, inaddition to providing a paraconsistent foundation for mathematics, also provides afoundation for and new directions in paraconsistent mathematics.

The intent of the chapter is not to interpret or philosophically defend at anylength the results; this is for chapter 7. The issues involved, for example the identityof theorems when moving from one language to another, are subtle and philosoph-ically controversial. Whether or not the entities being studied are really sets, whatit means for identity = or mebership ∈ to behave inconsistently, and therefore inwhat sense this work is a classical recapture, are all interesting questions. Butmathematics, controversial or otherwise, must first stand on its own; Zermelo hadto give his wellordering proof before we could argue about the axiom of choice.Before any useful discussion of the theory can take place, we must know what thetheory is.

The ultimate goal here is to provide a new, paraconsistent proof of Cantor’stheorem that there are orders of cardinal powers, by pinning down a rigorous theoryof ordinals. Whether in a paraconsistent background this has the same force as itdid over a century ago when Cantor discovered it, is a discussion I hope will occurelsewhere.

2. Logic

Last chapter we presented DLQ. Here we will officially add the hypotheticalsyllogism pair.1 I briefly recall support for this logic.

Comprehension is an extremely powerful principle and can only be channeled byweak logics. Still, we want a logic that respects some natural behaviors of sets. Forexample, given that two sets with all and only the same members are identical, itshould also be the case that sets with different members are not identical. Or again,when y = z : Φ(z) and ¬Φ(x), then it should be that x 6∈ y. These are fairly

1The addition officially makes the logic TLQ.

2. LOGIC 129

strong requirements on negation. When formalized (e.g. with a counterexampleaxiom VIII below), these demands mean that set negation excludes the middle,tertium non datur: Φ ∨ ¬Φ for any sentence Φ. This is to be expected. Anexhaustive universe is explicitly postulated by Cantor and Frege; it is the triggerof inconsistency in comprehension to begin with.

The language of first order set theory has primitives ∧,¬,→,∀, =, ∈, and :;variables x, y, z, ...; names a, b, c, ...; and formulae Φ,Ψ,Υ, ..., built up by standardformation rules. Shorthand is used: Φ ∨Ψ for ¬(¬Φ ∧ ¬Ψ); Φ↔ Ψ for (Φ→ Ψ) ∧(Ψ→ Φ); ∃ is ¬∀¬. (Taking these as definitions means that e.g, Φ∨Ψ→ ¬(¬Φ∧¬Ψ)is no more than an instance of axiom I below.)

2.1. Axioms. All instances of the following schemata are theorems:I Φ→ ΦIIa Φ ∧Ψ→ ΦIIb Φ ∧Ψ→ ΨIII Φ ∧ (Ψ ∨Υ)→ (Φ ∧Ψ) ∨ (Φ ∧Υ) (distribution)IV (Φ→ Ψ) ∧ (Ψ→ Υ)→ (Φ→ Υ) (conjunctive syllogism)V (Φ→ Ψ) ∧ (Φ→ Υ)→ (Φ→ Ψ ∧Υ)V I (Φ→ ¬Ψ)→ (Ψ→ ¬Φ) (contraposition)V II ¬¬Ψ→ Ψ (double negation elimination)V III (Φ→ Ψ)→ ¬(Φ ∧ ¬Ψ) (counterexample)IXa (Φ→ Ψ)→ [(Ψ→ Υ)→ (Φ→ Υ)]IXb (Φ→ Ψ)→ [(Υ→ Φ)→ (Υ→ Ψ)] (hypothetical syllogisms)X (∀x)Φ→ Φ(y/x)XI (∀x)(Φ→ Ψ)→ (Φ→ (∀x)Ψ)XII (∀x)(Φ ∨Ψ)→ Φ ∨ (∀x)Ψ

In axiom X, y is free for x in Φ. In axioms XI and XII, x is not free in Φ.

2.2. Rules. The following rules are valid:I Φ,Ψ ` Φ ∧Ψ (adjunction)II Φ,Φ→ Ψ ` Ψ (modus ponens)III Φ→ Ψ,Υ→ ∆ ` (Ψ→ Υ)→ (Φ→ ∆)IV Φ ` (∀x)ΦV x = y ` Φ(x)→ Φ(y) (substitution)

Modus ponens is also valid in the single premise form: Φ ∧ (Φ → Ψ) ` Ψ, notto be mistaken for the illegitimate axiom form. Substitution, similarly, is only validin rule form.

The following derived facts will be most helpful.Double negation introduction follows from axioms I and VI, and modus ponens:

(¬Φ→ ¬Φ)→ (Φ→ ¬¬Φ). Then the contraposition axiom VI can be rearranged to

130 5. ELEMENTS

(Φ→ Ψ)→ (¬Ψ→ ¬Φ). Counterexample gets its name from the contraposed formΦ∧¬Ψ→ ¬(Φ→ Ψ). From the instance of counterexample (Φ→ Φ)→ ¬(Φ∧¬Φ),by axiom I and modus ponens we have the law of non-contradiction: ¬(Φ ∧ ¬Φ).Then by the definition of disjunction follows the law of excluded middle, Φ ∨ ¬Φ.Contraposition on axiom V gives a schema for ∨-elimination: (Φ → Ψ) ∧ (Υ →Ψ) → (Φ ∨ Υ → Ψ). Then as theorems are the pair (Φ → ¬Φ) → ¬Φ and(¬Φ→ Φ)→ Φ, reductio and consequentia mirabilis. From axiom XI we have thescheme for existential instantiation, (∀x)(Φ→ Ψ)→ (∃x)Φ→ Ψ.

A useful derived rule is

Φ→ (Ψ→ Υ),Υ→ ∆ ` Φ→ (Ψ→ ∆).

The rule holds because Υ → ∆ implies [Φ → (Ψ → Υ)] → [Φ → (Ψ → ∆)] bytwo applications of prefixing; and then the conclusion follows by modus ponens.Similarly, an extra application of prefixing gives the extended

Φ→ (Ψ→ (Υ→ Γ)),Γ→ ∆ ` Φ→ (Ψ→ (Υ→ ∆)),

and so forth. These will be called derived prefixing rules.Like other relevant logics in its area, the deduction theorem fails for DLQ,

exactly because a demonstration that Ψ is provable from Φ is not sufficient to showthat Φ relevantly implies Ψ; often a good deal more information than Φ alone isneeded to get Ψ. So the difference between deductions marked by ` and implicationsmarked by → are stark. While the move from Φ → Ψ to Φ ` Ψ is modus ponens,the other direction does not follow. In general, → carries the burden of relevanceand theorems containing it are harder to obtain. To this end we have two morerules:

2.3. Meta-rule. If Φ ` Ψ, then Φ ∨Υ ` Ψ ∨Υ.

2.4. Quantified Meta-Rule. If Φ ` Ψ, then (∃x)Φ ` (∃x)Ψ.The idea of meta-rules is discussed in [Bra06](6). Importantly, meta-rules

soften the loss of the deduction theorem; in particular, the following argument, aderived meta-rule, is validated:

if Φ ` Υ and Ψ ` Υ, then Φ ∨Ψ ` Υ,

This is useful for disjunction eliminations when reasoning under hypothesis, as intheorem 5.41. We also obtain the useful substitution form, a = b,Φ(a) ` Φ(b).Another simple derived rule is: Φ ∧ (Φ → Ψ) ` Ψ, and its quantificational coun-terpart, Φ(a) ∧ (∀x)(Φ(x) → Ψ(x)) ` Ψ(a). So although pseudo-modus ponens isnot included, using the meta-rules we can still reason fairly naturally even underhypothesis.

2. LOGIC 131

A fusion operator cannot be added to this logic for naive set theory on painof triviality. Such an operator obeys the two way residuation rule

Φ Ψ→ Υ a` Φ→ (Ψ→ Υ),

(see [DR02](12)) but allows derivation of curry-like paradoxes [Bra06](35).Again, we have the connective t, the two-way rule Φ a` t → Φ, and the

enthymematic conditional Φ 7→ Ψ defined Φ ∧ t→ Ψ.In general for DLQ there is no way to convert a proof into an entailment.

Now, it would not be unreasonable to expect that 7→ to do this, e.g. if Φ ` Ψthen ` Φ 7→ Ψ. But this would in fact be a disaster. For instance, we know thatΦ,Φ→ Ψ ` Ψ. Yet ‘t-pseudo modus ponens’, ` Φ∧ (Φ→ Ψ) 7→ Ψ admits of curryparadoxes just as its t-free mate does.2 Thus 7→ is not to be used in this way.

In [Bra03](327), there is a promising approach For proofs to yield formulaeby restricted quantification, we could add more axioms to govern quantifiers of theform (∀xΦ)Ψ. The desired move from proofs to formulae is brought about by anextra primitive rule:

∃xΦ(x) ` (∀xΦ)Φ(x).

This gives us some nice flexibility, due to the derived equivalence

∃xΦ(x) ` (∀xΦ)Ψ(x)

iff

Φ(x) ` Ψ(x).

That is, a proof of Ψ from Φ, along with the supposition that there are Φs, licensesa restricted quantification. Brady shows that, with the minimal assumption thatthe restricting predicate is non-empty, a rule can safely be converted to a formula(and vice versa—assuming some further postulates).

Here, the set theory being developed is not couched in terms of quantifiers be-yond ∀,∃, because the general theme of this thesis is to investigate the relationshipbetween intension and extension via sets. Running proofs with restricted quantifiersto derive formulae is a promising approach, but in effect writes → out of the story.For example, using restricted quantifiers of this sort, the natural way to formulatea subset relation between X and Y would be (∀xx ∈ X)x ∈ Y , and then the exten-sionality axiom should say [(∀xx ∈ X)x ∈ Y ∧ (∀xx ∈ Y )x ∈ X)]→ X = Y . Giventhe philosophical study of intensional sets, this is not appropriate; more exactly,given our choice of logic, this is not an option (see last chapter). We will insteadsee how far →, with some help from the t constant, can take us.

2This is an unpublished result due to Brady.

132 5. ELEMENTS

3. Axioms

The formal language ofDLQ is augmented with a variable binding term formingoperator · : −. It remains an open question how to add term-forming symbolsconservatively in relevance contexts [Bra06](177), and is not a problem we wish toaddress here. The set concept is now characterized by two axioms.

Axiom 1 (Abstraction). x ∈ z : Φ(z, u) ↔ Φ(x, u),

Axiom 2 (Extensionality). (∀z)(z ∈ x↔ z ∈ y)↔ x = y.

By existential generalization on abstraction follows immediately the principle:

Theorem 5.1 (Comprehension). (∃y)(∀x)(x ∈ y ↔ Φ(x, u)).

The terminological distinction between abstraction and comprehension I takefrom [Lib04].

Abstraction and extensionality can be reconnected, as in Frege’s axiom:

Theorem 5.2 (Basic Law V). x : Φ = x : Ψ ↔ (∀x)(Φ↔ Ψ).

Under abstraction, the substitution rule is recast;

Theorem 5.3. x = y ` (∀z)(x ∈ z → y ∈ z).

Peano chose the ∈ sign to denote predication, from the greek verb εστιν, ‘tobe’. Since arbitrary predicates determine sets, then, in the comprehension principlethe occurrence of y in the predicate Φ is not ruled out. Following Routley, thisis completely unrestricted comprehension; without this, some sets would not beobtainable, e.g. the limiting case of diagonal sets, Z = x : x 6∈ Z, a subset ofwhich is used to prove Cantor’s theorem. The abstraction axiom keeps track ofthese cases viz

x ∈ z : Φ(z, u) ↔ Φ(x, z : Φ(z, u)).

Originally, tracking occurrences of sets terms appearing in their own collectivizingpredicate was accomplished with an extra ‘reflection’ axiom [BR89]. Here wefollow instead the simpler approach in [Bra06](177) and take Φ(x, z : Φ(z, u)) asa simultaneous substitution on Φ(z, u).

The full comprehension scheme is studied for three reasons. The first has al-ready been stated, namely, the philosophical conviction that all predicates, evenbaroque predicates, determine sets.3 Second is a pragmatic motive: Since DLQ is

3Priest and Routley: “The naive notion of set is that of the extension of an arbitrary predicate....

This is as tight an account as can be expected from any fundamental notion. It was thought to

be problematical only because it was assumed (under the ideology of consistency) that ‘arbitrary’could not mean arbitrary. However, it does.”[PRN89](499)

4. BASICS 133

so terribly weak, the proving power for theorems must come from somewhere. Un-restricted comprehension will allow easier construction of functions, diagonal sets,and other useful objects. The third motive, then, is to model easily some venera-ble fixed-point phenomena in set theory; see e.g. our recursive characterization ofthe natural numbers below. Non-well-founded sets like x = x have been studiedby Aczel and are easily reproduced (albeit without the condition that such single-tons are unique). Full comprehension axiomatizes important limiting, circular, andcontradictory cases.

It is frequently convenient to use names for sets, e.g. for a set a there is aset of all the subsets of a, called the powerset of a. The symbol P (a) is used todenote this set in the same way that ∅ is used to denote a particular set below;it is a name. Similar comments apply to complementation a, intersection, union,etc. This notation can be thought of, temporarily, as governed by instances ofcomprehension:

〈x0, ..., xn〉 ∈ y ↔ Φ(x0, ..., xn),

with ordered pairs 〈x, y〉 primitive. In later sections we will officially develop orderedpairs (prop. 5.24). Already without this we have

Proposition 5.4. y = z : Φ(z) ↔ (∀x)(x ∈ y ↔ Φ(x)).

Proof. By extensionality, y = z : Φ(z) ↔ (∀x)(x ∈ y ↔ x ∈ z : Φ(z).By abstraction, (∀x)(x ∈ z : Φ(z) ↔ Φ(x)).Then by transitivity, (∀x)(x ∈y ↔ Φ(x)). For the converse, we again invoke the abstraction scheme, where(∀x)(Φ(x) ↔ x ∈ z : Φ(z), so by transitivity (∀x)(x ∈ y ↔ x ∈ z : Φ(z). Andthis with the extensionality axiom completes the proof.

4. Basics

Proposition 5.5. Identity is an equivalence relation:

x = x,

x = y → y = x,

x = y ∧ y = z → x = z.

Proof. These follow directly from the properties of→ and ∧ (axioms I, II, IVand V), and extensionality.

As with proposition 5.11 below, identity also obeys an alternate form of tran-sitivity due to the hypothetical syllogism, namely

x = y → (y = z → x = z).

Proposition 5.6. (∀x)(∀y)(x ∈ y ∨ x 6∈ y).

134 5. ELEMENTS

Proof. With axiom I, a ∈ y → a ∈ y. Then by axiom VIII, a 6∈ y ∨ a ∈ y.Rule IV generalizes to the result.

Proposition 5.7. Sets that differ with respect to membership are not identical.In particular, (∃x)(x ∈ a ∧ x 6∈ a)→ a 6= a.

Proof. We prove the contrapositive.1 a = a→ (∀z)(z ∈ a↔ z ∈ a) Ext.

2 (∀z)(z ∈ a↔ z ∈ a)→ (b ∈ a↔ b ∈ a) 1, Ax.X3 (b ∈ a↔ b ∈ a)→ b 6∈ a ∨ b ∈ a Ax.II, V III

4 b 6∈ a ∨ b ∈ a→ ¬(b ∈ a ∧ b 6∈ a) 3, Ax.I5 a = a→ ¬(b ∈ a ∧ b 6∈ a) 1− 4, Ax.IV6 (b ∈ a ∧ b 6∈ a)→ a 6= a 5, Ax.V I

Existential generalization completes the result.

.When a set a is such that its membership is inconsistent, some b ∈ a and b 6∈ a,

then a is inconsistent. (The same is not said of b, since e.g. every set is both inand not in Routley’s set Z = x : x 6∈ Z, but we don’t think that all sets areinconsistent.) There are inconsistent sets in naive set theory and they have a neatcharacterization:

Proposition 5.8. (∃x)x 6= x.

Proof. By comprehension we have Russell’s set, R = x : x 6∈ x.1 (∀x)(x ∈ R↔ x 6∈ x) Comp.

2 R ∈ R↔ R 6∈ R 1, Ax.X3 R ∈ R→ R 6∈ R 2, Ax.II4 R 6∈ R ∨R 6∈ R 3, Ax.V III5 R 6∈ R 4, Ax.V6 R ∈ R 2, 5, RuleII7 R ∈ R ∧R 6∈ R 5, 6, RuleI

Since R differs from itself with respect to membership, by proposition 5.7,R 6= R.

From this novelty, Restall [Res92](427) infers by generalization the

Corollary 5.9 (Restall). There are at least two objects, (∃x)(∃y)(x 6= y).

These few facts already show us that the present set theory will have as theo-rems some propositions not contained by classical theory, and that classical theo-rems may be recaptured by distinctively non-classical means.

Definition 5.10 (Subsets). x ⊆ y is (∀z)(z ∈ x→ z ∈ y).

4. BASICS 135

This leads to a natural partial order; the converse of anti-symmetry actuallyholds, too, since this is just the axiom of extensionality rewritten.

Proposition 5.11. Subsets are reflexive and anti-symmetric,

x ⊆ x,

x ⊆ y ∧ y ⊆ x→ x = y.

They are also transitive:

x ⊆ y ∧ y ⊆ z → x ⊆ z,

x ⊆ y → (y ⊆ z → x ⊆ z),

and x ⊆ y → (z ⊆ x→ z ⊆ y).

Proof. Reflexivity and antisymmetry come from extensionality, axiom I andthe commutativity of conjunction. The forms of transitivity are direct results ofconjunctive syllogism and the hypothetical syllogism pair, respectively.

Proposition 5.12. Every set has an unrestricted complement, x = y : y 6∈x, and x = x.

Proof. Existence is by comprehension; its behavior is checked with doublenegation introduction and elimination: Since y ∈ x ↔ y 6∈ x, it follows thaty ∈ x↔ y ∈ x.

Moving from these simple but useful facts, we fix a top and bottom of theuniverse of sets, an emptyset and a universal set, using von Neumann’s definitionof set as an object that is a member of something. These will exhibit most of theexpected structure of a boolean lattice, except that x∩x = ∅ and x∪x = V do notgenerally hold. Note that the characterizations we are about to use are not thosegiven by Russell and Whitehead, which have since become standard: x : x = xfor the universe and its complement, x : x 6= x for the emptyset. We have alreadyseen in R 6= R that there are object not in, and in, these sets respectively. Nor,due to the relevance of the logic DL, would it be the case that y ⊆ x : x = xor x : x 6= x ⊂ y for every y. Our universe and emptyset, by contrast, are wellbehaved and have a very natural reading if sets are taken to be models of properties.

Proposition 5.13. The universe,

V = x : (∃y)x ∈ y,

exists, and (∀x)x ∈ V . The emptyset,

∅ = x : (∀y)x ∈ y,

exists, too, and is empty: (∀x)x 6∈ ∅.

136 5. ELEMENTS

Proof. Existence of both sets is by comprehension. To prove that (∀x)x ∈ V ,1 (∀x)(x ∈ V ∨ x 6∈ V ) prop.5.62 (∀x)(x 6∈ V → x ∈ V ) Abstraction, prop.5.12.3 (∀x)(x ∈ V → (∃y)x ∈ y) Ax.X,Ax.V I

4 (∀x)((∃y)x ∈ y → x ∈ V ) Abs.

5 (∀x)(x 6∈ V → x ∈ V ) 2, 3, 4The conclusion follows by ∨-elimination on line 1. To show that the emptyset

is empty, similarly, if x ∈ ∅ then (∀y)x ∈ y; then x ∈ ∅ and therefore x 6∈ ∅.

Proposition 5.14. (∀x)(x ⊆ V ) and (∀x)(∅ ⊆ x).

Proof. By existential generalization, z ∈ x → (∃y)z ∈ y. Then z ∈ x → z ∈V by abstraction, and therefore x ⊆ V . Similarly, z ∈ ∅ → (∀y)z ∈ y, and thenz ∈ x by ∀-elimination.

