studying the atlas of all possible polinomial equations with quantifier-prefixes

7/23/2019 Studying the Atlas of All Possible Polinomial Equations With Quantifier-prefixes

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Studying the Atlas of all possible

polynomial equations with quantifier-prefixes

Andrey Bovykin and Michiel De Smet

This version 3.0 is dated December 23, 2010 and will expire on January 10, 2011This draft is not for distribution (yet).


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Abstract

A prefixed polynomial equation (or a polynomial expression with a quantifier-prefix) is an equationP(x1, x2, . . . xn) = 0, where P is a polynomial whose variables x1, x2, . . . xn range over natural numbers, thatis preceded by some quantifiers over some or all of its variablesx1, x2, . . . , xn. Here is a typical generic exampleof such an expression, :

me N ab cd A X xy BCF fghijklnrpq

x (y+ f x) (A + m + f y) [(A + f d)2 + (dg+ g c + A)2 + (B+ h dx)2 + (dxi + i c + B)2+

+(C+ j dy)2 + (dyk + k c + C)2 + (B+ I+ 1 C)2 + (C+ n N)2 + (F+ r b(B+ C2))2+

+(bp(B+ C2) +p a + F)2 + ((F X)2 qe)2] = 0.

In this note we study the collection of all possible such expressions ( the Atlas), with the important equiv-

alence relation of being EFA-provably equivalent on its members. (EFA stands for exponential functionarithmetic and is, simply speaking, a theory where all usual concrete mathematics that does not rely on theuses of genuinely countably unapproximable mathematical notions takes place.) So, members of the same classare prefixed polynomial expressions that are EFA-provably equivalent to each other. The Atlas is partiallyordered by the following relation: the class of an expression A is smaller than the class of an expression B ifEFA proves that B implies A. Here is the first, abstract picture of the Atlas to have in mind:

on this picture, we emphasise the partial ordering by strength

Notice that the set of prefixed polynomial equations is arithmetically complete(i.e. every first-order arithmeti-cal formula is EFA-equivalent to a prefixed polynomial expression). In this sense, the Atlas is just anotherway of talking about first-order arithmetical statements.

First we fix a way of counting the length of prefixed polynomial expressions and notice how deep this prefixed


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polynomial equations template is: there are relatively short members in many non-trivial equivalence classes:we give a series of examples, producing elements of the following classes: 1-consistency ofI1 (the expression at the beginning of this abstract), 1-consistency ofI2, 1-consistency of full Peano Arithmetic, 1-consistencyof predicative mathematics ATR0, 1-consistency of ZFC + {there exists an n-Mahlo cardinal}n. So far wehave only aimed for very rough polynomial expressions, but we hope that later drafts will include more beau-tiful examples than what we have so far. The abundance of examples shows that these equivalence classesoccur surprisingly early in our next picture:

(on this picture, we emphasise lengths of polynomial expressions, the distribution of equivalence classes of a given length and show seeds)

Then we study an example of a phase transition phenomenon. We produce a polynomial expression that hastwo free variables mand n such that if m

n is smaller or equal to Weiermanns constant w 0.6395781750 . . .,

then the expression is EFA-provable, otherwise it is unprovable even by methods of predicative mathematics, forinstance in the theory ATR0(i.e. this is a phase transition between EFA-provability and ATR0-unprovability).

Also, we produce an example of a polynomial equation with quantifiers that is equivalent to the Graph MinorTheorem and hence has strength at least 1-consistency of 11-CA0, but its exact equivalence class is unknown.

In the process of building a point that belongs to the equivalence class of 1-consistency of ZFC + {there existsan n-Mahlo cardinal}n, we produce important technical by-products that will serve as building blocks ofmany future results: a polynomial expression that knows all values of all polynomials and a polynomial ex-pression that knows all values of all BAF-terms (terms in the language {+, , 2x, , log, 0, 1}).

Then we study a phenomenon ofhopping between classes, where a prefixed polynomial expression p(n) thathas a free natural number parameter n, visits many equivalence classes of the Atlas as the parameter n varies.


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(In 1975, J.P.Jones [8] introduced a first example of such an expression that visits all equivalence classes thatcontain a point ConT for some recursively axiomatized theory T.) We start with a presentation of a slightlyenhanced version of Joness theorem (with a slightly better polynomial), and proceed to ...........

Aboutmagic polynomials.

A seed is a prefixed polynomial equation that is of minimal length in its EFA-provable equivalence class. Weraise the first sequence of open questions about seeds. (See seeds on picture 2.)

We discuss the role of the Atlas and its possible partial implementation as a database that would map allprovable equivalence classes of all possible prefixed polynomial expressions of small sizes, and its impact onmathematical logic. We discuss Arithmetical Splitting (the research programme to find statements whosenegation is as good as the statement, without preference), logically inaccessible equivalence classes (first-orderarithmetical phenomena that we cant easily access via any axiomatisation), and the question of classifying un-provable statements (do all unprovable statements have to be equivalent to consistencies or n-consistencies?).

In the potential partial implementation of a fragment of the Atlas, the most important problem at the momentis to find a good normal form to store.

This note is divided into two parts. Part I will contain all results and the full discussion and is intended forreading. Part II, the Appendix, will contain all proofs and can be omitted by most readers.

This incomplete draft is dated December 23, 2010, and contains, perhaps, about one third of the future ma-terial that will be included in the final version. We hope that the final version should be ready by the end ofspring 2011.

This note is intended for the audience of logicians: mathematicians, computer scientists and philosophers alike,as well as thinkers of other backgrounds interested in unprovability. We shall insert a reminder of the basicsof Unprovability Theory at the end of this note.

Andrey Bovykin: [email protected]

Michiel De Smet: [email protected]

The first author would like to thank the John Templeton Foundation for financial support, and for its interestin Unprovability.


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Contents

I Results and discussion 1

1 Introduction 11.1 Five main definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Three possible polynomial templates: overN, Z, Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Diophantine equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Arithmetical completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 On the choice of our background theory EFA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.6 Why this study is now possible . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.7 The language of logic (an extra explanation for non-logicians) . . . . . . . . . . . . . . . . . . . . . . . . 61.8 Examples of equivalence classes and why we care . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.9 How to imagine the Atlas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 First three examples to guide us 92.1 Unprovability by primitive recursive means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Unprovability in two-quantifier-induction arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Unprovability in full Peano Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Going beyond predicative mathematics is not that difficult after all 12

3.1 A coarse polynomial expression equivalent to the Finite Kruskal Theorem . . . . . . . . . . . . . . . . . 123.2 A phase transition polynomial between EFA-provability and predicative unprovability . . . . . . . . . . 133.3 Graph Minor Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.4 Phase transition polynomial equivalent to the planar graph minor threshold . . . . . . . . . . . . . . . . 14

4 Magic polynomials 154.1 Pell equation as a rudimentary example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.2 Superexponentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.3 Ackermannianness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5 Universality and hopping between equivalence classes 165.1 Jones 1978 theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.2 Dyson-Jones-Shepherdson results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.3 More hopping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

6 Polynomials for Nash-Williams theory and Banach spaces 17

7 Values of polynomials and BAF-terms 187.1 A prefixed polynomial equation that knows values of all polynomials on all inputs . . . . . . . . . . . . 187.2 An exp-polynomial expression that knows values of all polynomials on all inputs . . . . . . . . . . . . . 197.3 The polynomial expression for values of BAF-terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197.4 A poly-exp-log expression that knows values of all BAF-terms on all inputs . . . . . . . . . . . . . . . . 217.5 Arbitrary programming languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217.6 About Katies pro ject . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22


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8 Polynomial expressions that cant b e tackled by ZFC and stronger theories 238.1 Friedmans Proposition E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238.2 An intermediate poly-exp-log-expression equivalent to 1-Con(ZFC+ Mahlo cardinals) . . . . . . . . . . . 24

8.3 A polynomial expression that is equivalent to 1-Con(ZFC+Mahlo cardinals) . . . . . . . . . . . . . . . . 258.4 What will be the final length after some optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278.5 A short poly-exp-log expression equivalent to 1-Con(ZFC+ Mahlo cardinals) . . . . . . . . . . . . . . . 27

9 Subtle cardinals 28

10 Seeds 2910.1 Provable and refutable cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3010.2 Some short known open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3010.3 What are seeds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3010.4 Quantifier complexity of seeds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3010.5 R eckless conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3010.6 No reason to expect that our examples are anywhere near seeds . . . . . . . . . . . . . . . . . . . . . . . 3110.7 Magic polynomials versus seeds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

10.8 S eeds in the poly-exp set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3110.9 C ounting what grows from seeds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

11 What use could a fragment of the Atlas be 3211.1 The Atlas as a knowledge database . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3211.2 Building some fragment of the Atlas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3411.3 Seed-size as a new invariant to measure simplicity of logical equivalence classes . . . . . . . . . . . . . . 3411.4 Canonical equivalence classes not easily accessible to humans via axiomatisation . . . . . . . . . . . . . 3411.5 The role of the Atlas in the search for Arithmetical Splitting . . . . . . . . . . . . . . . . . . . . . . . . 3411.6 Atlas in three languages: polynomials, poly-exp-log and Rich Language . . . . . . . . . . . . . . . . . . 3511.7 Almost all arithmetical statements are unprovable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3511.8 The story of Arithmetical Bifurcation formulated in terms of polynomials . . . . . . . . . . . . . . . . . 35

12 How to build a fragment of the Atlas 36

12.1 Feasibility matters: the question of normal form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3612.2 Automatic transformations of polynomial expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3612.3 Proofs get translated into mechanical transformations of polynomials . . . . . . . . . . . . . . . . . . . . 3612.4 Three calculi of polynomial transformations: I0, EFA, richer language . . . . . . . . . . . . . . . . . 37