What these theorems do not show—and indeed what probably cannot beshown—is the uniqueness of a universal or empty set. Indeed, one can show that(∀x)x ∈ z : z = z, since any x is self-identical by proposition 5.5 above. And wealso have z : z = z ⊆ V , since everything is. But it is formally irrelevant that(∃y)x ∈ y → x = x, and unlikely that the present system will produce this as atheorem. (Proving that this is not a theorem is a metatheoretic task.) Ergo it willnot follow that z : z = z = V . Similar arguments apply to ∅. The extensionalityof naive set theory in light of this phenomenon were discussed last chapter.

Proposition 5.15. V = ∅ and ∅ = V .

Proof. One direction of each is already proved. For V ⊆ ∅ we argue thecontrapositive (using only arrows): x ∈ ∅ → (∀y)x ∈ y, and (∀y)x ∈ y → x ∈ V .Then x 6∈ V . For V ⊆ ∅, x 6∈ V → ¬(∃y)x ∈ y, so (∀y)x 6∈ y; in particular, x 6∈ ∅.Double negation elimination completes the proof.

Note that ∅ is explosive: If anything is a member of it, triviality follows.

5. ZF

Here we retrieve all the axioms of Zermelo-Fraenkel set theory, except Fundierung.Since general comprehension induces sets that are not well founded, e.g. V ∈ V ,the foundation axiom is not a part of the present theory, as is expected from Re-stall’s results in [Res92]. That the other axioms are forthcoming is not especiallysurprising, since Zermelo in 1908 [Zer67] saw them as a consistent fragment ofthe naive theory. The axiom of infinity will be provable, too, showing that thenaive theory does not fail a reductive program in the same way as Russell andWhitehead’s system did.

In each proposition, the step of universally generalizing on free a, b is omitted.

5. ZF 137

Proposition 5.16 (Aussonderung). (∃y)(∀x)(x ∈ y ↔ x ∈ a ∧ Φ(x)).

Proposition 5.17 (Powerset). (∃y)(∀x)(x ∈ y ↔ x ⊆ a).

For any a, we use the name P (a) = x : x ⊆ a.

Proposition 5.18 (Pairing). (∃y)(∀x)(x ∈ y ↔ x = a ∨ x = b).

For any a, b we use the name a, b = x : x = a ∨ x = b. A special case isthe singleton a = x : x = a. For relevance purposes, sometimes singletons arerelativized to a particular set, e.g. x : x = a∧x ∈ b, so that ab ⊆ b when a ∈ b.

Proposition 5.19 (Union). (∃y)(∀x)(x ∈ y ↔ (∃z)(z ∈ a ∧ x ∈ z)).

Proposition 5.20 (Intersection). (∃y)(∀x)(x ∈ y ↔ (∀z)(z ∈ a→ x ∈ z)).

The standard names are adopted:⋃a = x : (∃z)(z ∈ a ∧ x ∈ z) and⋂

a = x : (∀z)(z ∈ a → x ∈ z) for the union and intersection of a, respectively.Note that, because the conditional in DLQ is not material, these are not necessarilyduals. On the other hand, a∪ b = x : x ∈ a∨x ∈ b and a∩ b = x : x ∈ a∧x ∈ bobey their usual algebra, save for the explosive a ∩ a ⊆ b. The complement of b ina is now easily construed as a− b = a ∩ b.

The next axiom is the axiom of infinity. Note that V is already a set containing∅ and its successors (since every set whatever is in V ), and that V will count asdedekind-infinite once that notion is defined. We will also prove that the naturalnumbers exist later. So the following is only for curiosity.

Proposition 5.21 (Infinity). There is a non-empty set i such that when x ∈ i,also xi ∈ i.

Proof. Consider i = x : x ⊆ i, a set that is its own powerset. Both i ∈ iand ∅ ∈ i, so i is not empty. For x ∈ i, also xi ⊆ i, and so xi ∈ i.

In the next definition and beyond, we take to writing some sets in a common-place way, as e.g. 〈x, y〉 : x ∈ a ∧ y ∈ b. As noted above, this is to be controlledwith instances of abstraction, in this case 〈x, y〉 ∈ z ↔ x ∈ a ∧ y ∈ b for some y,which we will name a× b.

Definition 5.22. An ordered pair is 〈a, b〉 = a, a, b. A cartesian prod-uct is a × b = 〈x, y〉 : x ∈ a ∧ y ∈ b. A relation is r ⊆ a × b, and its inverseis r−1 = 〈x, y〉 : 〈y, x〉 ∈ r. A function is a relation f : a −→ b with domaindom(f) = a and range rng(f) = b, such that

〈x, u〉 ∈ f ∧ 〈x, v〉 ∈ f 7→ u = v.

138 5. ELEMENTS

The composition of two functions f , g is fg = 〈x, z〉 : (∃y)(〈x, y〉 ∈ g∧〈y, z〉 ∈ f.The image of a under f is f ′′a = f(x) : x ∈ a. The restriction of f to x ⊆ a isf |x = f ∩ (x× a).

Proposition 5.23 (Replacement). Let f be a function with domain a. Thenf ′′a exists.

When 〈x, y〉 ∈ f and f is a function, then we want to write f(x) = y. For this,we can think of functional claims defined by two-place predicates Φ in the usualway: (∀x)(∃y)[Φ(x, y) ∧ (∀z)(Φ(x, z) → y = z)], though the details need checking.The next theorem shows that with some care similar details can be (tediously)worked through.

Proposition 5.24 (Ordered Pairs). 〈a, b〉 = 〈c, d〉 a` a = c ∧ b = d.

Proof. Right to left is substitution, a = c ` a ∈ 〈a, b〉 → c ∈ 〈a, b〉, andsimilarly for b = d. We prove left to right. For a = c,

1 〈a, b〉 = 〈c, d〉2 (∀x)(x ∈ a, a, b ↔ x ∈ c, c, d) def.5.223 ∀x)(x = a ∨ x = a, b ↔ x = c ∨ x = c, d) 2, Abs.4 a = a → a = c ∨ a = c, d 3, Ax.X5 a = c ∨ a = c, d →

(∀x)(x ∈ a ↔ x ∈ c) ∨ (∀x)(x ∈ a ↔ x ∈ c, d) 4, Ext.6 (∀x)(x = a↔ x = c) ∨ (∀x)(x = a↔ x = c ∨ x = d) 5, Abs.7 (∀x)(x = a↔ x = c)→ (a = a→ a = c)∧(∀x)(x = a↔ x = c ∨ x = d)→ (c = c ∨ c = d→ c = a) 6, Ax.V,Ax.IX

8 a = a ∨ c = c ∨ c = d→ a = c 7, Ax.V9 a = c 8, prop.5.5, RuleII

To prove that b = d, first we prove that b = d ∨ b = a. Then we prove thatb = d ∨ d = a. Once proved, these can be used in substitutions, and together willgive the result. Again from the definition of ordered pairs,

6. ORDINALS 139

1 a, b = a, b → a, b = c ∨ a, b = c, d2 (∀x)(x = a ∨ x = b↔ x = c)∨(∀x)(x = a ∨ x = b↔ x = c ∨ x = d) 1, Abs.

3 (b = a ∨ b = b↔ b = c)∨(b = a ∨ b = b↔ b = c ∨ b = d) 2, Ax.X

4 b = c→ b = c ∨ b = d Ax.II

5 b = a ∨ b = b→ b = c ∨ b = d 3, 4, Ax.IV, V6 b = b prop5.57 b = c ∨ b = d 5, RuleII8 a = c prop.5.249 a = c ∧ (b = c ∨ b = d) 7, 8, RuleI10 (a = c ∧ b = c) ∨ (a = c ∧ b = d) 9, Ax.III11 a = c ∧ b = c→ a = b prop.5.512 a = c ∧ b = d→ b = d Ax.II

13 b = a ∨ b = d 10, 11, 12, Ax.Vcompleting the first step of the argument. Now1 a = c prop.5.242 〈a, b〉 = 〈c, d〉3 〈a, b〉 = 〈a, d〉 1, 2, RuleV4 (∀x)(x = a ∨ x = a, b ↔ x = a ∨ x = a, d) 3, def.5.225 a = a → a = a, d ∨ a, d = a, b 4, Ax.X6 a = a, d ∨ a, d = a, b →

(∀x)(x ∈ a ↔ x ∈ a, d) ∨ (∀x)(x ∈ a, d ↔ x ∈ a, b) 5, Ext.7 d = d→ d = a ∨ d = b 6, Ax.X, V8 d = a ∨ d = b prop.5.5, 7, RuleII

completing the second step of the argument. To conclude, we adjoin the twofacts (b = a ∨ b = d) ∧ (d = a ∨ d = b); distributing possibilities,

(b = a ∧ a = d) ∨ (b = a ∧ d = b) ∨ (b = d ∧ d = a) ∨ (b = d ∧ d = b),

each of which implies that b = d, as required.

Most set theories include the axiom of choice; proving choice would recapturethe axioms of ZFC. We will see that choice is a consequence of a deeper Cantorianprinciple: that every set can be put into a wellorder. To show this, though, wemust first do a lot of work on the notion of order itself.

6. Ordinals

Although the theory of cardinals is our final destination, this section, in whichthe ordinals are defined and their properties developed, is logically speaking boththe most import and the most difficult. Cantor abstracted the notion of an ordinal

140 5. ELEMENTS

from wellordered sets, as the prototypical inductive structure; the ordinals formthe central pillar of set theory. Deriving a fixed metric is the key to a set theoreticfoundation for mathematics, and the heretofore unfulfilled task of the paraconsistentproject. While the definitions and theorems are, for the most part, standard, someproofs are not. Again, much of the novelty here is to show that the classicaltheorems can be captured, where until now it had appeared that this could not bedone.

This section also goes beyond the classical, and proves that the set of all ordinalsOn is itself an ordinal, confirming both that On is wellordered and that the Burali-Forti paradox holds. The most important property of ordinals is self-similarity,displayed in the recursive proposition that an ordinal is the set of all precedingordinals, or more subtly when ordinals are defined to be transitive both insideand out. The present approach confirms this fractal property even at the level ofabsolute comprehensiveness: It is only right that On itself be an ordinal. Thisshown, the Peano axioms are justified, and proofs of transfinite induction andrecursion given.

The theorems could be arranged to first study wellfoundedness, up to a proofof transfinite induction on any wellfounded set. That is, we could avoid discussinglinearity or transitivity and still have a good chunk of structure, as many of thestandard texts do. Useful classical references are [Dra74], [Lev79], and [Kun80].

Definition 5.25. A set a with respect to ∈ is:strictly ordered iff

x, y, z ∈ a→ x 6∈ x

∧ (x ∈ y ∧ x 6∈ x→ y 6∈ x)

∧ (y ∈ z → (x ∈ y → x ∈ z));

totally or linearly ordered by ⊆ iff a is strictly ordered and

x ∈ a 7→ (y ∈ a 7→ x ⊆ y ∨ y ⊆ x),

that is, ⊆-trichotomy holds;wellfounded, Wf(a), iff

y ⊆ a ∧ (∃z)z ∈ y 7→ (∃z)(z ∈ y ∧ ¬(∃x)(x ∈ z ∧ x ∈ y))];

wellordered, Wo(a), iff totally ordered and wellfounded;transitive, Tr(a), iff x ∈ a→ x ⊆ a;an ordinal iff a is a wellordered and transitive set of ordinals, ⊆-connected to

all other ordinals.

In summary, by (full) comprehension

6. ORDINALS 141

Proposition 5.26. There is a set of all ordinals, On = x : x is an ordinal ,such that

x ∈ On↔ Wo(x)

∧ y ∈ x→ y ⊆ x

∧ x ⊆ On

∧ y ∈ On 7→ (x ⊆ y ∨ y ⊆ x).

The definition of ordinal is adapted from von Neumann. We have added im-predicative clauses, to capture the recursive idea that the ordinals are the sameinside and out. The ordinals are an analysis of the concept of induction, gener-alized to the transfinite and refified. The hard work in the theory of ordinals isin locating the right definition. Then the properties of the ordinals, culminatingin the Burali-Forti theorem, should all follow, as it were, by logic alone. On thispoint, in his Calculus on Manifolds, Spivak memorably isolates “three importantattributes [of] many fully evolved major theorems: 1. It is trivial. 2. It is trivialbecause the terms appearing in it have been properly defined. 3. It has significantconsequences.” I hope that, like the classical definition of ordinals, the definitionhere comes close to delivering in this way.

A few more detailed comments. Our extra clause in the anti-symmetry con-dition for strict order, x 6∈ x, is due to the relevance constraint on implication, aswe will see in the proofs below. In the wellfounding clause, the rendering of a sethaving a least member is material; the intensional alternative would have been thatfor any non-empty y ⊆ a, there is a z ∈ y such that (∀x)(x ∈ z → z 6∈ y). Usingthis definition, however, makes it too difficult to prove that anything is wellfounded.Similarly, to gloss z as a least member if z ∩ y = ∅ would be almost impossible toconfirm, since one would have to show that x ∈ z ∩ y not only fails, but is absurd.It may be contradictory, but it is not absurd. For linear order, we are using the⊆ relation instead of the ∈ relation, and have built in an added clause into thedefinition of On based on this choice. There is, as far as I can try, no way to provethat On is linearly ordered by ∈.

On to the mathematics. Ordinals are written with lowercase greek letters.

Proposition 5.27. ∅ ∈ On.

Proof. This is because ∅ is explosive. First, ∅ ⊆ On, by prop. 5.14. Similarly,x ∈ ∅ → x ⊆ ∅, transitivity, and x ∈ ∅ → (y ∈ ∅ → x ⊆ y ∨ y ⊆ x), a linear order.To show that ∅ is ⊆-connected to all ordinals it again suffices that ∅ ⊆ x for any xat all. Finally, to show wellfoundedness, we have a ∈ y ∧ (x ∈ y → x ∈ ∅) → a ∈

142 5. ELEMENTS

y ∧ (x ∈ ∅ ∨ x 6∈ y), by counterexample. Again since ∅ is explosive we get

a ∈ y ∧ (x ∈ y → x ∈ ∅)→ a ∈ y ∧ (x 6∈ a ∨ x 6∈ y).

Generalization completes the proof.

Proposition 5.28. Subsets of a wellordered set are wellordered:

Wo(α), β ⊆ α `Wo(β).

Proof. Let α be wellordered and β ⊆ α. By proposition 5.11, β ⊆ α→ (y ⊆β → y ⊆ α). And (y ⊆ β → y ⊆ α) means that members of β behave just asmembers of α, giving wellorder. For example, linearity is proved using hypotheticalsyllogism and (the rarely seen) rule III:

1 x ∈ β → x ∈ α2 y ∈ β → y ∈ α3 (y ∈ α→ x ⊆ y ∨ y ⊆ x)→ (y ∈ β → x ⊆ y ∨ y ⊆ x) 2, hyp. syl.4 (x ∈ α→ (y ∈ α→ x ⊆ y ∨ y ⊆ x))

→ (x ∈ β → (y ∈ β → x ⊆ y ∨ y ⊆ x)) 1, 3Rule III5 x ∈ β 7→ (y ∈ β 7→ x ⊆ y ∨ y ⊆ x) 4,m.p.

For wellfoundedness, notice that

(*) Φ→ Ψ ` Φ ∧Υ→ Ψ ∧Υ;

this allows the following derivation.1 β ⊆ α.2 β ⊆ α→ (y ⊆ β → y ⊆ α). prop.5.113 (y ⊆ β → y ⊆ α). 1, 2,m.p.4 (y ⊆ β ∧ ∃zz ∈ y → y ⊆ α ∧ ∃zz ∈ y). 3, (∗)5 y ⊆ α ∧ ∃zz ∈ y 7→ ∃z(z ∈ y ∧ ∀x(x 6∈ z ∨ x 6∈ y)) def.5.256 y ⊆ β ∧ ∃zz ∈ y 7→ ∃z(z ∈ y ∧ ∀x(x 6∈ z ∨ x 6∈ y)) 4, 5Ax.IV

as required.

Proposition 5.28 extends immediately to ordinals, as these are wellordered sets.

Proposition 5.29. On is transitive, α ∈ On→ α ⊆ On.

Proof. This is a clause in the definition of being an ordinal.

Proposition 5.30. An ordinal α is the set of all preceding ordinals,

α = x : x ∈ On ∧ x ∈ α.

Proof. Let α ∈ On. The previous theorem and the axiomatic instance β ∈α→ β ∈ α shows that β ∈ α→ β ∈ On∧β ∈ α. The other direction is immediate.Therefore α = α ∩On.

Proposition 5.31. α ∈ On→ α 6∈ α.

6. ORDINALS 143

Proof. The idea is that, were α ∈ α, then still α 6∈ α.1 α ∈ On→ (∀x)(x ∈ α→ x 6∈ x). def.5.252 (∀x)(x ∈ α→ x 6∈ x)→ (α ∈ α→ α 6∈ α). Ax.X.

3 (α ∈ α→ α 6∈ α)→ α 6∈ α ∨ α 6∈ α. Ax.V III

4 α 6∈ α ∨ α 6∈ α→ α 6∈ α. Ax.VConjunctive syllogism completes the proof.

Proposition 5.32. α ∈ β ∧ α 6∈ α→ β 6∈ α.

Proof. α ∈ β ∧ α 6∈ α→ β 6⊆ α by counterexample, and β 6⊆ α→ β 6∈ α.

Proposition 5.33. β ∈ γ → (α ∈ β → α ∈ γ).

Proof. By the definition of transitivity on γ.

The last few propositions have showed, by virtue of self-similarity, that theordinals are strictly ordered. Now we need total and wellordering.

Theorem 5.34 (Trichotomy of Ordinals). Any two ordinals are ⊆-connected,

α ∈ On→ (β ∈ On 7→ α ⊆ β ∨ β ⊆ α).

Proof. This is a clause in the definition of ordinal.

This delivers, for a start, some miscellany like the following.

Proposition 5.35. α ∩ β ∈ On, and α ∪ β ∈ On.

Proof. We can show α ∩ β is wellordered and transitive independently oftrichotomy. Let x ∈ α ∩ β. Then x ∈ α ∧ x ∈ β; then x ⊆ α ∧ x ⊆ β; thenx ⊆ α∩β, showing transitivity. Meanwhile, α∩β ⊆ α; since α is wellordered, α∩βis wellordered, too.

Easier still is to notice that if α ⊆ β, then α ∩ β = α; and if β ⊆ α thenα ∩ β = β, ordinals both. The case for α ∪ β is just like this.

More weakly, we see that certain intervals of On must be empty. Since α∩β ∈On, we have α ∩ β 6∈ α ∩ β. Therefore either α ∩ β 6∈ α or α ∩ β 6∈ β. Thereforethere cannot be ordinals intervening between both α and α ∩ β, and β and α ∩ β.Observations like these are the spur for the classical proof that On is linearlyordered.

Proposition 5.36. The ordinals are wellfounded.

Proof. We have to show that a non-empty θ ⊆ On has a least member. Letβ ∈ θ. The idea is that either β is ∈-least in θ, or else the least member of β is theleast member of θ. Note that β ∩ θ ⊆ β; since β is wellordered, β ∩ θ is wellordered(by prop.5.28). Either β ∩ θ is empty, or not.

144 5. ELEMENTS

Suppose (∀y)(y 6∈ β ∩ θ). We infer by t-introduction that

θ ⊆ On ∧ β ∈ θ 7→ β ∈ θ ∧ (∀y)(y 6∈ β ∨ y 6∈ θ).

This says that β is a least member of θ, and generalizes:

θ ⊆ On ∧ ∃zz ∈ θ 7→ ∃z(z ∈ θ ∧ (∀y)(y 6∈ z ∨ y 6∈ θ)).

So far this shows (∀y)(y 6∈ β ∩ θ) `Wf(On). On the other hand, suppose (∃y)(y ∈β ∩ θ). Since β ∩ θ is wellfounded, trading ∃ for γ we have

γ ∈ β ∧ γ ∈ θ ∧ (∀y)(y 6∈ γ ∨ y 6∈ β ∨ y 6∈ θ).

As γ ⊆ β (since γ ∈ β), by contraposition y 6∈ β → y 6∈ γ. Therefore the aboveformula reduces to

γ ∈ θ ∧ (∀y)(y 6∈ γ ∨ y 6∈ θ),

which says that γ is ∈-minimal in θ. Again with t-introduction this generalizes to

θ ⊆ On ∧ ∃zz ∈ θ 7→ ∃z(z ∈ θ ∧ (∀y)(y 6∈ z ∨ y 6∈ θ)).