13 Incomparable consistency strengths 3813.1 Proofs by Montague, Shelah, Solovay and Solovay-Shavrukov . . . . . . . . . . . . . . . . . . . . . . . . 3813.2 A priority argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3813.3 A n infinite splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

14 Finding unusual equivalence classes (like 2-Con(NF)) 39

15 Final remarks 4015.1 M eaninglessnessization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4015.2 M eaningfulnessization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

II Appendix 42

Bibliography 61


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Part I

Results and discussion

1


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Chapter 1

Introduction

1.1 Five main definitionsDefinition 1.A prefixed polynomial equation (or a polynomial expression with a quantifier-prefix, or, occa-sionally, simply a polynomial expression) is an expression of the form

Q1x1 Q2x2 . . . Qnxn P(x1, x2, . . . xn) = 0,

wherePis a polynomial with integer coefficients whose variables x1, x2, . . . xnrange over natural numbers, thatis preceded by a block of quantifiers Q1, Q2, . . . , Qn over its variables x1, x2, . . . , xn. We shall allow Latin andGreek, capital and low-case letters with any subscripts to serve as our variables. It is important to interpretsubtraction correctly (our statement is evaluated over natural number inputs, and outputs are also naturalnumbers).

Definition 2. AtlasThe set of all possible polynomial expressions with quantifiers with quantifiers is called the Atlas.

We shall often refer to this Atlas as a template in the sense 1 that it is the set of all substitution instancesof a concrete polynomial Pand a quantifier-block into one fixed pattern.

Definition 3. EquivalenceWe say that two prefixed polynomial equationsPandQareequivalentor EFA-equivalentor EFA-provablyequivalent if there is a proof ofP Q in the theory EFA, the Exponential Function Arithmetic. This is anequivalence relation on the Atlas.

Definition 4. Length (or size)Let us fix the following method of counting the length of a polynomial expression with quantifiers. We ignorethe quantifier-prefix and the final = 0 and count only the length of the polynomial, as follows: everyoccurrence of a variable or multiplication or addition operation contrubutes 1 to the total size, coefficient ncontributes (n 1) to the total size, +n or nboth contribute n, and the power n contrubutes (n 1) to thetotal size.

This is the only moment in this paper that we make an arbitrary decision: we could of course instead countcoefficients and powers as contributing log2(n) or log10(n) or in other ways. Apart from this arbitrariness, thedefinitions and objects of this paper are very absolute.

1we borrowed the word template from Harvey Friedman and his explanations presenting Boolean Relation Theory as thestudy of a template

1


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Definition 5. Seed graA polynomial expression with quantifier-prefix that is shortest in its EFA-equivalence class is called a seed ofits equivalence class. There may occasionally exist several seeds of the same equivalence class of the same size.

The Atlas is not really a new object. Some people in the XXth century called the set of equivelence classesof closed formulas in any language under T-provability the Lindenbaum-Tarski algebra ofT. However, thispaper, to our knowledge, is the first systematic attempt to study the whole arithmetically-complete Atlas asone mathematical object.

1.2 Three possible polynomial templates: over N, Z, Q

There are at least three standard natural interesting templates involving polynomials with quantifiers whichwe could use in this project:

1. to let variables range over natural numbers (non-negative integers);

2. to let variables range over all integers;

3. to let variables range over all rationals.

All three set-ups are equally interesting mathematically. Of course the polynomials with quantifiers behavecompletely differently when the variables are allowed to range over these different sets. In building examples andproving various theorems, we can expect advantages and disadvantages of each of these templates and trade-offsbetween them. We chose to take template (1) and explore it thoroughly, but the other two are also excellenttemplates and can afterwards be dealt with, building on the experience of study of template (1). We chosetemplate (1) because it is comfortable and usual for logicians. Template (3) will be a shoot-off of a separateproject about Unprovability Theory in The Language of Rational Numbers (not necessarily polynomials, butmixtures of rational numbers and natural numbers are allowed). By Julia Robinsons theorem about rationals

[20], template (3) is arithmetically complete so there is some guarantee of success.

1.3 Diophantine equations

One important template was studied in the 1950s - 1970s that led to the solution of Hilberts 10th problem:the collection of all possible Diophantine equations.

Solvability of Diophantine equations (the set of all sentences of the form x1x2 . . . xn P(x1, x2, . . . , xn) = 0)is a 01-complete set of setences. It means that every

01 formula in the language of first-order arithmetic

is EFA-provably equivalent to a sentence from this set. Equivalently (and somewhat psychologically moreimportantly), every 01 sentence of arithmetic is EFA-provably (see [5]) equivalent to non-solvability of acertain Diophantine equation.

Should we type some brief history of MDRP or refer readers to classical accounts?

However, the restriction of having only one block of quantifiers makes this template metamathematicallyboring: it seems we cant encounter and study interesting metamathematical phenomena with this restrictionin place. We can do some good metamathematics if we dont have to be restricted by it.

Throughout the project we use many big and small methods and tricks developed by the community of peoplewho studied Hilberts 10th problem and related topics in the 1950s -1980s.


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1.4 Arithmetical completeness

Non-solvability of Diophantine equations is 01-complete. Similarly, our Atlas has a complteness property:

every first-order arithmetical formula is EFA-equivalent to a prefixed polynomial equation.

For example, the following prefixed polynomial expression:

...

...

is EFA-provably equivalent to . . ..

So, the Atlas of all prefixed polynomial equations is just another way of talking about formulas in the languageof first-order arithmetic. You can think of polynomial expressions as normal forms of first-order formulas.

1.5 On the choice of our background theory EFA

Ideas evolved for a centuryto rto msub

The search for the right background foundational theory for mathematics lasted for about a century, evolvingthrough several stages of understanding. It went from an initial naive desire to build a rigorous foundation formathematics through several generations of logical controversies before reaching the modern understandingof the roles of expressive strength, unprovability and logical strength and their relation to mathematics ofthe time, and, more controversially, mathematics of the future. Nowadays, the answer is usually EFA (orits variants I0+ exp or EA) or I1 (or its variants WKL0, RCA0 or PRA). Let us explain, very brieflyand crudely, why the answer is not ZFC or NF or Z2 or I0. Half of this discussion interlaps with Avigadsexplanations in [1], but not all.

Two early attempts: set theories and type theories

In the period between 1900 and 1931, two initial directions of thought emerged, both with the idea of buildinga foundation of mathematics, as they saw it at the time: the type-theoretic theories like that ofPrincipiaMathematica, the simple theory of types, etc and things that grew from them. On the other hand, anothergroup of thinkers was developing set theory, a series of theories and ideas building on the reckage of Cantorsinconsistent set theory, with the idea of prohibiting the Comprehension Scheme and substituting it by theextremely minimalist Separation Scheme and cummulative hierarchy. What is important to note at this stageis that both approches successfully showed how usual mathematics of that time can be embedded into theseset-ups. Both expressive toolsof these theories and their power to mimick usual mathematical arguments (i.e.strength) are absolutely adequate to imitate (or serve as foundation for) the fragment of mathematics thatexisted in early XXth century. So, if the goal was to build somefoundation of mathematics (as opposed to,for example, a possible goal of finding the true mathematics) then both approaches succeeded. It is also clearwhat are some of the the trade-offs they chose (the set-theoretic approach introduced a wealth of non-existent

entities (zfcsets, different infinities, etc) but gained quite some strength while type-theoretic approaches didntsink into non-existent notions but reached smaller strength).

Set theory fitted well with abstract mathematics somehow, but do you really needstrength?

With the rise of abstract mathematics, the idea of inevitability of set theory was somehow taking routeand most set-theorists of that time chose ZFC or ZFC+ extensions as the golden standard for mathematicalrigour. Bourbaki took a much weaker theory (see [12], [13]). Later some intermediate set theories were tried,for example MAC (see [14]). All these set theories are stronger or weaker manifestations of the same idea of


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minimising Comprehension, and producing a transfinite hierarchy by iterating set-generation according tocertain restricted rules along internal well-founded linear orderings. It feels very strange to read thinkers ofthat time since so many things changed our view of mathematics since then. (The biggest problem is thatthere is no proof, only opinions, why we should prefer those theories (like ZFC+ extensions) over billions offuture set-ups that will give us very different picture of mathematics.)

What is left of type-theoretic approaches today

Type-theoretic approaches (simple type theory, Ramseys type theory, NF and the several new set-ups usedby computer scientists in the emerging field of automated theorem proving and Mathematical Knowledgegeneration and storage) still exist and we shall mention them again later.

Arithmetization Revolution

Godel and Tarski in the early 1930s made a crucial discovery: arithmetisation of mathematics. It later tookseveral generations of logicians to pinpoint how it really works in terms of expressive means and in termsof strength. First it was arithmetisation of syntax and of computability in the 1930s (you can talk aboutlanguage, provability and computability in the language of first-order arithmetic). The important word hereis coding. Then Alan Turing, followed by Markovs school of constructive mathematics, introduced the ideaof arithmetising calculus, algebra, topology, etc. (A lot of this work was based on the initial, somewhat naiveand non-exact ideas of Brouwer 20-30 years before them.) So, in order to say sinx

x x0 1, you no longer

need to claim that you embed your notions in the language of set theory, and imitate the usual proof usingset-theoretic axioms. Instead, you give the encoded definitions in the language of arithmetic, and prove yourtheorem, perhaps, by mathematical induction. The new wave arithmetisation was completed by the Markov-Shanin school, where they showed that you can say even more complex statements of separable mathematicsin the language of arithmetic. 2 So, the language of arithmetic was very quickly shown to have adequateexpressive means to express all that is associated with separable or concrete mathematics.