This shows ¬(∀y)(y 6∈ β ∩ θ) `Wf(On).Both directions of an excluded middle lead to Wf(On). An instance of a

derived meta-rule,

if Φ ` Υ and ¬Φ ` Υ, then Φ ∨ ¬Φ ` Υ,

is now used, and we have a theorem. Any arbitrary non-empty subset of ordinalshas a least member.

Proposition 5.37. Any transitive set of ordinals, connected to all other ordi-nals by ⊆, is an ordinal.

Proof. Any set of ordinals is wellordered, by the previous theorems. Thenthe definition of being an ordinal is satisfied.

Theorem 5.38. [Burali-Forti 1897] On ∈ On.

Proof. By the transitivity of On, we have y ∈ On → y ⊆ On. By ∨-introduction, y ⊆ On → y ⊆ On ∨ On ⊆ y. With conjunctive syllogism, y ∈On 7→ y ⊆ On∨On ⊆ y as required. On is a wellordered, transitive set of ordinalsconnected to all other ordinals.

Corollary 5.39. On 6∈ On, and then On 6= On.

Proof. From prop. 5.31, and then prop. 5.7.

At this point the limitations of our logic leave us with an unsolved problem ofbasic relevant set theory. To see that On is closed under some expected operations,and that certain objects are ordinals, we need the following lemma:

6. ORDINALS 145

Conjecture. Let α, β ∈ On, let θ be a wellordered, tran-

sitive set of ordinals, and let α ⊆ θ ⊆ β. Then θ ∈ On.

To prove this, we need to show under the assumptions of the conjecture that(∀y)(y ∈ On 7→ y ⊆ θ ∨ θ ⊆ y); then all the conditions for θ being an ordinal aresatisfied. I do not have a proof that is adequate from a paraconsistent perspective.(Solatium miseris, socios habuisse malorum.) Any set of ordinals θ is such that∅ ⊆ θ ⊆ On; so if this conjecture were proved, then for any θ that is transitive andwellordered, θ is an ordinal. We will assume the conjecture for the duration.

Definition 5.40. α+ = α ∪ α is the successor of α.Ordinals α with no predecessor, ¬(∃β)(β ∈ On ∧ β+ = α), are called limits.

Proposition 5.41. α ∈ On ` α+ ∈ On.

Proof. All the members of α+ are ordinals, therefore α+ is wellordered. Fortransitivity,

1 x ∈ α+ → x ∈ α ∨ x ⊆ α.2 x ∈ α→ x ⊆ α.3 x ⊆ α→ x ⊆ α ∪ α.4 x ∈ α+ → x ⊆ α+.

To show that α+ is connected to all other ordinals, it suffices to notice thatα ⊆ α+ ⊆ On.

Proposition 5.42. On is a successor, and On is a limit.

Proof. To show successor: Since On is an ordinal, On+ is an ordinal. SoOn+ ∈ On, and by transitivity, On+ ⊆ On. And On ∈ On+ like all other ordinals,so On ⊆ On+ by transitivity again. Therefore On = On+. To show limit, we needβ+ = On ` β 6∈ On for all β. We can then apply this rule disjunctively to excludedmiddle, to obtain (∀β)(β 6∈ On ∨ β+ 6= On), which is equivalent to On being alimit. We prove β+ = On ` β = On from which β 6∈ On follows, since On 6∈ On bycor. 5.39. So let β+ = On; then β ⊆ On. Conversely, since On ∈ On by thm 5.38,also On ∈ β+ by def. 5.40. Then On ∈ β ∨On = β and in either case, On ⊆ β bythe transitivity of β. So β = On as required.

What Burali-Forti’s contradiction shows, in effect, is that the entire structureof the ordinals is recapitulated at the top. This fact provides the core of the othermain arguments to follow: the wellordering principle and thus the solution to thecardinal assignment problem, as well as the existence of distinct and inaccessibletransfinite cardinals, and a counterexample to the GCH. We get, then, quite a lotof work out of On; this is appropriate for two reasons. The first is just that it isnot at all an easy task to prove On ∈ On in a non-trivial setting, so the fact that

146 5. ELEMENTS

it is powerful shows that the logical machinery is in some sense ‘fuel efficient’. Thesecond, perhaps deeper reason is that since set theory was clarified as a legitimatemathematics in Cantor’s Beitrage, there has been intense concern over the statusof On, because it was recognized that there is a very serious paradox here; for acentury it has been thought so serious that were On ∈ On then not only does acontradiction follow, but triviality, too. What we draw out of the paradox is moreinformative and more nuanced.

Now we sketch the development of Peano arithmetic, followed by more generalordinal arithmetic.

Full comprehension induces

Proposition 5.43. There is an ω = x ∈ On : [x = ∅∨(∃y)(y+ = x)]∧x ⊆ ω).

Members n,m, ... of ω are called natural numbers, or just numbers. By defini-tion, ω ⊆ On; so it is a wellordered set.

Proposition 5.44. ω ∈ On.

Proof. Let n ∈ ω. Then by definition, n ⊆ ω, showing transitivity. As asubset of On, ω is wellordered. And n ⊆ ω ⊆ On, showing ⊆-connection to allother ordinals. So ω is an ordinal.

Theorem 5.45 (Peano’s Postulates). The following hold:0 ∈ ω;n+ 6= 0;n ∈ ω ` n+ ∈ ω;n+ = m+ ` n = m.

Proof. Since 0 ∈ On ∧ 0 = 0 ∧ (∀y)(y ∈ 0→ y ∈ ω), zero is a number. Sincen ∈ n+ but no n ∈ 0, zero is not the successor of any number. If n ∈ ω, then n+

meets all the requirements to be a number: n+ ∈ On, and has a predecessor, andall its members are either in n, in which case they are numbers, or else n, a numberagain. And if n+ = m+, then ∀z(z ∈ n ∨ z = n↔ z ∈ m ∨ z = m); then picking nand m for z respectively, (n ∈ m∨ n = m)∧ (m ∈ n∨m = n), which distributes to

(n ∈ m ∧m ∈ n) ∨ (n ∈ m ∧m = n) ∨ (n = m ∧m ∈ n) ∨ (n = m ∧m = n),

each of which implies that n = m. So the successor of every number is unique.

6. ORDINALS 147

Names of the first few natural numbers are

∅ = 0

∅+ = 1

∅++ = 2...

The fifth postulate is induction, proved in the general, transfinite case over On.

Theorem 5.46 (Transfinite Induction). Let θ ⊆ On. Suppose

(∀β)(β 6∈ α ∨ β ∈ θ)→ α ∈ θ.

Then ¬(∃α)(α ∈ On ∧ α 6∈ θ).

Proof. Suppose (∃α)(α ∈ On ∧ α 6∈ θ). Then there is a least such,

(∃α)[α 6∈ θ ∧ (∀β)(β 6∈ α ∨ β ∈ θ)].

But the hypothesis implies (∃β)(β ∈ α∧ β 6∈ θ)∨α ∈ θ, for every α, which negatesthis claim. Therefore there is no least, and so no ordinal not in θ, as required.

Transfinite induction will hold for any wellordered set, including ω. The basecase, 0, is covered automatically by the induction hypothesis: If ¬Φ(0)→ (∃x)(x ∈0 ∧ ¬Φ(x)), then Φ(0) by the explosiveness of ∅.

A close mate of induction is definition by recursion. There is something appro-priate in proving the recursion theorem, as we are about to, with a set containingitself in its defining condition.

Theorem 5.47 (Transfinite Recursion). Let h be a function from V to V .There is a function f from On to V such that

f(α) = h(f |α).

Proof. Take the set

〈x, y〉 ∈ f ↔ y = h(f |x).

Existence is immediate from full comprehension. This is a function because h is; if〈x, y〉, 〈x, z〉 ∈ f then y = h(f |x) = z.

The recursion scheme is used to define ordinal arithmetic. To deal with limitordinals, we look to least upper bounds.

Definition 5.48. Let X be a set of ordinals. The supremum of X, sup(X),is the least ordinal δ such that every x ∈ X is either in δ or identical to δ.

148 5. ELEMENTS

The usual identification of the sup(α) with⋃α does not seem to follow in

the logic DLQ, or at least not with ordinals as they are here defined, becausethe existential quantifier resists a proof that

⋃α is an ordinal. Nevertheless, or-

dinals including every member of X certainly exist—On, for example—and so bywelfoundedness a least exists, too. If there is any question about the uniquenessof a supremum, a choice function—developed in the next section independently ofordinal arithmetic—can be applied to make a functional selection.

Addition, taking h to be h(α) = α+, is

α+ 0 = α

α+ (β + 1) = (α+ β) + 1

α+ β = supα+ γ : γ ∈ β for limit β;

multiplication, taking h to be hβ(α) = α+ β, is

α · 0 = 0

α · (β + 1) = α · β + α

α · β = supα · γ : γ ∈ β for limit β;

exponentation, with hβ(α) = α · β, is

α0 = 1

αβ+1 = αβ · α

αβ = supαγ : γ ∈ β for limit β.

From here we will work a bit faster, and assume that the ordinals are linearlyordered by membership as well as subsets, although α ⊆ β ∨ β ⊆ α ` α ∈ β ∨ β ∈α ∨ α = β is unproved.

7. Global Choice

In 1977, Routley produced an argument for the axiom of global choice fromfull comprehension [Rou77]. Since then the claim that choice is a theorem of naivetheory has become part of paraconsistent folklore. However, on close examination,Routley’s proof relies on a faulty definition of function. On the one hand, he doesshow the existence of a global choice, but on the other hand, the choice is not func-tional. A second formulation, in [PRN89](374), accounts better for functionality,but nevertheless falls afoul, at least, of Curry’s paradox. A significant claim ofnaive set theory in paraconsistent logic, then, has been erroneous.

Routley’s details were awry, but his forecast was not. Answering a fundamentalquestion about the set concept, we here surpass classical theory and derive a globalchoice theorem without need of any further assumptions. In fact what is proved is

7. GLOBAL CHOICE 149

a weak version of Cantor’s wellordering principle, which he took to be a Denkgesetz(law of thought), “basic and consequential and especially remarkable because ofits general validity.” It was in service of proving this principle, essential to thetheory of transfinite cardinals, that Zermelo formulated his choice principle in 1904.The wellordering here is produced by injecting V into a particular subset of On,thereby giving a wellorder for every subset of V , that is, every set. The wellorderingthen provides the minimal resources for proving Hausdorff’s maximal principle; thislatter a structure for Zorn’s lemma; and only then does a global choice theoremobtain.

In the next section on cardinals, the choice principle is used to strengthenwellorderings.

Definition 5.49. A function f : a −→ b is:injective, or one-one, iff (∀x)(∀y)(x 6= y 7→ f(x) 6= f(y));surjective, or onto, iff (∀y)[y ∈ b 7→ (∃x)(x ∈ a ∧ f(y) = x)];bijective, or a one-to-one correspondence, iff f is injective and surjective.a ≤ b iff there is an injection from a to b.a < b iff a ≤ b and (∀f)[(∃y)(y ∈ b ∧ ¬(∃x)(x ∈ a ∧ f(x) = y)].

Theorem 5.50. The universe can be wellordered.

Proof. An injection f : V −→ On is required. Consider

〈x, y〉 ∈ f ↔ x ∈ V ∧ y ∈ On ∧ y = On.

That is, for each x ∈ V , f(x) = On. This is clearly a function. Now,

α ∈ On : α = On ⊆ On,

showing that the range of f is a segment of the ordinals and therefore wellordered.Intuitively, the Burali-Forti paradox indicates that the members of the range of fare discrete (corollary 5.39), of the form

... ∈ On ∈ On ∈ ...,

so On may be injected into by arbitrarily large sets, inducing a wellorder onthem. Formally, because On 6= On, (∀x)(∀y)(x 6= y 7→ On 6= On), so (∀x)(∀y)(x 6=y 7→ f(x) 6= f(y)). Therefore f is an injection. Thus

xf(x) : f(x) ∈ On

is a wellorder on V .

The proof is clearly not constructive; given the ordering aOn, bOn, cOn, ... onV with each On distinct, it is not said how to determine a first member. This isexactly the case with Zermelo’s choice principle, which is a pure existence claim.

150 5. ELEMENTS

A proof of a Cantorian “law of thought” will inevitably be by demonstration ofa bare existence of an ordering; and so here, it has been established that for anywellordered set there is a least member, and On is wellordered. The differencebetween Zermelo’s proof and our own is that no extra assumptions are needed toproduce the existence claim. It comes directly from the set concept.

Since subsets of a wellordered set are wellordered,

Corollary 5.51 (Zermelo 1904). Every set can be wellordered.

Definition 5.52. A ⊆-chain in a is a c ⊆ a such that, for x, y ∈ c, eitherx ⊆ y or y ⊆ x. An upper bound x ∈ a is such that for all y ∈ a, y ⊆ x. Amaximal element x ∈ a is such that for any y ∈ a, if x ⊆ y then x = y. A maximalchain b ⊆ a is such that if any c is both a chain in a and b ⊆ c, then b = c.

Below, a ⊆-chain is called just a chain.

Proposition 5.53 (Hausdorff’s Maximal Principle). Every set has a maximalchain.

Proof. By the wellordering theorem, let xξ : ξ ∈ α be a wellordering of a.From this ordering is built a maximal chain c ⊆ a. Define f by transfinite recursion:

f(xζ) =

xζ if xζ ∪

⋃ξ∈ζ f(xξ) is a chain;

∅ otherwise.

For relevance purposes, x here is xa = y ∈ a : y = x, so that x ⊆ a whenx ∈ a. Then it is to be shown that

⋃x∈a f(x) = c is a maximal chain in a. Note

that, by the law of excluded middle, c is not empty. To show c is a chain, considerany u, v ∈ c. Either u precedes v or v precedes u in the wellorder on a, or elseu = v. So either u = ∅ or uζ∪ c is a chain, and either v = ∅ or vζ∪ c is a chain.Since (∀x)(∅ ⊆ x), either u = v, or u ⊆ v, or v ⊆ u. Therefore c is a chain. To showc is maximal, let b be a chain in a such that c ⊆ b. Since b is a chain, for u ∈ c andv ∈ b we have either u ⊆ v ∨ v ⊆ u. Then v ∪ c is a chain, so v ∈

⋃x∈a f(x) by

definition of f . Then v ∈ c and b ⊆ c, giving b = c as sought.

Lemma 5.54 (Zorn). If every chain of some non-empty a has an upper bound,then a has a maximal element.

Proof. From Hausdorff’s principle, let c ⊆ a be a maximal chain in a, and exhypothesi let x be the upper bound of c. Suppose y ∈ a ∧ x ⊆ y. Then c ∪ y is achain in a with upper bound y. But c ⊆ c ∪ y; so c = c ∪ c by the maximalityof c. So, both being upper bounds of c on assumption, x = y. This proves that xis maximal in a.

8. CARDINALS 151

Deviating in notation, in the next proof σ and η denote not ordinals but func-tions, because they are rather special functions. A function σ is called a choicefunction on a iff σ(x) ∈ x when x ∈ a and (∃z)z ∈ x.

Theorem 5.55 (Choice). There is a choice function on every non-empty set.

Proof. Let c be the set of all choice functions on subsets of a. Let u ⊆ c

be all the choice functions totally ordered by inclusion: For any f, g ∈ u, eitherf ⊆ g ∨ g ⊆ f . Let

⋃u = η. For each 〈x, y〉 ∈ η, η(x) = f(x) for some f ∈ u, so η

is a choice function on subsets of a, showing η ∈ c. For f ∈ u, by definition f ⊆ η,showing η is an upper bound of u. Then by Zorn’s lemma c must have a maximalelement, σ, whence from f ∈ c and σ ⊆ f follows f = σ. We only need now thatthe domain of σ is a. Let x ∈ a be non-empty, and consider σ′ = σ ∪ 〈x, y〉 withy ∈ x. Then σ′ ∈ c and σ ⊆ σ′, so by maximality σ′ = σ. Thus σ is a choice on a

as desired.

Corollary 5.56 (Global Choice). There is a choice function on V . A fortiori,for every non-empty a there is a choice function σ such that σ(a) ∈ a.

We can complete the familiar circle and re-prove Zermelo’s theorem:

Proposition 5.57. (∀x)(∃α)(x ∼ α).

Proof. By recursion:

f(yζ) =

σ(x− yξ : ξ ∈ ζ) if there is one;

∅ otherwise,

again leads to a wellorder on x, a least ordinal θ such that f(yθ) = ∅.

Aside from settling long-standing questions, a global choice theorem provides asolution to a pressing problem of uniqueness for mathematics in intensional logicslike DLQ. The wellordering on V , and indeed on On, so far developed in theorem5.50 does not guarantee that there is only one least member of a given wellorderedset—the least member. Instead, we have an equivalence class of the minimal mem-bers of a. A functional choice σ(x) from the equivalence set x, though, can act asthe desired unique member, as the next section explores.

8. Cardinals

Here we define cardinal numbers and study their most basic properties, workingfrom the fundamental

Definition 5.58. a is equipollent to b, a ∼ b, iff there is a bijection betweena and b.

152 5. ELEMENTS

Chiefly we must solve the cardinal assignment problem posed, in essence, byCantor [Can95](482), of defining an operation | · | which satisfies (at least):

x ∼ |x|,

|x| = |y| when x ∼ y.

As is standard, we will use von Neumann’s solution, taking specific ordinals forcardinals, and requiring the axiom of choice.

The end goal is to prove Cantor’s theorem, that there are orders of transfinitepowers. This will show that the transfinite still has rich structure in a paraconsistentsetting, a point that has been in some doubt [Pri06b](253). To start, here is aclassic theorem of naive set theory, an adaptation of theorem 66 from [Ded88]proving the existence of an infinite set. Any set a is dedekind infinite iff (∃b)(b ⊂a ∧ b ∼ a). Then

Proposition 5.59 (Dedekind 1888). Some sets are dedekind infinite.

Proof. Consider f = 〈x, x〉 : x ∈ V . This is an injection of V into aproper part of itself. Therefore V is dedekind infinite.

Proposition 5.60. Equipollence is an equivalence relation.

a ∼ a,

a ∼ b→ b ∼ a,

a ∼ b ∧ b ∼ c→ a ∼ c.

The following forms of transitivity hold, too.

a ∼ b→ (b ∼ c→ a ∼ c),

a ∼ b→ (c ∼ a→ c ∼ b).

Proof. For reflexivity, the function is identity. For symmetry, if f is a bijectionfrom a to b, then f−1 is an bijection from b to a. For (conjunctive) transitivity, iff is a bijection from a to b and g is a bijection from b to c then f g is a bijectionfrom a to c. As in proposition 5.5, too, the implicational phrasings of transitivityare from hypothetical syllogism.4

For any a, therefore, we have the ∼-equivalence class x : x ∼ a. This isthe Frege cardinal of a. Frege cardinals are a plausible philosophical analysis ofthe concept of size, especially given the naive tie-up between set membership andpredication: The concept of a-sized is just the extension of a-sized objects. The

4If there is any doubt about this form of transitivity, note that it and prop.5.61 below which it

supports are disposable, since the proof of the more important prop.5.62 requires only conjoinedtransitivity.

8. CARDINALS 153

notion can be tightened by intersecting Frege cardinals with the ordinals. Let C bethe set of all ∼-least ordinals:

x ∈ C ↔ x ∈ On ∧ ¬(∃z)(z ∈ x ∧ z ∼ x).

This immediately delivers a weak form of an important counting principle:

Proposition 5.61. For every α ∈ C there is a set Cα = x : x ∈ C ∧ x ∼ α,such that, for β ∈ C,

Cα = Cβ a` α ∼ β.

Proof. Left to right, the premise is x ∈ C ∧ x ∼ α ↔ x ∈ C ∧ x ∼ β. Thenα ∼ α implies that α ∼ β. Right to left, let f : α −→ β be a bijection. Then ifg : x −→ α is a bijection, f g : x −→ β is a bijection: hence x ∼ α→ x ∼ β. Nowsuppose x ∈ Cα. Then x ∈ C ∧ x ∼ α; then x ∈ C ∧ x ∼ β, and therefore x ∈ Cβ .Ergo Cα ⊆ Cβ . Mutatis mutandis to show Cβ ⊆ Cα.