First-order, second-order, third-order language of arithmeticLogical strength

Now that the question of expressive means is so much more settled, comes the question of strength. It wasgradually realized how strong the axioms of PA are (these were held, by logicians, as the golden standard ofmathematical reasoning until mid-1970s). Soon, PA gave way to I1 (see, for example [16]) and, currentlylogical opinion usually oscillates between I1 and EFA as the theory where usual mathematics takes place.

Shouldnt we choose a very strong theory?

Should we go for a very strong arithmetical theory as our background theory? How about the set of first-order arithmetical consequences of ZFC+ there is a huge cardinal? Or the set of first-order arithmeticalconsequences of Z2? Even leaving aside the question of possible inconsistency of strongest known theories

3,

here are the two main objections against fixing a very strong theory as the background theory.

1. Arbitrariness.Why this strong theory and not another one? What if they contradict each other on anarithmetical statement? And even if we convince ourselves that a certain theory gives us true arithmeti-cal statements, maybe the next generation of mathematicians will invent a billion better theories, andbetter reasons, to prefer a theory that contradicts our theory. See a discussion of the research programmeof Arithmetical Splitting in [4].

2At that time, they were also extremely sensitive to the logic used in the proofs of these results, and the many distinctionsyou obtain in the absense of the Law of Excluded Middle. We shall be blind to these distinctions in this paper.

3we shall ignore this question throughout the paper


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2. Blindness to important distinctions. A strong theory dramatically factorizes the Atlas, hiding themost important distinctions. From the point of view of ZFC+ there is a huge cardinal, the Paris-Harrington Principle, 7-consistency of ZFC+ there are infinitely-many measurable cardinals and 0 = 0are equivalent. We dont want this level of blindness because we know this is the kind of distinctions wecare about.

Example of Ramsey Theorem and the Hales-Jewett theorem

How about another extremely strong theory I1? This is a more reasonable candidate. But from the pointof view ofI1, the Finite Ramsey Theorem, the Hales-Jewett Theorem and the Prime Number Theorem areequivalent statements. However, mathematicians sense that these distinctions are important. Erdos-Radoupper and lower superexponential bounds for the Finite Ramsey Theorem are a signal that this statementhas a somewhat different character from what is done in, say, analytic combinatorics. The Hales-Jewetttheorem, whose first proofs, alongside van der Waerdens theorem, involved the Ackermann function (andhence were beyond the reach ofI1) is on yet another, higher level of logical complexity and was even for awhile conjectured to be I1-unprovable until Shelah [21] found the unusual lower and upper bound in terms of che

lowbou

the iterated tower (or tower of towers) functions. These are important mathematical distinctions, whichwe would like to make, this is why we shall go below I1, namely to EFA.

Shouldnt we choose a very weak theory?

How about fixing a very weak theory, below EFA, for example Robinsons Q or polynomial-function arithmeticPFA or I0? Or perhaps one of the theories studied by the subject of weak arithmetics, like I0+x

logx

exists? Here, the objection is:

- Irrelevance of distinctions. In the theories that describe mathematics without exponentiation, lotsof things become unavailable: you cant arithmetize well, you cant deal with the syntax well, there aremany things that are simply missing. Some of these theories cant prove commutativity of addition. Wecould of course decide to fix one of these theories to be our ground theory. Then the equivalence classes of

our Atlas would explode in number, by dividing each equivalence class into neighbouring distinct chunks,whose equivalence that weak theory cannot prove, because some common everyday things are missing(for example inability to concatenate two arbitrary words). We understand that in some situationsthese distinctions matter, for example in the interrelation between weak arithmetics and computationalcomplexity. Also in Sazonov-style philosophy of computer science. We chose, like most of the rest ofmathematics, to ignore these distinctions and accept totality of exponentiation.

Summary

In summary: choosing a background foundational theory for mathematics, and for the Atlas, we could, reason-ably, choose any theory between EFA andI1. We decided to chose the most sensitive one, EFA. IncidentallyEFA happens to be an exremely powerful foundational theory that formalises all usual concrete mathematics.H. Friedman says Probably every theorem published in Annals of Mathematics whose statement involves

only finitary mathematical objects (i.e., an arithmetical statement) can be proved in EFA. More discussionof EFA can be found in [1].

Lets say it again: the reason for our choice was based on whether the unprovabilities of the theory signal thatthere is an important mathematical phenomenon that governs that unprovability or whether these unprovabil-ities are there simply because the theory is a silly one that is not equipped with tools for normal backgroundreasoning.


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6

1.6 Why this study is now possible

Why has the project become possible now in 2010? It is the extent of development of modern Unprovability

Theory: from the Paris-Harrington Principle and Kruskals theorem to Harvey Friedmans Boolean RelationTheory and beyond. It is important to understand that the theoretical possibility of studying the Atlas was(or could have been) understood a long time ago, but modern Unprovability Theory didnt exist yet, so thetools were not available.