Is Cα is the cardinality of α? This assignment does not meet the requirementthat α ∼ Cα, the useful criteria that x ∼ |x|. For this we need to be even morespecific about the object called a cardinal.

A set x is a soft cardinal iff there is some ∼-equivalent part of C having x asa member:

C = x : (∃α)(x ∈ Cα).

By the wellordering theorem, for every set a there is an ordinal α such thata ∼ α. So for every a there is an x ∈ C such that a ∼ x and x is least of itssize; write C(a) for such an x. In a sense, this is a sufficiently robust notion ofcardinality: It solves the cardinal assignment problem, yielding easily that a ∼ b

when C(a) = C(b).On the other hand, the paraconsistent logic used here is not like classical logic,

and the members of C are not demonstrably unique. A fixed point property doesnot obtain: If x ∈ C, we cannot show (although we don’t have a disproof) thatx = C(x). Nor does C(a) = C(b) follow from a ∼ b, which is awkward.

To be even more exacting, then, we can assigning a specific ordinal to be thecardinal of a set. Let σ be a choice function. The set of the cardinals are:

C = x : (∃α)[x = σ(Cα)].

Obviously C ⊆ C, and in many cases there is no cause to distinguish the two. The(proper) cardinals, however, do deliver a function: The cardinality of a is a function| · | : V −→ C,

〈x, κ〉 ∈ | · | ↔ κ ∈ C ∧ x ∼ κ.

To show that cardinality is functional, it suffices to check that |x| = κ ↔ κ ∈C∧x ∼ κ, which follows from the functionality of the choice σ. It is similarly basic

154 5. ELEMENTS

to check that cardinality is a fixed point: κ ∈ C ∧ κ ∼ κ↔ |κ| = κ. Theorem 5.50means that for every a, there is a κ ∈ C such that |a| = κ.

Proposition 5.62. |a| = |b| a` a ∼ b.

Proof. Suppose |a| = |b|. Let |a| = κ = |b|; then by definition a ∼ κ ∼ b.Then by transitivity, a ∼ b. In the other direction, suppose that a ∼ b. If |a| = κ

then a ∼ κ, and by transitivity, b ∼ κ too, showing that |b| = κ. Then |a| = κ = |b|,implying |a| = |b| as required.

Proposition 5.63 (Hartogs). |a| ≤ |b| ∨ |b| ≤ |a|.

Proof. Let |a| = κ, |b| = λ. Since κ, λ are ordinals, trichotomy under ∈ holds.If κ ∈ λ, then κ ⊆ λ, and then f(x) = x is an injection from κ to λ. Mutatismutandis for λ ∈ κ and κ = λ.

Proposition 5.64. a ⊆ c ⊆ b, |a| = |c| ` |a| = |c| = |b|.

Proof. Usually the axiom of choice makes this very easy; but even withoutchoice (or any attendant easiness), a proof by standard back-and-forth constructiongoes through without paraconsistent difficulties. Define by recursion c0 = c, cn+1 =f ′′c, and b0 = b, bn+1 = f ′′b. Let dn = cn ∩ bn for every n. Then define g from c tob as

g(x) =

f(x) if x ∈⋃n∈ωn=0 dn;

x otherwise,

and we have that g is a bijection, showing |c| = |b|, and then by transitivity, theresult.

Proposition 5.65 (Cantor-Schroder-Bernstein). |a| ≤ |b| ∧ |b| ≤ |a| ` |a| = |b|

Proof. Let f : a −→ b and g : b −→ a be injections. By definition, g′′(f ′′a) ⊆g′′b ⊆ a, and g−1 : g′′b −→ b is a bijection. Now, g f : a −→ g′′f ′′a is an injection.By prop.5.64, therefore, there is a bijection h : a −→ g′′b. And then g−1 h is abijection between a and b.

The most spectacular feature of set theory is the existence of distinct transfinitecardinals, as Cantor discovered in 1873. His argument is recast at the end ofthis chapter. Interestingly, one can prove the substance of the theorem with avery direct reflection argument. The following general lemma, in fact, suggeststhat inconsistency is as closely tied to infinity as Aristotle originally suspected.Distinctive of some paraconsistent mathematics, here is how to count inconsistentsets.

Lemma 5.66. b ∈ a ∧ b 6∈ a ` (∀z)|z| < |a|.

8. CARDINALS 155

Proof. Let b ∈ a ∧ b 6∈ a. The idea is that b is in a but outside the range ofany mapping into a, because b is not in a. Let f : z −→ a be a function,

f ⊆ 〈u, v〉 : u ∈ z ∧ v ∈ a.

For each u ∈ z, we can at least have 〈u, b〉 ∈ f be a functional injection, as in prop.5.50; this shows |z| ≤ |a|. Because b 6∈ a, also ¬(u ∈ z ∧ b ∈ a), for any u. Then

(∀u)〈u, b〉 6∈ f.

Thereforeb ∈ a ∧ ¬(∃u)(u ∈ z ∧ 〈u, b〉 ∈ f).

Then by generalization, (∀f)(∃y)(y ∈ a ∧ ¬(∃x)(x ∈ z ∧ f(z) = y)). Therefore a isstrictly larger than any set.

The first application is to prove that the ordinals are a cardinal, which makessense, given that On is a limit ordinal, to say the least. The last ordinal is thegreatest size:

Proposition 5.67. On ∈ C.

Proof. There are two main conditions to satisfy: being an ordinal, and beingthe least ordinal of its size, ¬(∃α)(α ∈ On ∧ α ∼ On). Well, On ∈ On. SinceOn 6∈ On, also (∀u)〈u,On〉 6∈ f, for any u, as in lemma 5.66. If (∀z)|z| < On, thenthere is no surjection from z to On, and so no bijection, showing (∀z)(x 6∼ On).Therefore (∀x)(x 6∈ On∨x 6∼ On). It is further desirable that On be a cardinal, therepresentative of its size-class. Now, it seems plausible that if any least ordinal of itssize be unique, then On would be. Is On = σ(COn)? Well, unlike any other ordinal,we know that On is maximal: that On ∈ α 7→ On = α, for every α, including anyα ∈ COn. This is a unique property for On, so here we can make the choice functionspecific and stipulate that σ′(COn) = x ↔ (∀α)(α ∈ COn 7→ (x ∈ α 7→ x = α)).This makes On a cardinal.

Corollary 5.68. For every cardinal, there is a strictly greater cardinal.

Proof. Because (∀x)|x| ≤ On, by generalization (∃y)(∀x)|x| ≤ |y|. Then(∀x)(∃y)|x| ≤ |y| as required.

A statement and proof of the more well-known form of Cantor’s theorem isbelow, theorem 5.73.

We have shown that inconsistent sets are absolutely infinite—vindicating thelimit on size intuition, that all inconsistent sets are overlarge:

Proposition 5.69. If (∀z)|z| < |a|, then |a| = On.

156 5. ELEMENTS

Proof. Let (∀z)|z| < |a|. Then (∀z)(|z| ≤ |a|). Meanwhile, |a| ≤ On. SinceOn is a cardinal, the Cantor-Schroder-Bernstein antisymmetry prop. 5.65 applies.

Definition 5.70. Let κ ∈ C. Cardinal successor is a function | · |′ : C −→ C

satisfyingκ′ = λ↔ κ < λ ∧ ¬(∃y)(y ∈ λ ∧ κ < y).

By recursion, the alephs are

ℵ0 = ω

ℵα+1 = ℵ′αℵλ = supℵκ : κ < λ

for λ a limit ordinal.

Cardinal arithmetic works as follows. If a and b are disjoint sets, ¬(∃x)(x ∈a ∩ b), then cardinal sum and product are

|a|+ |b| = |a ∪ b|

|a| · |b| = |a× b|

In general, let κξ : ξ ∈ α be a set of cardinals; then sum and product are,respectively, ∑

ξ∈α

κξ = |supκξ : ξ ∈ α × ξ|,

∏ξ∈α

κξ = |∏ξ∈α

κξ|.

To define exponentation, let ab = f : f is a function from b to a. Then

|a||b| = |ab|.

Lemma 5.71. ℵOn = On.

Proof. Since On is a limit (prop. 5.42),

ℵOn = supℵα : α ∈ On.

As On ∈ On, it follows by the definition of supremum that ℵOn ∈ supℵα :α ∈ On. So by substitution of identicals, ℵOn ∈ ℵOn. These are ordinals, soℵOn 6∈ ℵOn, too. Then by prop. 5.66, 〈On,ℵOn〉 6∈ f for all f , showing On ≤ ℵOn.Since ℵOn ∈ On, also ℵOn ≤ On, and the result follows.

Theorem 5.72 (Cantor 1899). Every cardinal is an aleph.

8. CARDINALS 157

Proof. This is an adaptation of an argument from Cantor’s 1899 letter toDedekind [Can67] that makes use of inconsistent sets and so shows some affinityof technique with our own. Let |M | be a power not an aleph. By wellorder, then,the whole of the alephs are injectable into |M |; in particular, On ≤ |M |. But also,like everything else, |M | ≤ On, showing that |M | = On = ℵOn, an aleph afterall.

Theorem 5.73 (Cantor 1892). |a| < |P (a)|.

Proof. The function taking each z ∈ a to x ∈ a : x = z establishes aninjection, |a| ≤ |P (a)|. Cantor’s original argument for strictness employs propertiesof implication outside DLQ. Nevertheless, ours shares with Cantor the constructionof a diagonal set that pushes its powerset beyond the reach of any one-one mapwith a. Consider

r = x ∈ a : x 6∈ r.

Some properties of r are forthcoming. Since (x ∈ r → x 6∈ r) → x 6∈ r, we have(∀x)x 6∈ r. Further, by definition of r, (∀x)(x ∈ a ∧ x 6∈ r → x ∈ r). Ergo everymember of a both is and is not in r.

The argument considers an arbitrary f from P (a) to a and finds that it is nota surjection, establishing that all such functions are not onto and hence that thereis no bijection, proving strictness. Suppose that there is a one-one correspondencef between a and P (a). Then for any y ⊆ a there is an z ∈ a such that f(z) = y.Either (∃x)x ∈ a ∨ ¬(∃x)x ∈ a. The meta-rule allows us to eliminate cases overconsequence. In the second case, for every y ⊆ a there is no z ∈ a such thatf(z) = y, so there is no bijection and the theorem holds. So now we focus only onnon-empty a. Then for some z ∈ a, f(z) = r, which means by extensionality that

(∀x)(x ∈ f(z)↔ x ∈ r).

We show this equivalence fails and then so too does the map f . For every x, eitherx ∈ f(z) ∨ x 6∈ f(z). Let z ∈ a. If the former, then since always z 6∈ r, we have

z ∈ f(z) ∧ z 6∈ r,

so by counterexample, (∃x)¬(x ∈ f(z) → x ∈ r), so f(z) 6= r after all. If insteadz 6∈ f(z), since z ∈ a we know that z ∈ r,

z ∈ r ∧ z 6∈ f(z)

which by counterexample again gives ¬(∀x)(x ∈ r → x ∈ f(z)), and again showsf(z) 6= r. Thus f(z) 6= r for z ∈ a, proving there is no bijective f , quod eratdemonstrandum.

This leads directly to Cantor’s paradox:

158 5. ELEMENTS

Corollary 5.74. |V | < |P (V )|, but also |V | = |P (V )|.

Proof. The first is an instance of Cantor’s theorem. The second follows fromproposition 5.65.

The generalized continuum hypothesis GCH conjectures for all cardinals κ, λthat ¬(κ < λ < |P (κ)|).

Theorem 5.75. The GCH fails at On, and an instance of GCH holds at On.

Proof. For a failure of the GCH: The cardinal On provides a counterexample.From Cantor’s theorem, On < |P (On)|. But On < On < |P (On)|. For an instanceof GCH, let λ be a cardinal On < λ < |P (On)|. Being a cardinal, ¬(On < λ). Thus¬(On < λ < |P (On)|). In fact, because 2On = On (by the maximality of On andCantor’s theorem), and On = ℵOn, and On = On+ 1, then by a few substitutionsand existential generalization follows

(∃α)(2ℵα = ℵα+1).

CHAPTER 6

Reflection and Large Cardinals

Is it you,or is that a fault-lineon the horizon

a second sky?

—Matthew Francis, Ocean

This chapter proves the existence of some large cardinals, based on a reflectiontheorem. The ordinals were developed combinatorially in chapter 5. Now, usingwhat in standard terminology would be called proper class arguments, they areenriched to produce a clean description of some features unique to Cantor’s absoluteand to diagonal inclosures in general. Connections are also drawn with more recentmainstream mathematics, particularly Kunen’s result on Reinhardt cardinals.

1. Introduction

Definitions by recursion on the ordinals are well-understood—from the bottomup. We define a base case and a method for succession. What about from the topdown? As Barwise and Moss point out [BM96](216), expressions like

f(α) =⋂β∈α

f(β)

are not well-understood, because to get the recursion going would require a greatest,as opposed to a least, fixed point in the ordinals. The issue arises in non-well-founded set theory, where least members are not in general guaranteed; and someinteresting work is done there on corecursion, defining downward from the greatestfixed points of operators. Here we have a straightforward and natural startingpoint: The fixpoint On itself.

As discussed in chapter 2, the ordinals may be approached from below andfrom above. By building upward from below, the existence of further and furtherordinals, and their limiting points, the cardinals, has been a matter of postula-tion. From above, especially from a well-defined starting point like On, the processof approaching these objects is significantly streamlined. Working out the details

159

160 6. REFLECTION AND LARGE CARDINALS

here will give some precision to the analysis that “actually conceived, the abso-lute bounds the entire ordinal sequence, whereas potentially conceived, the ordinalsequence is absolutely unbounded.”[Jan95](383)

Since at least 1897 it has been clear that there is something special aboutthe totality of the ordinals. This is evinced by the conflicted treatment that theordinals receive. On the one hand, it is a truism that the ordinals are the centralorganizational index for set theory, that ordinals as the bearers of wellorder arethe essence of the discipline. “When we look at the general pattern of questionsdiscussed in set theory,” writes Levy, “we see that one main theme is the studyof the structure of well ordered classes, or, what amounts to the same thing, thestudy of the structure of the class of the ordinals.”[Lev79](292) To this end mosttwentieth century research occurred at the limiting case of this attitude, spent tryingto reduce the entire set theoretic universe to no more than the wellfounded universe.On the other hand, most research in the twentieth century was preoccupied withblocking and denying the resilient inconsistency of On, because it was believedthat to admit On ∈ On would be trivializing, proving everything. The totality ofordinals is important and inconsistent.

The driving idea in the present chapter is to find a point of moderation betweenthese extreme tendencies of “angst and awe”, the tendencies of both privileging andscorning the class of ordinals—to learn more than classical theory by following theordinals all the way up, but by the conviction that in doing so we infer less thanex falso quodlibet suggests. Naive set theory can prove quite a lot without provingeverything.

This is new mathematics, very much related to old. The ideas are connectedwith some more recent mainstream research in higher set theory. To show this, Iwill quote some classical theorems which, though they may not be part of naivetheory, make for important points of comparison and corroboration with the pictureof sets being developed here—namely, inconsistent, interesting activity at the endof the ordinal line.

1.1. Axioms of Infinity. The deep question about transfinite ordinals, andso of this chapter, is their extent, a theme Levy identifies as a main question of settheory [Lev79](289). The question therefore finds focus at the limiting regions—the place at which the question can be answered. And insofar as ordinals are meantto extend as far as possible, the question is broader, about the totality of all setsV . Reflection principles are for isolating particularly distant ordinals, called largecardinals, in order to shed light on the question: How far does the universe go on?A natural, perhaps naive, strategy for answering this question is simply to try andfind the extremity; this will supply, by direct inspection, a concrete answer.

1. INTRODUCTION 161

Paul Cohen [Coh66] showed that ZF is incomplete. For want of more theo-rems, classical set theorists must posit more axioms: in Godel’s idiom, axioms ofinfinity. In [G64](264), Godel explains that

the axioms of set theory by no means form a system closed initself, but, quite on the contrary, the very concept of set onwhich they are based suggests their extension by new axioms ....These axioms can be formulated also as propositions assertingthe existence of very great cardinal numbers (i.e. of sets havingthese cardinal numbers). The simplest of these strong ‘axiomsof infinity’ asserts the existence of inaccessible numbers (in theweaker or stronger sense) > ℵ0).

Similarly, “the belief in the existence of inaccessible cardinals,” Tarski [Tar62]writes,

seems to be a natural consequence of basic intuitions underly-ing the ‘naıve’ set theory and referring to what can be called‘Cantor’s absolute’. ... What is regarded by many as one ofthe main aims of research in the foundations of set theory [is]the axiomatization of increasingly large segments of ‘Cantor’sabsolute’.

The axioms assert that there are more and more cardinals, each prescribing itsown transcendence over all the cardinals below [Kan94](xi). These successivelystronger assumptions are linearly ordered by consistency strength, and have beenshown by Kunen to come to an abrupt, contradictory end at Reinhardt cardinals[Kun71]. The aim of these posits has been to settle outstanding questions of settheory, e.g. the generalized continuum hypothesis. To date, the posits have notbeen successful in this regard.

For classical set theory, large cardinal axioms are an attempt to regain theconceptual closure that is lost in the iterative hierarchy. They also encode the factthat any closure can be transcended. So large cardinals can be explained as aninclosure phenomenon; the part of the diagonal function is played by the variousdefinitions of larger and larger cardinals. Classically, the result at the ultimateclosure level is a brute contradiction. In naive set theory this conflict is put todirect use in fueling new proofs.

Because the last century has been dominated by the cumulative concept of set,the wellfounded sets, V0 = ∅, Vα+1 = P (Vα), Vλ =

⋃κ<λ Vκ with λ a limit, has been

taken as the answer, as the universe simpliciter.

In such arguments ‘universe’ is sometimes replaced by ‘class ofall ordinals’. Such replacement is quite natural given the tie


up between ordinals and the universe through the cumulativehierarchy. Moreover, this also fits with Cantor’s view that theordinals are natural ‘expression’ or ‘representation’ of the Abso-lute. [Hal84](116)

Now, we can indeed stratify V and index the levels with ordinals, so talk of Vα andeven VOn is entirely definite. And in adapting the large cardinal work of classicaltheory, it will make most sense to deal in ordinals and the fragment VOn, ratherthan, as is claimed in e.g. [Kan94], the universe V itself. But the iterative setsalone are, we know, not the end of the story. The full universe V is properly morevast even than the closure level VOn; this is clear since on the naive view the axiomof foundation is false, since one can prove the existence of e.g. a such that a = a.So the wellfounded idea that V = VOn certainly fails; a fortiori, if this identity wereprovable, it would probably be explosive, since V 6∈ V is.

On the other hand, there are a great many ordinals, and it is hard to see howVOn = xα : α ∈ On could have left anything out. Indeed,

Proposition 6.1. |V | = On.

Proof. Like everything else, On ⊆ V ; so On injects into V , showing |On| ≤|V |; and |V | ∈ On, so |V | ⊆ On, and so |V | ≤ On. Thus they are identical, byproposition 5.65.

There is then some temptation to elide between the overlarge sets VOn and On.But a great deal of care is needed here, since V is fragile. Reflection as we describedit in chapter 2, where a set M reflects Φ if some proper subset of M is Φ, is toocrude. Suppose V reflects in this way; then if V = V , there is some proper subsetM = V . But from V = M ⊂ V , it follows by the definition of ⊂ that some x is notin V , whence triviality. A very precise formulation of reflection is needed to retaincogency while also providing the information we seek about extent. In the sectionof the chapter on Kunen’s inconsistency theorem, we will be in position to askwhether he has identified the end of the classical universe, or the end simpliciter.

Whatever the details turn out to be, our derived ‘axioms’ of infinity ought toanswer the question on how far the ordinals go on in most obvious way: To theend.

2. Reflection Theorems

An explicit reflection principle is Russell and Whitehead’s axiom of reducibilityin their Principia. This was a device introduced to augment expressive power,compensating for the heavy losses incurred from the vicious circle principle andtheory of types. The axiom asserts that any property occurring at a higher level of

2. REFLECTION THEOREMS 163

the type hierarchy already occurs at the first level [RW13](55). In the end, it wasreducibility that most overtly exposed the Principia’s failing of logicism, becausesuch an axiom is not self-evident, as Russell knew.

Reflection principles have become mainstays of modern set theory and are usedas heuristic justifications for cardinal postulates beyond ZFC: Reflection is themeans by which the truncated and intuitively shorn classical theory hopes to re-connect with more expansive truths. We have already met the intuitive form ofreflection in chapter two, where I suggested its connection to Cantor’s absolute.Kanamori claims that, as reflection principles manifest in large cardinals, they arethe “rightful heirs” of early set theory, the “trustees of an older tradition.” Still,Russell’s trouble remains the trouble: Reflection is a subterfuge, a nostalgia for theknowledge abandoned after the trauma of the antinomies; and indeed, as we will see,all agree that a reflection principle in its most basic form is multiply inconsistent.

The classical reflection theorem proved by Levy in 1960 deals in the model-theoretic satisfaction relation |=, and reads

(∀α)(∃β > α)[V |= Φ↔ Vβ |= Φ].

To see how Levy’s principle works, here is a proof sketch. Beginning with thelowest segment Vα that models Φ, define a function f on α and use the axioms ofreplacement and infinity to prove the existence of a higher set Vf(α) that modelsΦ. Upward reflection is classically equivalent to the axioms of replacement andinfinity, both of which are theorems here. For full detail see [Dra74]. The theorembears some resemblance to the δ-ε arguments in calculus, used to eliminate talk ofinfinitesimals; upward reflection can go ‘arbitrarily’ high. As Wierstrauss developedthat technique to sterilize the conceptual foundations of his subject, so too herethe idea is to approximate V without straying into paradoxes. Tait has a gooddiscussion of second-order reflection in [DO98]. Reflection is a device to speakabout totalities indirectly. Naive set theory can stare into these totalities directly;and so we can use reflection as a powerful device to produce not hypotheses buttheorems.

The next appearance of reflection will provide us with the most useful formula-tion, a dialethic reflection theorem suggested by a proof sketch in [Dra74] of Godel’ssecond incompleteness theorem. This establishes that, were ZF able to prove theexistence of its own model (equivalently, by Godel’s completeness theorem, to prov-ing itself consistent), then a contradiction follows. Classically, everything follows.

If ZF proves that (∃M)M |= ZF ,then ZF proves that (∃M ′)[M ′ < M ∧M ′ |= ZF ].

Suppose ZF proves that (∃M)M |= ZF . Then M satisfies this fact, so ∃M ′ ∈ Msuch that ZF provesM ′ is a model of ZF . These wellfounded models are indexed by


the ordinals, though, and given any stretch of ordinals, there is always a least. So letM be the least provable model of ZF . Then there would be a lesser, some M ′ ∈Mthat witnesses ZF proving the existence of its own models—which contradicts theminimality of M . Classical reasoning demands that the premise of the argumentbe rejected, that ZF cannot prove the existence of its own models.

Since the contradiction arrived at is not necessarily explosive, though—in fact,it is quite likely to be a property of naive model theory—Godel’s modus tollensmay be our modus ponens. Godel’s theorem shows that closure leads to infinitedescent. Since naive comprehension principles lead to closure, here we investigatethe possibility that the subsequent descent is rich and illuminating. The limitationimposed by Godel’s second theorem reemerges as a powerful mechanism. For nowwe work just with the ordinals, leaving VOn for later.

Let |On| = Ω. Since On = |On|, this is just another name for the set ofall ordinals, used to mark only a difference in emphasis. We use the conditionalΦ 7→ Ψ, defined Φ ∧ t→ Ψ, to state

Theorem 6.2 (Reflection). (∀X)[Ω ∈ X 7→ (∃Θ)(Θ < Ω ∧Θ ∈ X)].

The notion of reflection all along has been meant to capture absoluteness;all the hard work of founding the ordinals in a paraconsistent setting done, ourmathematical characterization is now, appropriately, delivering reflection almostfor free.

Proof. Since Ω < Ω (lemma 5.66), then Ω ∈ X ∧ t → Ω < Ω ∧ Ω ∈ X. AndΩ < Ω∧Ω ∈ X → (∃Θ)(Θ < Ω∧Θ ∈ X). Transitivity of the conditional completesthe proof.

In absence of a worked out model theory, we are taking advantage of the tie upbetween ∈ and predication to state the theorem; X is playing a second-order role.Since naive comprehension makes membership a synonym for bearing a property,the reflection theorem for Ω can be abbreviated without loss: If X(Ω), then, sinceΩ < Ω, there is some ordinal Θ < Ω such that X(Θ). (A capital letter suggeststhat Θ is within conceptual striking range of Ω.)

Downward reflecting could just as easily be directed upward; again because Ω <

Ω, there is an Ω > Ω such that if X(Ω) then X(Θ), too. Downward reflection—thatany property enjoyed by Ω is enjoyed by some lesser set—is an old idea that receivedits clearest articulation from Reinhardt [Rei74] who calls it the naive reflectionprinciple, because in this simple form simple contradictions can be derived. (Takeany property satisfiable by nothing but Ω.) Since inconsistency occurs naturallyin Ω, though, this is less than reason to turn from the basic reflection principle.Rather, this is incentive.

3. ULTRAFILTERS 165

Reflection has attracted much attention in the last decades, for both its intuitivejustification and expansive proving power. Classically, large cardinals mark blindspots: If Vα |= ZF then α is inaccessible in the sense of Godel’s incompletenesstheorem [Lev60]. Symmetry would suggest, and our theorems will confirm, thatthese blind spots, these gaps, are really gluts under naive set theory. The time hasnow come to build some high-impact machines, and use them to find some overlargesets.

3. Ultrafilters

Ultrafilters, ultraproducts, and ultrapowers are basic tools of higher set theory,and model theory, too; [BS71] is an elegant introduction. For the first part ofthis section, we will give an appropriate definition of these objects, and make someadapted applications of Zorn’s lemma to prove their fundamental properties.

The theory of ultrafilters was founded at the height of classical logic. Funda-mental theorems are proven by use of the disjunctive syllogism. More, the definitionof filter as usually given (with a material conditional) fails relevance considerations,as the following example shows. A filter F is is closed under intersection: x∩y ∈ Ffor x, y ∈ F . How is this to be formalized? While x ∩ y ∈ V for any x, y, theentailment x, y ∈ V → x ∩ y ∈ V is irrelevant. But V − ∅ is a very naturalcandidate for being a filter. To work around these limitations, as with the ordinalnumbers, much hinges on our choice of definitions; using the t connective avoidsmuch of the difficulty.

Conversely, naive set theory can handle important portions of these conceptsvery naturally, because ultrafilters in general and ultraproducts in particular pickout large sets.

Definition 6.3. Let M be a set, P (M) its subsets. F ⊆ P (M) is a filter onM , Flt(F,M), iff

∅ 6∈ F,

x ∈ F ∧ y ∈ F 7→ x ∩ y ∈ F,

x ∈ F ∧ x ⊆ y ⊆M 7→ y ∈ F.

F is maximal in M just in case: F ⊆ G∧Flt(G,M) 7→ F = G. F is an ultrafilteron M iff F is maximal. F is a principal filter on M iff F = x ⊆ M : y ⊆ x forsome non-empty y ∈M .

A chain is a set totally connected by ⊆.

Proposition 6.4. Let C be a non-empty chain of filters over M . Then⋃C

is a filter over M .


Proof. Let ⋃C = x : (∃F )(Flt(F,M) ∧ F ∈ C ∧ x ∈ F ).

There are three conditions to satisfy. One: If ∅ ∈⋃C then (∃F )(∅ ∈ F ∧ F ∈ C).

But each such F ∈ C is a filter, whereby ∅ 6∈ F , so ∅ 6∈⋃C by contraposition. Two:

If x, y ∈⋃C then there are filters F,G ∈ C with x ∈ F and y ∈ G. Because C is

a chain, either F ⊆ G or G ⊆ F ; therefore either x, y ∈ F or x, y ∈ G. Thereforeeither x ∩ y ∈ F or x ∩ y ∈ G, so it follows that x ∩ y ∈

⋃C. And three: If x ∈ F ,

and x ⊆ z ⊆⋃C, then z ∈ F because F is a filter and therefore z ∈

⋃C.

Theorem 6.5 (Tarski). Every filter is contained in an ultrafilter.

Proof. Let F be a filter in M . Let

F = G : Flt(G,M) ∧ F ⊆ G.

Note that F ∈ F, so F is not empty. If C is a non-empty chain in F then⋃C is

a filter by the previous proposition. Also, F ⊆⋃C because F ∈ C. Therefore⋃

C ∈ F. And since G ∈ C implies G ⊆⋃C,

⋃C is an upper bound of C in F.

By Zorn’s lemma there is a maximal element U ∈ F. Because U is maximal, U isan ultrafilter containing F .

This is all the development needed for now. There remain mostly unansweredquestions about ultrafilters U in naive set theory. For example, whether they areexhaustive: For any x ⊆M , does it hold that either x ∈ U∨M−x ∈ U? Classicallythis is equivalent to U being maximal. Delicacies of definitions and the behaviorof → however, make the matter non-trivial here. Nevertheless, we can proceedwithout deciding.

4. Axioms of Infinity

“Mathematicians and other children often play the following game: We taketurns naming numbers, and see who can name the largest one.”[Bar77](396) SoKunen introduces the climb we are about to undertake. We will have an advantageover the classical strategy in this game, though, since we are starting from the topand descending, using our downward reflection theorem to establish features of thehigher infinite.

Generally, we study the properties of Ω and ascribe some of these properties tolower sets, proving the existence of various large cardinals. Comparing against theclassical postulation method,

We look for justification for these axioms from the point of viewof the cumulative type structure, where we want to say that thecollection of levels, which is indexed by the ordinals, is a very rich

4. AXIOMS OF INFINITY 167

structure with no conceivable end. ... It always seems a plausiblestep, in view of the reflection principle, to take a property of thewhole universe and postulate that it already holds at some levelVα. [Dra74](124),

working naive sets directly extends such “justification” so that the postulates be-come theorems. The arguments are streamlined reflection proofs, achieved just byexistential generalization; the proofs end by appealing to reflection, theorem 6.2,as a heuristic. Since higher varieties of cardinals are of the lower varieties, too, e.g.if κ inaccessible and κ < λ then λ is inaccessible, the whole of this part can besummed up: On is the largest cardinal, and exists; so every type of large cardinalexists.

The ordering of the higher infinite, at least at its initial segments, correspondsto the historical sequence in which it was discovered. First we consider some ideasof Hausdorff from 1906-8, and then of Mahlo from 1911-1913. For excellent in-troductory exposition of much of this material, see [Dra74]. More advanced is[Kan94].

Definition 6.6. Let α, β be ordinals and α < β. If α is unbounded in β, inthe sense that

(∀x)[x ∈ β 7→ (∃y)(y ∈ α ∧ x ≤ y)],

then α is cofinal with β, cf(α, β). The cofinality of β, cf(β), is the least suchcofinal α,

cf(β) = α↔ cf(α, β) ∧ ¬(∃y)(cf(y, β) ∧ y < α).

A cardinal β is singular when cf(β) < β. When cf(β) = β, β is a regularcardinal. When β < cf(β), β is defined as vanishing.

The finite numbers, for example, are all singular cardinals. If there is morethan one least α cofinal with β, use choice to obtain a unique cofinality. So we areassured that cf is a function.

Confinality is a fixed point,

cf(β) = cf(cf(β)),

since α is the least cofinal set iff α is the least least cofinal set. The cofinality of βis always dominated by β, cf(β) ≤ β.

All this is classical. A point of novelty, however, is to lift regular cardinals fromdefinition to theorem without any additional axioms.

Proposition 6.7. There are regular cardinals.

Proof. Let Θ = cf(Ω). Then for every ordinal x, some y ∈ Θ is such thatx ≤ y. Since Ω is an ordinal, (∃y)(y ∈ Θ ∧ Ω ≤ y). Since y is an ordinal, y ≤ Ω.


So y = Ω; so Ω ∈ Θ by substitution. But the only ordinal with Ω as a member isΩ itself, because ordinals α are transitive:

(α ∈ Ω→ α ⊆ Ω) ∧ (Ω ∈ α→ Ω ⊆ α)→ (α ∈ Ω ∧ Ω ∈ α→ α = Ω).

Therefore Θ = Ω. Ergo Ω is regular. By reflection, there is some κ below Ω suchthat

κ = cf(κ)

as desired.

Novelly, we have the additional and decidedly non-classical case:

Proposition 6.8. There are vanishing cardinals.

Proof. Since Ω = cf(Ω), if Ω < Ω, which it is, then by substitution of identi-cals Ω < cf(Ω). Reflecting down, some κ < cf(κ), as required.

Appropriately enough, though, these cardinals extinguish themselves:

Proposition 6.9. There are no vanishing cardinals.

Proof. By contraposing the definition of wellorder (def. 5.25), if a set ofordinals has no least member, then it has no members at all. If there were anyvanishing cardinals, there would be a least; call it ε. So ε < cf(ε), and for anyδ < ε, δ 6< cf(δ) by the minimality of ε. Except since

ε < cf(ε) ≤ ε

it follows that ε < ε. So ε is not the least vanishing cardinal. Then there is noleast, and so none at all.

Regular cardinals > ω are inaccessible because such a cardinal would be suffi-ciently large to model ZFC. If we can prove such cardinals exist, then it is verylikely that naive set theory can model ZFC. “In general,” Hausdorf observes,

whether there are regular initial numbers with limit indices isvery problematic; in any case, the smallest among them is al-ready of such an exorbitant magnitude that all sets consideredup till now and all sets still to be taken into consideration areprobably exceeded.[Hau05](179)

Definition 6.10. Call λ inaccessible iff κ < λ 7→ 2κ < λ.κ is 0−hyperinaccessable iff κ is inaccessable;κ is α+ 1−hyperinaccessable iff there are κ α-hyperinaccessable cardinals below κ.

Proposition 6.11. There exist inaccessible cardinals.

4. AXIOMS OF INFINITY 169

Proof. All ordinals precede Ω; in particular for any κ < Ω, also 2κ < Ω. So

2κ < Ω ` κ < Ω 7→ 2κ < Ω.

Modus ponens and reflection on Ω give the result.

In the spirit of alpinism the climb can be continued, by establishing the ex-istence of, say, an Ω-inaccessible. This is straightforward. Rather than focus onthese, we go on to show the existence of the next qualitatively bigger powers, usingthe idea of a normal function.

Definition 6.12. A function f : On −→ On is normal iff f is monotonic

x < y 7→ f(x) < f(y),

and continuousf(κ) = supf(x) : x < κ,

for κ a limit.κ is a Mahlo cardinal iff every normal function on κ has an inaccessable fixed

point.

Proposition 6.13. The identity f(x) = x over the ordinals is a normal func-tion.

It is expected that there are other normal functions. Cantor [Can15] observedthat there is an isomorphism (defined in section 5 below) between the set of allinfinite cardinals and the set of all ordinals; see [Lev79](90). This is a monotonicfunction by the definition of isomorphism. The continuity condition is met by thestructure of the ordinals. The proof at [Lev79](117) gives the idea, though it needssome modification to go through in DLQ.

Proposition 6.14. There are Mahlo cardinals.

Proof. By a reflection argument. Let f be any normal function on Ω. By thecontinuity of f ,

f(Ω) = supα<Ωf(α).

Note that f(Ω) ∈ On since normal functions are on On. Because Ω ∈ Ω, thereforef(Ω) ∈ f(Ω) by monotonicity and modus ponens; but being an ordinal, f(Ω) 6∈f(Ω), showing f(Ω) 6= f(Ω). Therefore f(Ω) is absolutely large (lemma 5.66),|f(Ω)| = Ω. Since we’ve already shown that Ω is inaccessable, we have the neededfixed point.

The next cardinals to consider arise out of the Lebesgue-measure problem.


Definition 6.15. An ultrafilter U over M is α-complete iff

(∀ξ)(Xξ ∈ U ∧ ξ < α 7→⋂Xξ : ξ < α ∈ U).

A cardinal κ is measurable iff κ > ℵ0 and there is a non-principal κ-completeultrafilter over κ.

Proposition 6.16. Ω is a measurable cardinal.

Proof. Let UΩ be the set of all subsets of the ordinals that have Ω as amember,

UΩ = X : X ⊆ Ω ∧ Ω ∈ X.

Intuitively, the members of UΩ are exactly all the ordinal properties borne by Ω. Sothis is a direct reflection argument: that UΩ is the requisite ultrafilter. The mostimportant facts to observe is that Ω ∈ UΩ, since Ω ⊆ Ω ∧ Ω ∈ Ω, and similarlythat x : x ∈ Ω ∧ x = Ω ∈ UΩ. First it is to be seen that UΩ is a filter. ∅ 6∈ UΩ

because Ω 6∈ ∅. If X,Y ∈ UΩ, then X ∩ Y ⊆ UΩ ∧ Ω ∈ X ∩ Y , so X ∩ Y ∈ UΩ.If X ∈ UΩ and X ⊆ Y ⊆ Ω, then Ω ∈ Y and therefore Y ∈ UΩ. Thus, a filter.To show it is an ultrafilter, suppose UΩ ⊆ G and G is a filter over Ω. Then everymember of G is a subset of the ordinals. And since Ω ∈ UΩ ⊆ G on assumption,X ∩ x ∈ Ω : x = Ω ∈ G for any X ∈ G because G is a filter; so Ω ∈ X, andtherefore X ∈ UΩ, showing G = UΩ as required. For Ω-completeness, consider thewellordering of our ultrafilter, Xξ : ξ ∈ On; the intersection

Y : (∀X)(Xξ ∈ UΩ ∧ ξ ∈ On→ Y ∈ Xξ)

has Ω as a member and is a subset of the ordinals, and is therefore in UΩ. Finally,we need that UΩ is non-principal—which follows if ¬(∃Y )(Y 6∈ Ω∧UΩ = X : X ⊆Ω ∧ Y ⊆ X). Well, suppose

Y ∈ Ω ∧ (∀X)(X ∈ UΩ ↔ X ⊆ Ω ∧ Y ∈ X).

Since Ω 6= Ω, also Ω 6⊆ Ω; so Ω ∈ UΩ ∧ Ω 6∈ UΩ, proving UΩ 6= UΩ, regardless of Y .This completes the proof.

If the foregoing seems a bit strange, one should remember that this is not anaverage ultrafilter, but one generated by a notoriously inconsistent object. “Thesubject of paradoxical assertions is one full of surprises. However that it should beso is not particularly surprising.”1

Measurable cardinals are due in part to Ulam, also to whom in part is due thehydrogen bomb. The existence of measurable cardinals is very suggestive; these are

1“After all ... we all normally assume that we are not reasoning about a paradoxical situation:

when we meet a contradiction we take it as a sign that something has gone wrong and refuse to

go further. ... It is precisely when we do go further that our familiar world disappears and we findourselves in strange new surroundings. The new terrain clearly needs to be explored.”[Pri79](240)

5. THE END OF THE ORDINALS 171

the most important of the large cardinals and resolution of their details would settleoutstanding questions. For example, Vitali’s use of the axiom of choice to producea non-Lebesgue-measurable set of reals has been a point of vexation. Another openavenue is Scott’s demonstration that there are measurable cardinals exactly wherethere are elementary embeddings of initial segments of the well-founded universe,which in turn impacts Godel’s hypothesis that the universe is constructible.

5. The End of the Ordinals

The last of the known cardinals are extendible cardinals, including a virulentstrain due to Reinhardt called (if inconsistent objects are called anything, classi-cally) Reinhardt cardinals. Like most other large cardinals, Reinhardt’s arise bylooking for closure properties; but his ideas go substantially beyond, unabashedlytaking On as a given totality capable of supersession [Jec74]. That the full ar-ticulation of this idea ends in inconsistency, then, should not be a surprise: Rein-hardt is working with the full set concept, looking for the ultimate sky of Cantor’suniverse.[Kan94](311)

This section does not prove any theorems of naive set theory. Instead I describesome mainstream results, involving the |= relation, with the aim of suggesting thatineluctable invariance that independent mathematics tends to show. The sameactivity at the end of the ordinal line that we recorded last chapter has been in-dependently observed by classical set theorists working at the extremity of theirsubject. For this and the next section, we study not On but the wellfounded uni-verse,

VΩ =⋃

α∈OnVα.