1.7 The language of logic (an extra explanation for non-logicians)

It is widely known and felt that mathematical theorems are not all equivalent to each other, that there areissues of relative strength and unprovability there. The currently most advanced way of speaking about thesemetamathematical phenomena comes from logic. For readers not fluent in modern logic, we are including thissection with extra explanations, to illustrate some important points.

Theorem 1. Over EFA,- the finite Ramsey Theorem for allm, n, c there is Nsuch that for every colouring ofn-element subsets

of{0, 1, 2, . . . , N 1} into c colours, there exists a homogeneous subset of size m;

- 1-Con(EFA), i.e. the arithmetised sentence 1(PrEFA() ), whereis a provability formula;

- the statement superexponentiation is a total function

are equivalent. Neither of them is EFA-provable.

Proof. We couldnt find any reference in literature, so we provide a simple model-theoretic proof in the Ap-pendix.

So, this mathematical rigid phenomenon has different names, which are interchageable, depending on

the context. The usual mathematics, conducted in EFA will only yield functions that are computable intime bounded by some fixed power of exponents. 19th century mathematics never left that class, and the firstescapes from it happened only in early XXth century4 When your mathematics falls into this new, higher EFA-equivalence class, you notice that something slightly unusual happens: you would occasionally have to hit withan unexpected argument that will produce a superexponential lower bound. In model-theoretic constructions,you will see the statement 1-Con(EFA) necessarly used, as a big weapon to deal with the situation. So, themain point we make here is: there is no need to be afraid of logical names for metamathematically-interestingcombinatorial phenomena. There is usually a combinatorial name for it just round the corner. (1-Con(EFA)is one of the lowers classes, so the explanation with superexponentiation is clear and simple. When you gohigher - explanations of what this phenomenon really is become more complex.) What is important for usis the thing that carries this strength invariant: the EFA-equivalence class. We shall not dwell on particularexamples manifesting a phenomenon, but will usually refer to the class (often under its logical name).

On lower bounds for van der Waerden theorem and Hales-Jewett, as another example. Text will be here.

Here is a picture (from [4]) of some of the logical theories we may mention later, ordered by the followingrelation: a theoryT1is below another a theory T2(we shall say has smaller arithmetical strength) if the set offirst-order arithmetical consequences ofT1is a proper subset of the set of first-order arithmetical consequencesofT2. So, ZF and ZFC are denoted by the same point (they have the same arithmetical consequences). We putZFC-branch, NF-branch and PST-branch apart because it is not known whether any two of them contradicteach other on arithmetical formulas.

4historical question: apart from Hardys text with Hardy hierarchy, have there been concrete examples before Ramsey Theorem?


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7

Notice that we do not care about such issues as whether a theory already has a neat axiomatisation orwhether its axioms have been justified. (Everything that has been justified and accepted falls below thepreference horizon). What is most important is the theorys set of first-order arithmetical consequences, itsarithmetical fragment.

1.8 Examples of equivalence classes and why we care

Let us give a few basic examples of EFA-equivalence classes, to get the first view of the complexity of theobject (the Atlas) we are facing. We shall not be assuming any philosophy in this study. In particular we shallnever assume or imply that prefixed polynomial equations are somehow divided into true and false ones.

Our main aim in this study is to approach the Atlas as mathematicians, and scientists, and to discover whatis really going on.

Another (somewhat marginal) aim here is to develop a new language or a new way of looking at the Atlasthat is different from the 20th century realist explanations about true but unprovable statements. Weare craving for a more adequate way of thinking, suited for the real purposes of modern metamathematics.However, this aim, of catching the adequate way of looking at things, is marginal here because we dont thinkthat anyone can really accomplish it at this stage in the history of logic, before a certain hard mathematicaldiscovery (arithmetical splitting) has been made.

Here are some examples of differentequivalence classes in our Atlas:


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8

- theorems of EFA;

- negations of theorems of EFA;

- totality of the fifth branch of the Ackermann function;

- van der Waerden Theorem;

- 3-Con(ZFC);

- 17-Con(I2);

Notice that the last two examples are incomparable classes. The statement 3-Con(ZFC) can, over EFA, implyonly new 04 arithmetical formulas, so not the

018formula 17-Con(I2). The statement 17-Con(I2) implies

all 017 consequences ofI2, but not some simple consequences of ZFC, like the Paris-Harrington Principle.For the same reason Con(ZFC) is of no help in proving Ramsey Theorem.

Here is another example of incomparable classes: this time each of them will contain a 01 formula. Let T1

and T2 be such that EFA + Con(T1) doesnt prove Con(T2) and EFA + Con(T2) doesnt prove Con(T1) (seefor example [22]). Then clearly Con(T1) and Con(T2) belong to incomparable classes.