To discuss all this we need defined some notions from model theory. Here weintroduce the idea, following closely [BS71](73 - 75).

Let A = 〈A, r〉,B = 〈B, s〉 be two structures, with A,B sets and r, s relations onthe respective sets. A mapping j : A −→ B is a homomorphism iff, for a0, ..., an ∈A,

〈a0, ..., an〉 ∈ r ↔ 〈j(x0), ..., j(xn)〉 ∈ s.

If j maps A onto B, then j is an isomorphism. When there is such a j, the structuresare isomorphic, written A ∼= B.

A is a substructure of B, written A ⊆ B, iff A ⊆ B and r = s∩A, the restrictionof s to A. An equivalent definition could have been: A ⊆ B iff A ⊆ B and theinjection i : A −→ B, i(a) = a, is a homomorphism.

Two structures are elementarily equivalent, A ≡ B, iff for all sentences Φ,

A |= Φ↔ B |= Φ.


A is an elementary submodel of B, the latter called an elementary extension of theformer, iff A ⊆ B and A |= Φ↔ B |= Φ for any formula Φ.

An elementary embedding is an isomorphism j from A to B such that for anyformula Φ(x0, ..., xn),

A |= Φ(a0, ..., an)↔ B |= Φ(j(a0), ..., j(an)),

with a0, ..., an ∈ A. As some notation, j : X ≺ Y means that j is an elementaryembedding from X to Y , and if there is such a j, that X is elementarily embeddableinto Y .

An uncountable cardinal κ is extendable iff there is a β and a non-trivial (i.e.not the identity) elementary embedding j : Vκ+α −→ Vβ for α < κ, with κ the leastordinal moved by j.

Reinhardt’s conjecture, stated at the end of his dissertation and described at thesymposium on axiomatic set theory in 1967 [Jec74], is that there is an elementaryembedding:

(∃j)j : VΩ ≺ VΩ.

(Because Reinhardt reckons V = VOn, he omits the subscript.) There would be nofurther closure property on wellfoundedness than this. The conjecture states, asfar as I can see, that Ω itself is extendable.

“At times in the past,” Kunen writes in reference to his own 1971 proof, “plau-sible large cardinal assumptions have turned out to be inconsistent with ZFC.. . . But maybe ZFC is inconsistent.”[Bar77](399) Kunen proved Reinhardt’s con-jecture is unstable. Kunen’s theorem, which will be spelled out in more detailin a moment, says that no j can embed the wellfounded universe into itself: Ifj : VΩ ≺ VΘ, then there is an x such that if x ∈ VΘ then a contradiction follows;therefore Kunen reasons that x 6∈ VΘ, and consequently that VΘ is not universal.This was interpreted to mean Θ 6= Ω.

Realizing Kunen’s proof in naive set theory is beyond current capabilities. Anoutline of the classical proof serves more philosophical purposes: Kunen’s result willbe that, if there exists a sufficiently large cardinal, then there is an extension j whichboth does and does not extend. Structurally, this is a much more sophisticatedexplanation of what happens in the Burali-Forti construction, and, interpretedfrom naive set theory, an anticipation from mainstream mathematics of some ofthe more unusual behaviors of Ω studied here.

Recall that f ′′(x) = f(y) : y ∈ x is the image of x under f . Let [x]α be theset of all y ⊆ x such that y ∼= α. The following lemma is due to Erdos and Hajnal.

For any ordinal ξ, there is a function f : [ξ]ω −→ ξ

where for all y ⊆ ξ, if |y| = |ξ| then f ′′[y]ω = ξ.

5. THE END OF THE ORDINALS 173

Any f meeting these requirements is called ω-Jonsson. Here is a sketch of theproof: For x, y ∈ [ξ]ω, we find an equivalence relation ≡ such that

x ≡ y ↔ x− α = y − α

for some α <⋃x. By choice, we get a distinguished member z≡ from each equiv-

alence class E(x) = y : y ≡ x. For x ∈ [ξ]ω and z≡ ∈ E(x), take the minimalβ such that x − (β + 1) = x≡ − (β + 1); the desired g is g(x) = β. Now show,by reductio, that there is a κ ∈ [ξ]ξ where for any λ ∈ [κ]ξ, κ ⊆ g′′[λ]ξ. So thelemma makes use of the axiom of choice, in order to fix a function g from which therequisite function f may be derived. To prove Kunen’s theorem only a special caseof the above scenario is needed, namely when ξ is a strong limit cardinal cofinalwith ω.

Kunen’s answer to Reinhardt’s question, which appeared in a short paper[Kun71], launches from the Erdos-Hajnal result.

If j : VΩ ≺ VΘ is any elementary embedding except identity, thenΩ 6= Θ.

A few different proofs have been given, each lending different insights [Kan94](320).An outline Kunen’s original argument will show the kind of structural affinity be-tween classical (?) higher set theory and naive theory I’ve suggested.

Suppose j : VΩ ≺ VΘ. Let

j0(x) = x

jn+1(x) = j(jn(x))

for n finite. By induction, j is increasing, x < j(x). Further, let x0 be the firstpoint moved by j; similarly, letting⋃

jn(x0) : n ∈ ω = λ,

then λ is the least fixed point for j above x0. Then it is to be shown that j′′(λ) 6∈VΘ, meaning that Θ is below Ω. Let f be ω-Jonsson with domain the set offunctions from ω to λ and range λ. Since j′′(λ) = j(ξ) : ξ < λ, if j′′(λ) ∈ VΘ

then some g is a function from ω to j′′(λ) ∩ VΘ, with (j(f))(g) = x0. A subtlecombinatorial argument [Kun71](408) then establishes that, given such a g, x0 isin the range of j. Except that j is increasing, so x0 is also not in the range of j—acontradiction completing Kunen’s proof. In a rather dramatic turn of phrase, andwith a sensitivity to the role of choice in the proof, Kanamori remarks:


Kunen’s result can best be viewed as an ultimate limitation im-posed by the Axiom of Choice on the extent of reflection pos-sible in the universe. ZFC rallies at last to force a veritableGotterdammerung for large cardinals! (324)

What is shown is that if there is an increasing embedding from VΩ then it is alsonot increasing. Embeddings become inert when large enough ordinals are involved.At VΩ, there is simply nowhere left to go—unsurprisingly, since this fragment of Vis so large. As our dialethic investigations have been showing, though, Ω is indeednot the end of the ordinals. At the very least, it is a successor, Ω = Ω + 1, whereΩ < Ω. And in the next section indications that there are limit ordinals aboveΩ will be given, too. So Kunen’s observation that a non-trivial embedding mustbe both increasing and not, without further reductio, seems quite correct from theperspective of naive set theory. Kunen’s is a sophisticated classical confirmation ofthe more bluntly stated naive propositions.

On the other hand, the universe V really is comprehensive; nothing can falloutside its province on pain of triviality. In light of there being a wellorderingon V , and the fact that Ω is its ordertype, one might think that Kunen’s resultis stronger, that there is no (non-trivial) embedding j : V ≺ V , again on painof triviality. (Or worse, there is such a j and naive set theory is trivial.) Again,V 6= VΩ. But Kunen’s result does give pause. What more can be said on this is anopen question.

6. Reflecting the Universe

Cantor’s theory brought new meaning to the infinite, and his followers haveexplored the terrain. Naive theory offers a most precise expression of endlessness.Downward reflection is inconsistent because it implies a bottomless chain of ordi-nals, as predicted by Godel’s second theorem. Now we go further by consideringself-reference. If everything about Ω is reflected, then this very property shoulditself be reflected at a lower level. Reflection reflects.

Aside from its intrinsic interest, the construction we give follows up on Rein-hardt’s extendable cardinals, which demarcate the end of the consistent segmentof the ordinals. These come about through the upward extension of elementaryembeddings; is there a clearer picture of where these embeddings continue afterconsistent theory is exhausted?

First, a reflection theorem for VΩ itself.

Proposition 6.17. When VΩ ∈ X, also (∃Θ)(Θ < Ω ∧ VΘ ∈ X).

Proof. Again because Ω < Ω.

6. REFLECTING THE UNIVERSE 175

Let R be the set of all sets that reflect,

R = M : (∀X)[M ∈ X 7→ (∃M ′)(M ′ < M ∧M ′ ∈ X)].

More generally, now, we can see that any structure satisfying reflection must itselfreflect.

Proposition 6.18. M ∈ R ` (∃M ′)(M ′ < M ∧M ′ ∈ R).

Proof. Since Ω reflects, R is not empty; let M ∈ R. Then [M ∈ R 7→(∃M ′)(M ′ < M ∧M ′ ∈ R)]. (Note that this is not contraction, since M really is inR).

Repetition of this argument is iterated reflection.

Corollary 6.19. There is a countable chain of sets that satisfy reflection; ifM ∈ R then countably many other M0,M1, ...Mω are in R, too.

Note that full comprehension provides a sharp modeling of these facts with aset

R = M : (∃M ′)(M ′ < M ∧M ′ ∈ R).

A question arises as to the size of R. While it is easy to repeat the basicreflection iteration up to ω, further checking is needed to confirm that the processioncan continue.

Proposition 6.20. |R| = Ω.

Proof. Ω ∈ R. But Ω = Ω ∧ ¬(∃Θ)(Θ < Ω ∧Θ = Ω). So by counterexample,Ω 6∈ R. Ergo R 6= R and is absolutely large, as required.

Because there is an unlimited pool of reflecting ordinals,

Corollary 6.21. Iterated reflection can be continued up to any ordinal. Let-ting

RΩ = Θ ∈ On : (∀X)[X(Ω)→ X(Θ)],

then |RΩ| = Ω.

Structure emerges. Let σ be a choice function. Define by induction

M0 = Ω

Mα = σ(R− Mβ : β < α)

Mλ =⋃κ<λ

Mκ

with λ a limit; and to complete,

M =Ω⋃α

Mα.


Since the principle mechanism involved is Ω < Ω, we now have a direct pointof comparison to Reinhardt’s initiative. Exactly as before, since Ω < Ω, we cansee that Ω is extendible, (∃Θ)Ω < Θ. Here there can be embeddings j : VΩ ≺ VΩ,attendant extendable cardinals, and almost certainly many, many other types ofcardinal for investigation.

Having tread this close to an edge, it seems appropriate to recall Nietzche’senthusiasm in “The Wanderer” in Zarathustra[Nie76]:

Thus must you mount even above yourself—up, upwards, untilyou have even your stars under you! Yea! To look down uponmyself, and even upon my stars: That only would I call mysummit. That hath remained for me as my last summit!

7. Conclusion

The purpose of this chapter has been to begin applying the technical apparatusof naive set theory to areas for which it is particularly fit, and in doing so tocorroborate the philosophical theses put forward in part one. This continues nextchapter in more generality, towards which we make an outgoing remark.

There is a non-empty subset of the ordinals RΩ such that

(∀y)[y ∈ RΩ 7→ (∃z)(z ∈ y ∧ z ∈ RΩ)].

There are sets of ordinals with no least member. This turns out to be an extremelyfecund fact. This is also contradicted, via the counterexample axiom, by a theoremof chapter 5: For any u ⊆ On such that (∃x)(x ∈ u),

(∃y)(y ∈ u ∧ ¬(∃z)(z ∈ y ∧ z ∈ u)).

So the proposition that the ordinals are wellordered is a dialethia; this has beenproved several times over, since confirming Burali-Forti’s paradox last chapter. HereI want to conjecture, in accord with the limit on size doctrine (but now for clearerreasons than were available in part one), that such subsets occur only at the top ofthe structure, at On.

And if that is right, then various existence theorems in this chapter, as well asthe cumulative M hierarchy defined above, the large cardinals and perhaps evenall the transfinite cardinals whatever, are all over Ω and nothing else. The richreflection phenomenon is just this, and no less rich for it: Ω reflects itself.

To understand this and the overall meaning of these mathematics is the taskof the next and final chapter.

CHAPTER 7

On the Transconsistent

At the end, some outstanding philosophical debts need repaying through acohesive interpretation of the preceding work. The mathematical material is newenough that I do not pretend to have penetrated beyond some observations. Thereare, with respect to the set theory here, a few clear points of possible concern orconfusion as to the meaning and integrity of the mathematics. These are to bemet by returning to the philosophical issues of part one and producing a unifiedaccount, which is broadly as follows.

The paradoxes of naive theory are all due to a single, powerful truth: Aninconsistent fixed point stands where none can be, at the meeting of extensionand intension, at absolute infinity. The fixed point arises quantitatively when weconsider ordinals and cardinals. The fixed point also arises analytically when weconsider the comprehensive extensions of predicates. For both Frege and Cantor,the paradox occurs at junction of the universe and its powerset. Godel and Tarskiidentified the fixed point at the limits of proof and truth, respectively, and morerecently, Priest generalizes to show that the fixed point arises conceptually whenwe consider totalities. The fixed point arises even when we try to deny, as we must,that there is any such vertex here: That is exactly to affirm the paradox itself.

Using the mathematics of the last chapters, this view of the inconsistent infinitewill be given a precise characterization. The same technique is then used to broachlogical questions asked in chapter 0, questions about how to isolate true contra-dictions, as opposed to simply false ones. I show that these are intimately relatedquestions, and use some simple paraconsistent logic to provide an answer: Cardinalmaximality is necessary for both. We will see, though, that this is a new kind ofparadox analysis; and so whilst the cardinal answer is a good start, it carries withit paradoxes of its own.

Because I am forging a precise connection between higher set theory and logic,I will resurrect a term from the end of Priest’s [Pri06b] for the non-trivially incon-sistent: the transconsistent. An objective of the following is to precisely demarcatethe transfinite from the overlarge or absolutely infinite, and to show that this alsohelps to demarcate the transconsistent from the absurd or absolutely inconsistent.Dialethic paraconsistency provides a fuller and more exact characterization of the

177

178 7. ON THE TRANSCONSISTENT

infinite than any non-paraconsistent mathematics could, and this mathematics inturn provides a map of where the limits of the transconsistent lie.

Then a discussion of the mathematics is presented, arguing that the basic driv-ing mechanism behind the transfinite is contradiction; classical logic casts this as aprolix structure, whereas paraconsistent logic makes plain the paradox of infinity—out of which the familiar transfinite can be discerned, to be sure, but now recontex-tualized, as a basic consequence of a dialethia. In his work on paradoxes, Bolzanoopened by quoting de Morgan, himself quoting Aaron Hill; and we repeat:

Tender-hearted stroke a nettle,And it stings you for your pains;Grasp it like a man of mettle,and it soft as silk remains.If there be in mathematics a nettle danger out of which has beenplucked the flower safety, it is speculation on 0 and ∞.

The philosophical concerns of chapter 2 are amenable to our inconsistent mathe-matics, which can demarcate the absolute from the transfinite.

1. Characterizing the Absolute

Cantor distinguished three categories: the finite; the transfinite; and the ab-solute. Dedekind produced a precise means of distinguishing the finite from thetransfinite. A still-outstanding problem in set theory, however, is finding similardemarcation between transfinite and absolute; without such a method for drawinga clear line, the doctrines of limit on size and proper classes are underdetermined. Ipresent a way to characterize the absolutely infinite—a criteria to differentiate theabsolute from the transfinite—as maximal.

Paradoxes are fertile sites for new discoveries. In the nineteenth century, thevery same phenomenon blocking the way to a science of the infinite—various coun-terintuitive, inconsistent, and even ineffable properties of the infinite—were con-verted to definitions and theorems by Dedekind and Cantor. Dedekind’s inversionof Galileo’s paradox is the motivating example: Dedekind stipulated that a set isinfinite iff it bijects with a proper part of itself; and correspondingly, that a set isfinite iff there is no such bijection. In symbols, sets M bearing the property

(∃X)(X ⊂M ∧ |M | = |X|)

are dedekind infinite. Every radial through two concentric circles cuts each circum-ference at exactly one point, setting up a bijection between the two circumferences;but isn’t the circumference of the inscribing circle longer than the inscribed? Yes;its points are infinite. So Galileo’s paradox, rather than discrediting, characterizesthe infinite.

1. CHARACTERIZING THE ABSOLUTE 179

Similarly here, can we take a heretofore poorly understood phenomenon, that ofoverlarge or absolutely infinite sets, and interpret their strange features as a precisecharacterization? With the paradoxes of the absolutely infinite openly on displayand a cogent mathematics to work from, we can straightforwardly read off somemaximal properties and show that they serve precisely and provably to demarcateabsolute from transfinite in the appropriate way. This would be to incite an updatefrom Russell’s [Rus37] comments, to show that while “[set theory], though it cannotwholly dispense with [absolute] infinity, has as few dealings with it as possible, andcontrives to hide it away before facing the world,” that in fact “like many skeletons,it was wholly dependent on its cupboard, and vanished in the light of day.” Theaxiom of choice makes the definition of infinity precise; the axiom of comprehensionmakes the definition of the incomprehensible possible.

The matter of stating necessary and sufficient conditions that a set be abso-lutely infinite, or absolute, is not simple. Whereas in our motivating case, Galileo’scontradiction is simply resolved by making it a condition for infinitude, here therecan be no hope of such resolution. The reason is that the absolute is inherentlycontradictory; this is one of its defining characteristics. And the contradiction is aself-perpetuating one: something like

X is absolute iff X cannot be characterized.

This makes it remarkably difficult a priori to state conditions that, on the one hand,do serve as cogent criteria for absoluteness, and on the other hand respect the natureof the object under study—namely that the object under study cannot be captured.By comparison: When a set is dedekind infinite, it may still offend against certainintuitions (e.g. that wholes simply cannot be the same size as their proper parts)and so remains paradoxical in the etymological sense of the word. But as a matterof pure logic or mathematics, there is no such problem. The particular rules oftransfinite arithmetic exert a calm control of the dedekind infinite, with surprisesbut without contradiction, and that is the end of the problem. The absolute,on the other hand, is essentially paradoxical. Any appropriate definition of theabsolute would still by necessity incorporate some of its mystery; being intractablymysterious is in part constitutive of being absolute. We will satisfy this in part,because the naive view of sets is unitary: Any collection at all is a set. The naiveapproach is exactly to treat the universal set or the set of all ordinals as sets likeany other, so absolutely infinite sets are also transfinite; no sets are absolute. Andin a sense, this just is the paradox. The absolute is the domain of all objects, theinfamous universal set of all sets; and in line with our naive unitary conception, it isa set. But the absolute is also “aliquid quo nihil maius cogitari possit,” in Anselm’sidiom, and is not amenable to mathematical operations or extensions, pulling it


apart from all the other sets in its province. This fact plays out in two directions inthe Cantorian universe, as we saw in chapter 2, in generalizations from below andreflection arguments from above; the former highlights problems of extendability,the latter problems of incomprehensibility.

If a set is inconsistent (i.e. it has members that are also non-members) thenit is absolute, as demonstrated in lemma 5.66; but not conversely: V is absolutelyinfinite, yet is consistent, on pain of triviality. So inconsistency is not necessaryfor absoluteness. One cannot even pick out peculiar, even necessary properties ofthe absolute in order to isolate it; for example, by the conjecture that a set X isabsolute iff it reflects:

If Φ(X) then ∃X ′ such that X ′ ⊂ X and Φ(X ′).

This fails as a definition as we saw at prop.6.1: The universe V is absolute, andV = V , yet there is no set X ⊂ X = V , on pain of triviality; so absoluteness doesnot imply this kind of reflection. Assuming the universe is absolute—and if not V ,then what?—something else is required.

Let Abs(X) mean that X is absolutely infinite. We have seen that Abs(X)implies that X reflects; but reflection cannot characterize absoluteness. A charac-teristic relating to, but not synonymous with, reflection and inconsistency is needed.I propose the following.

Definition 7.1. A set X is absolute iff X is maximal with respect to cardi-nality,

Abs(X)↔ (∀Y )(|X| ≤ |Y | 7→ |X| = |Y |).

And X is transfinite iff it is infinite and extendable,

Tn(X)↔ ℵ0 ≤ |X| ∧ (∃Y )(|X| ≤ |Y | ∧ |X| 6= |Y |).

Here 7→ is the preferred conditional because the consequent may not followrelevantly from the antecedent.

Let us show that this is the right definition, by proving that it delivers theexpected properties of absoluteness.

Proposition 7.2. Some sets are absolute; in particular, Abs(Ω), and Abs(V ).

Proof. Since the property of absoluteness was inspired by Ω, it is no surpriseto see that if Ω ≤ |X|, then since |X| ≤ Ω for any X at all, it follows that Ω = |X|.And since |V | = Ω, by substitution the universe is absolute, too.

Since all inconsistent sets are Ω-sized (lemma 5.66), this gives the immediate

Corollary 7.3. If X is inconsistent, X is absolute.

Proposition 7.4. All absolute sets are transfinite.

2. CHARACTERIZING THE TRANSCONSISTENT 181

Proof. Let Abs(M). Then, by ∀-elimination, |M | ≤ Ω 7→ |M | = Ω. Sinceevery cardinal is less than Ω, by modus ponens follows |M | = Ω. By substitution,since Ω 6= Ω, it follows again by modus ponens that |M | 6= |M |. Now, |M | ≤ |M |,so adjoining we have |M | ≤ |M | ∧ |M | 6= |M |, from which follows by ∃-introductionthat Tn(X).

Proposition 7.5. Transfinite sets are not absolute.

Proof. If X is transfinite, then it is extendible: X is less than, and not equalto, some Y . Then by counterexample, it is not absolute.

Proposition 7.6. Every infinite set is transfinite. No set is absolute.

Proof. For every X, we have |X| ≤ Ω∧|X| 6= Ω. Generalization gives the firstresult. Then, transfinitude implies non-absoluteness, and all sets are transfinite,proving the second claim.

2. Characterizing the Transconsistent

Sets and logic have a long history together, and their junction is a notoriouslocus for paradox. Classical logic and classical set theory arose in the late nineteenthcentury largely as a single foundational enterprise, and the fact that cardinalityand consistency are deeply intertwined has been obvious to most, even if not wellarticulated by many. It is the connection between cardinality and consistency, butstudied for the first time at their limiting fixpoint, exploited here.

In the opening set theoretic excursion of chapter 0, we cited a distinction be-tween rejection and denial, and found that, whilst some propositions are dialethic,and so a proof of p does not necessarily rule out ¬p, nevertheless implication of ab-surdity was enough to assure the consistent behavior of some theorems. If we provep, and ¬p→ ⊥, then p is consistent; recalling da Costa’s consistency operator, thiscan be marked p.

Discerning consistent from inconsistent information would be very useful. In-ferences that are generally invalid, like disjunctive syllogism, could be revived,presumably in the form:

p ∨ q,¬p, p ` ¬q

Certain theorems which are not forthcoming in DLQ might then be provable. Forexample, letting α, β be ordinals, we have not yet established the basic theoremα ⊂ β ` α ∈ β, as the standard argument makes indispensable use of the disjunctivesyllogism. (Suppose α ⊂ β. Then there is a least γ ∈ β − α. To show thatγ ⊆ α∧α ⊆ γ, for α = γ and thereby α ∈ β, one ends up considering x 6∈ β∨x ∈ α,wanting to assert x ∈ α on the basis that x ∈ β. But this inference is invalid.) If(x ∈ β), we could use Priest’s notion of quasi-validity, as discussed in [Pri89] and


[Pri06b], and could allow paraconsistent reasoning to wax classical in consistentsituation.

As usual, though, matters become more subtle when we move from the propo-sitional level to the predicate calculus. In particular for naive set theory, one wouldneed to know when ∈ behaved consistently; and obtaining this knowledge is ingeneral not possible. Now, ∅ and V are consistent on pain of triviality, but whatabout some ordinal β, as in the example above? If β = On, then we know β isinconsistent and not suitable for quasi-inferences; but if β 6= On, we still don’tknow much, since On 6= On so β may yet just be On. And similarly, we have aproof that if (∃z)(z ∈ β ∩ β) then |β| = On, and then β = On, but again since|On| 6= On, this limit on size theorem cannot serve to reliably demarcate consistentordinals.

According to the logic DLQ, all propositions are either true or false. Somepropositions are both true and false. According to DLQ, all propositions are con-sistent. For any Ψ, that is, it is not the case that Ψ ∧ ¬Ψ, so the consistentpropositions are all the propositions and dialethism is itself a dialethia. This isanalogous to the unitary view we took with the absolute. There, no sets are abso-lute, so the transfinite exhausts all the infinite sets. Again, then, where is a workingcriteria that at once picks out all and only the dialethic propositions? Accordingto the logic, which has non-contradiction as a theorem, all propositions, all sets, allordinals are consistent!

Identifying the true contradictions, as opposed to the simply false ones, ishighly analogous to, and just as difficult as, identifying the absolutely infinite sets.Inconsistency, except in special cases like ∅, cannot be ruled out—the exclusionproblem[Ber07](ch14). We have obtained some techniques for demarcating incon-sistent sets—namely, that the transfinite has lapsed into the absolute by showingthat cardinality has hit Ω. Still, there is no algorithmic check for consistency or in-consistency, even with criteria for knowing when a set is absolute, since the absoluteitself is transfinite, too. This paradox, then, is harder than Galileo’s to appropriatefor the foundation of a new science, because one horn of the contradiction is thatthe subject—the dialethic absolute—does not even exist.

Here is the start of a mathematical gauge of inconsistency; we tread intometatheory, which has been outside the interests of the present work, but whichcan be used to sketch a technique.

Let Φ be any formula. Let p·q be a name-forming operator, e.g. a suitable godelcoding that could be provided with some basic arithmetic and recursion theory. Bycomprehension, ν(Φ) = x : Φ, the set of witnesses for Φ, can be used to define atruth predicate:

ν(Φ) ∈ ν(Φ) is T (pΦq).

2. CHARACTERIZING THE TRANSCONSISTENT 183

It is straightforward to check that

T (pΦq)↔ Φ,

showing that the truth predicate satisfies the T -schema. This will be our point ofentry into the connection between cardinality and consistency.

The more vexing results of mathematical logic have been in the vicinity of truthand provability predicates, to prove limitive theorems. Of metatheory, Minskywarns that “while the logic is easy to follow, one feels that the whole thing is asort of extended joke or pun.” [Min72](169) The punchline of Minsky’s joke isobtaining the diagonal lemma, which captures the fixed point phenomenon that wehave been studying throughout. For every formula Φ(x) of one free variable, thereis a sentence Λ such that

Φ(pΛq)↔ Λ.

A proof (which will eventually be able to be carried out within naive set theoryitself) is as follows; see [BJ89](173), [Pri06b](49).

Let Φ(x) be any formula with only x free. The diagonalization of Φ(x) isΦ(pΦ(x)q). Let k be the godel code of Φ(x), pΦ(x)q = k, so the diagonalization ofΦ(x) is Φ(k). There is a function δ such that

δ(k) = pΦ(k)q.

Consider Φ(δ(x)). Let pΦ(δ(x))q = n. The diagonalization of Φ(δ(x)) is Φ(δ(n));let pΦ(δ(n))q = m. The Leibniz identity principle is

δ(n) = m ` Φ(δ(n))↔ Φ(m).

And δ(n) = m. Therefore Φ(δ(n)) is what was sought,

Φ(pΦ(δ(n))q)↔ Φ(δ(n)).

From the diagonal lemma, it is then easy to show that truth is inconsistent.Consider the formula of one free variable, ¬T (x). By the diagonal lemma, there isa proposition

Λ↔ ¬T (pΛq)

Either Λ ∨ ¬Λ. Then Λ → T (pΛq), and this by contraposition on the T-schemebrings ¬Λ; so ¬Λ by the laws of dialethic logic. Yet ¬Λ→ T (pΛq) by contrapositionon Λ itself, which by the T-scheme implies Λ; so Λ. Therefore Λ∧¬Λ, and thereforeTpΛq ∧ ¬TpΛq.

And from this, if our coding device is working, it follows that truth is absolutelyinfinite, |pΦq : Φ| = On, by lemma 5.66. This insight is now applicable to thequestion at hand.


If Φ is true, then (∀x)x ∈ y : Φ, and if ¬Φ is true, then (∀x)x 6∈ y : Φ. Andin the dialethic case,

TpΛ ∧ ¬Λq↔ (∀x)(x ∈ ν(Λ) ∧ x 6∈ ν(Λ)).

This shows that membership in ν(Λ) for inconsistent Λ is inconsistent, and thereforethat ν(Λ) is absolute—that

|ν(Λ)| = |On|.

Thus we have the promised connection: All true contradictions make set theoreticcontact with the absolutely infinite.

What is still missing is a sufficiency condition; having an overlarge witness setis only necessity for dialethias. But this, like our account of the absolute, gets someessential properties correct. For a start, contrapositively, if |ν(Λ)| is not absolute,then it is not the case that TpΛ ∧ ¬Λq. But nothing is absolute, as we saw inthe last section; so no contradictions are true. This agrees with our paraconsistentlogic, and the dialethic nature of dialethsim.1

3. Characterizing the Mathematics

When the idea of changing the logic, rather than the set concept, was raised,I wonder if even its enthusiastic proponents knew exactly what they were propos-ing. The naive view of sets was expected to be: the classical view of sets, withsome additions—an extra layer at the top of the cumulative hierarchy was the wayReinhardt put it. It was to be a closure on an otherwise good theory. In gen-eral the expectations of paraconsistent inconsistent mathematics were essentiallynot revisionary, not even particularly weighty. Though his views have always beentreated as heretical extremism, in fact Priest from the start has been conservative,maintaining mostly orthodox structures and interpretations, affixing dialethias atthe edges; his logic LP is a straightforward example: In the cases of classical truthvalues, LP is just classical logic.

Like those before, I do not want or expect that dialethic paraconsistency willlead to the rejection of any core mathematics. As remarked from the outset, di-alethism is born out of and obeys the same basic rules as any other philosophy;DLQ has all classical tautologies as theorems, and if → is interpreted materially,then DLQ is a proper fragment of classical logic. But the force of that mathe-matics, that philosophy, that logic, is different. Changes in context make changesin meaning. If there were living microbes discovered on Mars, this would not inthe slightest change any biological structures here on Earth. The discovery would,

1As formulated in DLQ. In [Pri06b], chapter 4, Priest formulates the T -scheme with a non-contraposable conditional, so that Tp¬(Λ∧¬λq holds without ¬TpΛ∧¬Λq. Given our embeddingof the truth predicate in set theory in DLQ, which does contrapose, Priests’ analysis does not fit

here.

3. CHARACTERIZING THE MATHEMATICS 185

though, irrevocably alter the meaning of biological structure on Earth. So too withmathematics, philosophy and logic, it should be expected that the introductionand acceptance of entities as singularly exotic as dialethias will irrevocably alterthe context and so the meaning of mathematics, philosophy logic. Nothing willchange the fact that Godel proved the diagonal lemma; what follows from thatproof, though, we have yet to see in full.

Some of this is, of course, forecast by and even made a feature of the paracon-sistent program. The extent of the epoch-shift, though, is as yet unknown. Myclaim has been that, in the face of paradox, paraconsistency is a means to advance.In part two a naive set theory was developed, so at the most basic level, we knowthat there can be some paraconsistent contribution to foundational mathematics.How, though, does the naive set theory of this volume measure up to our adequacycriteria, of being explanatory? Even a cursory reading will show that it is unprece-dented and unusual in some respects, and is therefore not easily judged. “Therealm beyond the consistent,” after all, “is a continent on whose shore we have justalighted.”[Pri06b](209)

3.1. To make a reduction of mathematics to set theory worthwhile, recall, it isnecessary that sets be better understood than more complex objects like numbers,else we fall afoul of Hausdorff’s obscurum per obscurius. Is naive set theory descrip-tive and explanatory? Naive set theory is based on the idea of sets being predicatesin extension; but there is little hope of arguing in isolation that the naive concept isnatural. Extended apologies that p is ‘obvious’ are necessarily rather self-defeating.For what it is worth:

I’ve argued that the set concept, as much as it can be discerned, is inheritedfrom Cantor, Dedekind and Frege, who were only a few of those who took theconcept to be obvious; that no other concept has provided an adequate replacement;and that in fact the natural set concept is still the one implemented in mathematicalpractice. Routley and Priest write,

No one has ever explained what is wrong with instances of ab-straction which fail in the cumulative hierarchy. If it could beshown that the cumulative hierarchy was the essential core ofthe theory of sets ... the solution would still not be adequate.For, why we should ever have thought that every condition de-fines a set, remains a complete mystery. It doesn’t even looklike a plausible claim. How could such a mistake be made? No:the genuine conception of set is that given by the unrestrictedabstraction scheme.... [PR89a](506)


Adding a more constructive note to this point, I’ve endeavored to show that thereis much to be learned from the inconsistent aspects of sets. Yet, even if it is nowdemonstrated that naive set theory provides the start of an ontology for mathemat-ics, whether it is the correct, authentic ontology of mathematics is still undecided.Uncontroversially, naive set theory is the description of something; let us try tounderstand what it describes.

There are many theorems in this work, and I think they are true. Some of thetheorems are also inconsistent, and as pointed out in the last section, there is noway to fix unequivocally, sub speciae aeternai, that they are not all false. This isthe question of the force of dialethic truth, and a point at which it seems to departfrom the more widely accepted notion of truth.

This is the exclusion problem recast: A consistent truth p has a small witnessset, |ν(p)| strongly less than Ω; every size is less than Ω; there is no way to pick thesentences that are true only. Chapter 0 goes to some lengths to reconnect dialethismwith the wider community, but there is a basic stopping point here: A mainstreammathematician will not feel that an inconsistent theorem (which all theorems hereat least possibly are) is true. I think the theorems of naive set theory are true. Tojustify this sense, and in part to begin to address the exclusion problem discussedabove, we offer a more expansive explanation of mathematical truth, following upon Priest and Routley: “The cumulative hierarchy is exactly what it appears tobe with a little historical perspective: a consistent substructure of the inconsistentuniverse of sets, masquerading as the whole thing.”(506)

I do not claim that mathematicians, as a matter of practice, believe in di-alethism. With some notable and mystifying exceptions (for instance, see Woodin’sdisconcerting “Tower of Hanoi” in [DO98]), mathematicians do not seem to en-tertain the possibility that, upon encountering a contradiction, they have madeanything more than a mistake. So naive set theory is not a cultural study of math-ematicians, and, on this conception, I respect the point put to me by one skeptic(in conversation): “Inconsistent mathematics is not mathematics.” But this is onlyso much anthropology and does not rule out that naive set theory is explanatoryof actual mathematical truth, nor does it deprive naive set theory from descriptivestatus. Let me explain.

The technical details suggest, and I wish to argue further, that classical mathe-matics is an undisturbed fragment of the wider edifice, which by its nature must beparaconsistently dialethic. In this sense naive set theory is explanatorily descrip-tive, by presenting a well-motivated whole of which the familiar is a part. This isnot a particularly new claim—Priest has been making it about paraconsistent logicfor years, e.g.


Going paraconsistent is a gain. ... A paraconsistent logician mayanalyze every situation that a classical logician may analyze inexactly the same way. But they can also analyze more. Paracon-sistent logic therefore has excess power. ... Paraconsistent logicis therefore preferable to classical logic for exactly the same rea-son that classical logic was preferable to traditional logic. Itis a more powerful and flexible inference engine. Why preventyourself from exploring what is beyond the consistent, when youlose nothing by this? Why shackle yourself with the necessity ofconsistency, when you don’t have to? ... Logicians of the worldunite! You have nothing to lose but your chains. [Pri00b](232)

The point can only be pushed so far, but it is worth asking what this supercessionidea comes to.

There is a realist thought here, an optimistic induction. To take a well-wornexample as the base, the pythagoreans believed the world is explicable in termsof rational numbers. One can learn quite a lot about pythagorean triangles, righttriangles with only natural number length sides (Sierpinski has a nice little book onthe topic), but these are not all the triangles. Irrational numbers are a part of thelarger, more useful structure that is incommensurable with a rational-number world-view. Importantly, that the pythagoreans did not believe in irrational numbers, iscompletely irrelevant to the existence, meaningfulness, cogency or applicability ofirrationals. Similar comments could be made about negative, or much later, com-plex numbers. So real analysis does not, per se, describe pythagorean practise.But because mathematics is not a revisionary science, neither does analysis imputeagainst the positive aspect of the pythagorean belief set; progress adopts what hasalready been discovered, and then says more. The pattern has the emergence of anincommensurable magnitude or other insolubilia, its integration into a wider sys-tem, and then a retrospective explanation and description of what has come before.The main failure of the cumulative hierarchy is that it retains insolubilia; there isstill too much it cannot explain.

Now, by turns, we can ask if naive set theory really is a reworking of thenineteenth century systems, and then what points of contact naive set theory haswith the other systems of the twentieth century. At the start of his [She00] Shelahasks of his own complex research, “Could Cantor have read it?” and so too here.Again, I do not think Cantor was a dialethist; but I wonder whether he would haveregarded naive set theory as later explorations of the same universe he discovered,or whether it would be regarded as a different subject altogether. A case studymay help bring out what is at issue: the intertranslatability of theorems.


3.2. Routley states that

In order to sustain the ultramodal challenge to classical logicit will have to be shown that even though leading features ofclassical logic and theories have been rejected, ... by going ultra-modal one does not lose great chunks of the modern mathemat-ical megalopolis. ... The strong ultramodal claim—not so farvindicated—is the expectedly brash one: we can do everythingyou can do, only better, and we can do more. [Rou80](927)

In calling for the recapture of classical results with paraconsistent logic, Routleyseems to have assumed that theorems are somewhat logic-independent—that aclassical theorem proved paraconsistently is still the same theorem. For example,Routley’s 1977 proof with DKQ of the axiom of choice does indeed validate theassertion:

There is a function f such that, if x is a non-empty set, thenf(x) ∈ x.

But the definition of ‘function’ in [Rou77] is a relation that is either univocal orempty. Now, this latter, as Routley notes, is the case classically by virtue of aparadox of material implication: If f = ∅, then from 〈x, y〉 ∈ f follows anythingat all, including functionality. So Routley brings to the classical arrangement anon-classical logic, and argues that the axiom of choice is true, essentially (thoughthe details differ) by presenting the set

f = 〈x, y〉 : R ∈ R,

where R is the Russell set. Since R 6∈ R, f is empty, so it is a ‘function.’ SinceR ∈ R, any ordered pair at all is in f ; so of course f(x) ∈ x for non-empty x.But this is not, by any stretch of the imagination, what Zermelo’s axiom of choiceasserts.

Unreflective assumptions about the faithful transposition of mathematical con-cepts, then, pose real danger. I have been taking myself to be studying, not justnaive paraconsistent set theory, but set theory simpliciter. Naive paraconsistent settheory just is set theory. What comes out in these examples, though, is that thesame syntactic strings, e.g. ‘there is a measurable cardinal,’ may mean differentlyin different contexts. I have proved Cantor’s theorem, but not using the same diag-onal set; the set used here exists only by full comprehension. I have proved Peano’spostulates, but using a mobius-like representation of ω, with ω appearing in its owndefining predicate. One may worry that, whilst all agree that 1 + 1 = 2, my plusis really a wittgensteinian quus—that these are not the same mathematics. More,


some of the ideas and techniques in naive set theory are novel, e.g. at theorem 5.50,and may seem too different from the familiar canon.

The core mechanism of the results here obtained in naive set theory with respectto infinite numbers has been the non-self-identity of On, the set of all ordinals.More than any of the inclosure paradoxes (Russell’s set, for example, has playedno more than a nominal role), the Burali-Forti contradiction takes centre stage—from a demonstration of the wellordering theorem, to proof of limitation on sizeto the existence of all transfinite cardinals, to breaking the generalized continuumhypothesis. This is quite a lot. And the burden of the remainder of this section is toshow that this is as it should be. The paradox of all the ordinals was always regardedas pathological; but “it is always a mistake to think of anything in mathematics asmere pathology, for there are no such things in mathematics.”[For95](11) I submitthe following explanation.