Yet further examples will arise from Arithmetical Splitting: once two good theories T1 and T2 explicitlycontradict each other on an arithmetical statement then and belong to very interesting equivalenceclasses. . exp

1.9 How to imagine the Atlas

Every EFA-equivalence class can be thought of as surrounded by a cluster of other classes that EFA fails toidentify, but I1 does. Then come the further classes such that PA identifies them with our class, etc. Thestronger theory we use, the more blind it will be to the important metamathematical differences we care about.But also, keep in mind the disconnectedness of the Atlas: if and witness Arithmetical Splitting thenthere will be no theory to ever believe they belong to the same class, so clusters dont eventually all fusetogether.


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Chapter 2

First three examples to guide us

Let us start by briefly glancing at three particular points in our Atlas. All three points belong to differentequivalence classes. It is important to note at this stage: the three expressions below can be re-written into(or proved to be EFA-equivalent to) billions of other polynomial expressions of comparable lengths, so there isno need to concentrate attention on these particular expressions, but think of them as a mere representativesof their equivalence classes.

2.1 Unprovability by primitive recursive means

Here is a first element we come up with.

Theorem 2. For everye > 2, the following statement 1(e) is equivalent to 1-consistency ofI1 and henceis unprovable in I1.

m N ab cd A X xy BCF fghijklnrpq

x(y +fx)(A+m+fy)[(A+fd)2 +(dg +g c+A)2 +(B +hdx)2 +(dxi+ic+B)2+

+(C+j dy)2 + (dyk + k c + C)2 + (B + I+ 1 C)2 + (C+ n N)2 + (F+ r b(B + C2))2+

+(bp(B+C2) +p a+F)2 + ((F X)2 qe)2] = 0.

Proof.

See Appendix for the full proof. The proof goes by demonstrating equivalence with PH2e, the Paris-HarringtonPrinciple for pairs and e colours, so for every concrete e >2, this statement is unprovable in I1.

This statement seems to have quantifier complexity 0

6. However, the last four blocks of quantifiers can bebounded by some exponential expressions, which can be struggled with and eliminated using the methods from[11], so the formula is equivalent to a 02 formula). We didnt do any of it because it would blow up the sizeof the resulting polynomial.

With our method of counting length (see page 1), the polynomial above has size 131. The number of variablesis 25 (or 24 if you substitute e by 3).

Here, there are probably many possibilities to simplify the polynomial, perhaps to half of the current size, byre-using variables and clever combinatorial equivalences during the proof. Also the choice of coding tricks andthe way to arrange the colouring can transform this polynomial. No attempts have been made yet to simplify

9


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CHAPTER 2. FIRST THREE EXAMPLES TO GUIDE US 10

this raw polynomial expression. In this sense this was a very naive attempt. And there are no reasons to thinkthat we are anywhere near the shortest member of this equivalence class.

It may be possible to have a much shorter and simpler expression equivalent to the expression above if we useadditional means: exponentiation or both exponentiation and logarithm. It is interesting to see how muchbetter we can do with these extra means.

2.2 Unprovability in two-quantifier-induction arithmetic

Theorem 3. For each number e > 1, the following statement 2(e) is equivalent to 1-consistency ofI2, andthus is not provable in I2.

m N ab cd A X xyz uvw BCDGH fghijklnpqrst F G

x(y+Bx)(z+By)(A+m+Bz)[(A+fd)2+(dg+gc+A)2+(B+hdx)2+(dxi+ic+B)2+

+(C+jdy)2+(dyk+kc+C)2+(D+ldz)2+(dzn+nc+D)2+(B+F+1C)2+(C+p+1D)2+

+(D + q N)2 + (H+ r bt)2 + (bts +s a + H)2 + (B +C2 +D3 t)2 +((X H)2 Ge)2] = 0.

Proof.

For full proof see the Appendix. Again, we show that for everye, the statement above is equivalent to PH3e,the Paris-Harrington Principle for triples and e colours. Already for e = 2, the statement is equivalent to1-consistency ofI2.

The statement may look like 06 but is actually EFA-equivalent to a (much longer) 02 formula. The

statement has one free variable e (the number of colours) and 34 bound variables. With our method ofcounting, the polynomial has size 174. Again, this is a somewhat naive first attempt. We are sure that

by more delicate method we can find an example half this size. What is the possible way to use the extraexpressivity if we have exponentiation and logarithm available?