3.3. The paradox of absolute infinity, exemplified at the top of the ordinals,is the core subject matter of Cantor’s mathematics. Set theory is the study ofreflection properties of On. This is disguised by consistent reasoning and madeplain by the paraconsistent proofs. Generally, consistency preserving theories takethe shape of hierarchies; and this is what Cantor’s transfinite looks like. Here wehave the hierarchy, but also a unity, which is what the naive view entails. Putdifferently, set theory trades in ordinals (to paraphrase one of Forster’s metaphors,that the currency in which ZF trades information is cardinality), and we have allthe ordinals. It should not be surprising what an influx of such wealth does to theset theoretic economy.

In their excellent essay on unrestricted quantification, Shapiro and Wright ad-mit a certain frustration: There is just no denying that

Ω is the set of all ordinals—all possible order-types. Not all theordinals except those that come after, the ‘proper’ ordinals, orhigher-ordinals, or whatever. All of them. There may be a con-sistent formal theory, but it does not sustain the intended inter-pretation except at the cost of informal paradox—the same oldBurali-Forti paradox—and for good measure, some additionalvariations on the theme—that was there all along. Let us stoprunning in circles and step back.[SW06](292)

The authors then outline what they take to be the five possible responses to thesituation, each with an associated cost. Each cost is reckoned to be too high. Thelast option is to proceed with the natural, intuitive use of quantifiers and predicates,and “just accept that there are ordinals that come later than all the ordinals. Cost :none—unless one demurs from the acceptance of contradiction.”(293) Here we have


not demured and now count the benefits. The point will be detailed by recallingsome main characteristics of Cantorian theory: a wellordered hierarchy of transfinitepowers pegged to the ordinals, expositing maximal information from the universeof sets by naming maximal ordinals. The properties provided by On square nicelywith these features and suggest that direct study of On makes for more direct, butstill essentially the same, set theory.

Consider the setθ = α ∈ On : |α| ≤ ℵ0,

the collection of all finite or countable ordinals. This is a transitive set of ordinals,so θ ∈ On. Therefore θ 6∈ θ. Therefore (the informal argument goes), θ is uncount-able. Taking closure, by recognizing the set of countable cardinals, gives us highercardinals. And this is the same structure we get at the end of On itself, except atthat point the collectivising properties have tapered to a point and there is nowhereleft to go. In fact in chapter 6 we saw that an inclosure is just what Kunen found ofReinhardt cardinals—that for sufficiently large values of κ any extension will bothbe an extension and not. That Kunen concludes there must be no such κ, though,is only the most incredulous reading of data. In mathematics progress is made byasking the right questions; this is how Cantor came to the transfinite.

More, just by observing this structure at On, an inclosure structure, we are ableto infer the lower, more mainstream cardinal powers, too. When in e.g. [Kan94]large cardinals are studied, there is the expectation that these are discrete, in-dependent, and meaningfully different entities, e.g. that a measurable cardinalqualitatively transcends inacessable cardinals. In naive set theory, measurablesqualitatively transcend, too. And an inaccessible cardinal may well be a measur-able cardinal, since all these are initially shown to exist by virtue of the samecardinal, Ω. This makes sense, because large cardinals are, classically, expressionsof faith in consistency. ZF trades information in cardinality; ergo Godel’s proposednew axioms aimed at resolving outstanding question do not posit new properties ofsets, but more limit ordinals. In naive set theory we have reported the same facts,now theorems, by taking the closure of the ordinals and finding repetition in anarray of reflection phenomenon.

Seen in this way, the proof of Zermelo’s wellordering theorem by appeal to afunction into On is wholly appropriate. From Burali-Forti’s contradiction weknow that On itself is wellordered. What the paradox says, when taken seriously,is that the entire structure of the ordinals is recapitulated at the top. Like allordinals, On is itself the set of all preceding ordinals; but unlike other ordinals,which in their very naming insist on further extensions, On can only, and does,


extend into itself. The fact that

On = On ∪ On,

which is not a property of every self-membered set in DLQ but is instead a resultof On being an ordinal (proposition 5.39), makes plain that an injection into Onis an injection into a rich structure. The structure is wellordered because it isa subset of the ordinals. Cantor took wellorder to be a basic correlate of God’somnipotence. The mathematics of naive set theory, in absentia deo, finds that theonly basic assumption one needs is the set concept, and this alone. The intensionalgrasp, the Begriff, of the infinite induces wellorder.

Aside: Not all inconsistent sets are wellordered. Take T = pΦq : Φ, whichcan be seen as the set of all truths. Due to the liar and other dialethias, T 6= T . Butthe subset T is not well founded, since it is not the case that (∀y)(y 6= T ∨y 6∈ T ).

The unifying idea has been to observe the absolutely infinite, replete with itscontradictions, and to write down what can be discerned. Having shown that On isa cardinal, it is a matter of elementary logic—little more than asking the question—to show that there are orders of infinite cardinals: Just existentially generalize onthe discrete sequence

...On < On < ... < On...

Cantor’s diagonal argument, where an inconsistency-insistent subset is named andused to foil any surjection from a to P (a), follows the same principle: On thebasis of inconsistency, discern non-identity: |a| 6= |P (a)|. From the fact that theiridentity is contradictory, take the two sizes to be distinct. In this way Cantor wasable to see a significant aspect of the structure of infinite cardinals. Nothing provenin naive set theory disturbs his theorem, made by reductio; only naive set theoryusing the improved paraconsistent apparatus sees more, and says what it sees:

The tracts beyond the end of the number line are infinite because they areinconsistent.

3.4. Infinity is dialethic; mathematics has heretofore been insistent on consis-tency; therefore closure on infinity is denied and transcendence delivers a tower ofcardinals. Consistent reasoning reads hierarchical structure off an inherently incon-sistent edifice. And naive set theory is also consistent—or at least, all propositionsof the form Φ ∧ ¬Φ are (at least) false. Therefore, naive set theory should be ex-pected to and has produced the same transfinite tower. Cantor’s theorem holds innaive set theory.

It may yet turn out that all transfinite cardinals are identical. If there were aninconsistency in finite arithmetic, for example, as some have speculated based onGodel’s theorem, then in the way we have set them out here, all numbers above


the least inconsistent would both collapse and not. (If n 6= n for n ∈ ω thenn = Ω and so too does every α > n. Models for a collapse in the finite case,at some n = n + 1, are considered in [Pri97], and more generally in [Pri00a];or in [Pri06b], chapter 17.) The implications for such an inconsistency are vast,beginning with the countability of all sets [Pri94]. If sets really are just predicatesin extension, though, then since there are only countably many predicates, thisbegins to make some cohesive sense; a conjecture:

There is a one-to-one function between ω and V .

Throughout we have not proven, because we cannot, that all the transfinite cardi-nals do not collapse back into a single, ineffably infinite singularity. Even though|a| < |Pa|, and therefore |a| 6= |Pa|, still it may be that |a| = |Pa|. Above ω thereare different cardinals, but these cardinals may yet collapse back into one. Scottforesees this day:

I still feel that it ought to be possible to have strong axioms,which would generate ... models as submodels of the universe,but where the universe could be thought of as something ab-solute. Perhaps we could be pushed in the end to say that allsets are countable (and that the continuum is not even a set[i.e. is inconsistent]) when at last all cardinals are absolutelydestroyed.[Bel05](xv)

The generalized continuum hypothesis is, by the Godel-Cohen independenceresult [Coh66], for all intents and purposes a truth-value gap in the classical the-ory. Here in naive set theory we have a straightforward refutation, or at least acounterexample to GCH, at On itself. There we also have an instance where GCHholds. And were it to transpire that all infinite cardinals are identical, then theGCH would eo ipso be true, too. Based on some well-observed dualities betweenthe paraconsistent and the paracomplete, between gluts and gaps [Mor95](chapter13), this might be what we should expect. The GCH is an exciting question forconsistency-based set theory; its answer would still be informative for paraconsis-tent set theory; but once we can see the inconsistent whole apart from its consistentparts, as it were, the emphasis and meaning of the questions change.

3.5. In classical set theory we grow accustomed to forced models in which theaxiom of choice is true, and forced models in which it is not; and so we learn tobe resigned that there is no real answer here. On the question of the size of thelinear continuum—Hilbert’s first problem, on the real points in a familiar interval—we learn from [Coh66] not to press too hard. All of this is simply gone in naivetheory—or, if not gone, then shifted in interest: Now we wonder, if the GCH hasa counterexample at On, whether or not it continues to hold. Because of classical


training it is hard to think of advanced set theory, the independence proofs thatare the centre of most textbooks e.g. [DS96], as not centrally concerned with whatare in fact very idiosyncratic bits of classicality. In naive set theory we have back,in full, the set concept, and have given evidence that the ordinals and cardinals,the core of mathematics, can be built on it. A century of anxiety to the contrarycan fade away. Now there are new worries, about what to prove next, and how theproof will go.

“In many affairs,” Russell reminds us, “it’s a healthy thing now and then tohang a question mark on the things you have long taken for granted.” To the pointhere, the transfinite: Didn’t Cantor see a rich generalization of the natural numberline and demonstrate its discrete, consistent extension into the absolute? Unequiv-ocally, yes—with the caveat that claiming consistency is an overreach. Cantor’stheory was clearly coherent, and the tie up-between coherence and consistency ledmany to assert the latter, even in the face of paradoxes. Now under the paracon-sistent context shift, the meaning of the transfinite changes, exactly because itsstructure remains the same.

A mark of truth is being both surprising and obvious. A paradigm example isTarski’s account of truth itself, which on the one hand seems too banal to be worthstating, but on the other was never stated so clearly before, and has had indelibleimpact since. The brilliance of Tarski’s scheme is not in the answer it provides, somuch as the fact that Tarski thought to ask the question at all. Similarly, Cantorsimply took the actual infinite as a legitimate mathematical object; it is the factthat there is any mathematics there at all, and not so much the mechanics of, say,cardinal arithmetic, that transfixes us. In both Tarski and Cantor’s case, manyformerly perplexing issues are discredited, e.g. the liar sentence is unequivocallygrammatical, Galileo’s paradox is not a contradiction. Many new and more per-plexing issues follow. In the dialethic case, we simply take absolute sets and treatthem as mathematical objects, and as a result lay to rest perplexities about the fullpicture of set theory. And many new and more perplexing issues follow.

Conclusion

Let us conclude by looking at Pandora’s Box, and asking a question.

No sentence in the box is true.

Such a box (conceived of by van Frassen in 1986 and discussed in [HP00](14))contains a sentence both true and false. Now, any language that can express everyproposition can express the Pandora proposition. By classical reckoning, under thelaw of non-contradiction, it follows that, since Pandora is a contradiction, there canbe no such ultimate language. Here we have gone as far as embarking on a closedtheory of sets. Pandora’s box is now open; and as with the eponymous box in themyth, hope remains.

Cantor started us on an intellectual journey. One can peel offat any point, but one should not make a virtue of doing so. ...Pandora’s box is indeed open: Under what conditions should weadmit the extension of a property of transfinite numbers to bea set—or, equivalently, what transfinite numbers are there? Noanswer is final, in the sense that, given any criterion for whatcounts as a set of numbers, ... there would be no grounds fordenying [it] is a set. ... We go on. In the foundations of settheory, Plato’s dialectician, searching for the first principles, willnever go out of business.[Tai00](283 - 4)

The set concept, channelled by the axioms of comprehension and extensionality, isvolatile. How far can we go?

There are many open avenues for further mathematical research. In terms ofpure set theory, I have only put some initial stakes in the ground. Rather thanlist open questions in set theory proper, though, I want to look farther ahead,to the metamathematical questions of consistency, categoricity, and completeness.The pressing need is to devise an operational paraconsistent metatheory; Shapiropoints out that

An important advantage—perhaps the major advantage—of thedialethic program is the possibility of a single, uniform seman-tics. There is no need for a separate meta-language, since the

195

196 CONCLUSION

envisioned language is semantically closed. The language we useto talk about the (object) language is just a part of the very (ob-ject) language we are talking about. ... Just as we do not needto keep running through richer and richer meta-languages in or-der to chase our semantic tails, we also need not keep runningthrough stronger and stronger theories in order to chase Godelsentences. We embrace some contradictions in the semantics,and get it all from the start. Or so says Priest. [Sha02](818)

It is most important for the dialethic program, now that the set theory toolkit isopen, to turn this conjecture into a suite of theorems.

For example, the theorems of chapter 6 are motivated by and easiest to graspthrough the satisfaction relation of model theory, |=. Classical model theory grewup in a climate of paradoxes and is inured in various idiosyncrasies, such as theobject language/metalanguage distinction, which fall afoul of semantic closure andare not parts of the dialethic program. To keep every theorem about |= a theoremof naive set theory (NST ) requires completely recasting model theory so it can berepresented in NST itself—which involves godel coding, faithful representation ofrecursive functions with NST -proofs that the representation really is faithful (as in[BJ89], [Dav82], or [Sho67]), and a working definition of satisfaction. To do thiswith DLQ is a significant task; and while I have made progress on it, there is stillmore to do; at time of writing it is too rough to include here. See [Pri06b], chapter9, for a sketch of what is involved in constructing a semantically closed langauge,there stating an absolute as opposed to model-relative definition of truth with thelogic LPQ.

Among the first tasks one might carry out is a non-triviality (absolute con-sistency) proof for NST , within NST itself. One method would be to emulateBrady’s magisterial construction [Bra06]; once model theory is recast to paracon-sistent standards, we might reproduce a similar, but non-classical, proof in DLQ.There would be, however, no need. Another, simpler approach would be to defineand check the properties of provability and truth predicates, prove soundness withinDLQ, and then argue in one line: Since 0 6= 1, it is not true that 0 = 1, so bysoundness, it is not provable that 0 = 1. And thereby the system shows itself to benon trivial. The novel, autobiographical flavor of this argument perhaps indicatesmore general features of metatheory within NST . Once DLQ can reproduce itsown non-triviality results, what would such a theorem mean?

A question tied to the philosophical issues we’ve been considering, which wouldalso rely on some model theory, is this. A rigorous way to approach the questionof whether NST is the genuine theory of sets would be to determine if NST iscategorical : Does the theory have only one model up to isomorphism? With the

CONCLUSION 197

move to classical first-order model theory came unintended models. Without placingtoo much stock in homophony, we have put the ‘intent’ back into an intensional settheory. It is reasonable to speculate that, as Dedekind proved arithmetic categoricalusing naive set theory, so our theory could be categorical too.

Almost certainly the structure N = 〈V,∈〉 is a model for naive set theory—it isthe intended universe of study. A simple way to get categoricity would be to showthat any other model of NST is isomorphic to N , and then by the transitivity of ∼=,any two models are isomorphic. Indeed, classical theory has, for all the wrong rea-sons, espoused that any inconsistent theory is cateogorical. (The fallacious twofoldreasoning is, first, that inconsistent theories have no models. And then, by the badprinciple that ¬Φ ` Φ → Ψ, follows that all inconsistent models are isomorphic.)But the classical prediction will be immediately false here: NST is not categorical,because an isomorphism is a one-one map preserving structure; sets of differentsizes cannot be in a one-one map; two structures of different cardinality cannot beisomorphic. 〈V,∈〉 is a model of NST ; but

|V | 6= |V |,

because |V | = |On| and |On| 6= |On|. Therefore the canonical model of NST is noteven automorphic (isomorphic to itself), although of course it also is.

Is this just a dialethic trick? Is there still some means to a categoricity result?To a negative answer, there is stronger evidence to be had. An intriguing functiondeveloped to some extent by Quine, which Whitehead suggested be termed anessence (Forster calls these Boffa atoms), is:

E(a) = x : a ∈ x.

With sethood understood as predication, the essence of a is the collection of allproperties borne by a. Almost immediately follows an observation of Boffa—“themost arresting miscellaneous combinatorial fact in NF ...which has the status not somuch of folklore as legend.”[For95](35). Because a ∈ b↔ b ∈ E(a)↔ E(a) ∈ E(b),

a ∈ b↔ E(a) ∈ E(b).

Let E denote the set of all essences. From Boffa’s observation, it would seemthat 〈E ,∈〉 = E is a model of NST . For along with membership, identity is alsomodeled: If a = b then E(a) = E(b) by substitution, and E(a) = E(b) → a = b.

(Suppose a ∈ x ↔ b ∈ x. Since a ∈ a, so too b ∈ a. So a = b.) Thereforethe two atomic term-forming operators of set theory, ∈ and =, are modeled. Thismakes even dialethic categoricity implausible, for the following reasons.

198 CONCLUSION

Since E = E ′′V = E(x) : x ∈ V , there is no ⊂-minimal model of NST . For ifM = 〈M, e〉 is a model of NST , then E ′′M is, too. And because x ∈ E ′′M → x ∈ Eand x ∈M → x ∈M , then E ′′M ⊆M . But E is a proper substructure of M.

As a corollary, it would follow that carrier sets for models of NST need notbe universal. Since E is a carrier for a model of NST , suppose for reductio adabsurdum that (∀x)x ∈ E . Then V ∈ E , so (∃U)E(U) = V . But ∅ ∈ V , implying∅ ∈ E(U), and then by definition, U ∈ ∅, which is absurd. (So this (vaguelydisturbing) result is not only true, but true only, on pain of triviality.) Ergo E is anon-universal model. So much for the intended model, which was the universe.

Say that NST is Ω-categorical iff |M | = Ω, where M is the carrier set for anymodel of NST . Open question: Is NST Ω-categorical?

These would answer questions of non-triviality (‘absolute consistency’) and cat-egoricity. The third and perhaps most exciting possibility is completeness. Given aworking T-scheme, we will have that either Φ is true or ¬Φ is true, since excludedmiddle is part of the logic DLQ. What about provability—for any sentence Φ inthe language of NST , is either Φ provable or ¬Φ provable? Classically, a theoryis complete iff any two of its models make all and only the same sentences true(elementary equivalence). Because there will be no object language/metalanguagedistinction, there is some reason to suspect that, even without categoricity, any twomodels of NST are indeed elementarily equivalent. On the other hand, complete-ness is still a ways off; given sets like

R = x : x ∈ x

then the question of whether or notR ∈ R suggests no means of deciding [BR89](468).Hilbert in his landmark address said:

Is this axiom of the solvability of every problem a peculiaritycharacteristic of mathematical thought alone, or is it possibly ageneral law inherent in the nature of mind, that all questionswhich it asks must be answerable? ... This conviction of thesolvability of every mathematical problem is a powerful incentiveto the worker. We hear within us the perpetual call: There isthe problem. Seek the solution. You can find it by pure reason,for in mathematics there is no ignorabimus...

In investigating Socrates’ schema, the hypothesis was that dialethic paraconsis-tency, particularly as it supports the inclosure schema, offers some natural Archimedeanpoints. And indeed, by evaluating otherwise familiar structures such as the ordi-nals from external fixpoints, such as On itself, many new and powerful theoremshave emerged. The unnameable, “whereof we cannot speak,” has been named andspoken of.

CONCLUSION 199

Yet it has also been seen that absolutely comprehensive sets like V cannotbe inclosure-carriers: anything not in V implies ⊥. Anything inconsistent is Ω-sized, V sized. There is, interestingly enough, an ultimate extent to transcendence,even for dialethic paraconsistency. There is a trivializing limit, absurdity negation−p = p → ⊥, which cannot be dialethic. The world truly is all that is the case:There is a final outpost of reason beyond which is only chaos, the absurd. Whereis the place to stand, to move the world?

We have met universal sets, like U = x : x = x, which can be and are tran-scended by diagonalization. And we have done some work exploring the reflectionproperties of VΩ, discerning structure. So it is reasonable to speculate about apenumbra at the limit, as desired all along: A place to stand that is outside, and soArchimedean, but also inside and so available. The point, structurally, lies some-where between U and V , approached and extended by On. The work done here,the founding of a richer and stronger mathematics of the metaphysical, is a startat understanding this structure. Here is a means to study and stand at this fixedpoint, where none can be, and is.

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Paradox and foundation

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