2.3 Unprovability in full Peano Arithmetic

Theorem 4. Consider the following statement (n, e) with two parameters n and e. For everyn > 1, e > 2,the statement is equivalent to 1-consistency of In1. So, with the quantifier prefix ne, this statement isequivalent to 1-consistency of Peano Arithmetic and hence is unprovable in Peano Arithmetic.

m N ab cd xy zwit fg hklpqruv X ABCDEF j GI sH

[i (n + f+ 1 i) ((g + f yi)2 + (yih + h x + g)2 + (g + l wi)2 + (wik + k z + g)2)+((p + q b(x2 + y))2+

+(b(x2+y)r+ra+p)2+((b(z2+w)j +j a+p)2s1)(pb(z2+w)1s))2][(z+xd)2+(yd+yc+z)2+

+(t(z+m+ft)((g+ldt)2+(dtk+kc+g)2+(h+pd(t+1))2+(d(t+1)q+qc+h)2+(g+r+1h)2+(g+sN)2))+

+((u+1X)2+(X+vn)2+(A+CX)2+(X D+D+A)2+(B+E(X+1))2+((X+1)F+F+B)2+

+((A dG 1 H) ((dGI+ Ic +A)2H 1) G (z +m+ H G) (B + H+ 1 A))2) ((u+ A b(2+ ))2+

+(b(2 + )B+ B a + u)2 + ((u w)2 ve)2)] = 0.


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CHAPTER 2. FIRST THREE EXAMPLES TO GUIDE US 11

Proof.

For full proof see the Appendix. We prove that for all n e the expression above is equivalent to the Paris-Harrington principle in dimension n and number of colours e.

Again this is a somewhat naive theorem, without fine-tuning or clever tricks, and again we expect much simplerpolynomials, half the current size, to be quickly achieved using extra tricks. Again, all quantifiers after thefirst two blocks of quantifiers can be made bounded by some exponential functions, and the famous battleagainst the bounded quantifier (Chapter 6 of [11]) can reduce the statement to its true 02 shape, although atthe cost of losing the current small size. In the current formulation, we have 2 free variables n and e and 36bound variables. The above polynomial is of size 352.

Corollary 5.For every n, there is a prefixed polynomial equation of length 352 + 2n that is equivalent to 1-Con(In).

So, the sizes of seeds of 1-Con(In) are bounded by a linear function ofn. We can try to think of some

pigeonhole argument now, for the next step: perhaps for all even (or odd?) nstarting from some point there isa seed of some 1-Con(Im) of size n (unless they all have to be divisible by 4, etc...) We want some regularityor order here.


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Chapter 3

Going beyond predicative mathematicsis not that difficult after all

The readers could have thought for a moment that the three relatively neat examples from Chapter 2 are dueto pure luck and that it is much harder to reach high impredicative equivalence classes. We also thought thatfor a while until proving the theorems of this chapter.

3.1 A coarse polynomial expression equivalent to the Finite KruskalTheorem

Theorem 6. The following polynomial equation with quantifiers K is equivalent to Finite Kruskal Theoremand hence is unprovable in predicative mathematics, for example in the theory ATR0:

K Mab ijcdefhk lmnpq A grst BFGIJLOPQW XY Z uvxyz CDHN TERS U V[(ic1)2+(i+dM)2+(w+1t)2+(t+Xq)2+(g+1s)2+(s+Y+1r)2+(r+Zq)2+((p+l2bi1B)(l+Bp)

((biA+Aa+p+l2)2B1)((K+iq)2B1)(upr1E)((prC+Cl+u)2E1)(vps1E)((psD+Dl+v)2E

(xpt1E) ((ptH+ H l +x)2E1) (u+Ev) v (q+E+1z) (((vNu)2E1)2 +(vRx)2+(uSx)2)

(ypz1E)((pzT+Tl+y)2E1)((ERu)2+(ESv)2+((EUy)2V1)2))2][mni(mn)(K+i+r+1m)

(K+j+r+1m)(j+ri)(M+r+1j)((f+e2+rbi)2+(bis+sa+f+e2)2+(k+h2+tbj)2+(bjW+Wa+k+h2)2+

+(k+X+1h)2+(F+Yf m)2+(f mZ+Ze+F)2+(F+F2G2+gdm)2+(dmB+Bc+F+F2G2)2+(kOR+Rh+G)2+

+(G+EkO)2+(S+1OIPQ(ef))2+(O+VKj)2+(J+f n)2+(f n+e+J)2+(dn+c+J+J2L2)2+

+(J+J2L2+dn)2+((LG)21)2+(P+P2Q2+dI)2+(dI+c+P+P2Q2)2+(I+Ki)2+(P F)2+(P J)2+

+(P F + J)2 + (Q G)2 + (Q L)2 + (Q G+ L)2)] = 0.

This polynomial has size 648. It has 66 variables.

Discussion and commentary will be here.

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CHAPTER 3. GOING BEYOND PREDICATIVE MATHEMATICS IS NOT THAT DIFFICULT AFTER ALL13

3.2 A phase transition polynomial between EFA-provability andpredicative unprovability

Consider the following quantified polynomial equation A(m, n), with the two free variables m and n, whichwe show in bold font:

KMab ijcdefhk lmnpq A grst BFGIJLOP QWXY Z

uvxyz CDHN T E RS klmnop U V

[(((i)21)(()21))2+(((j)21)(()21))2][(2+2)((1+)2+(++1)2+

+(+)2+(+++)2+(+k+1i)2((+k)

2+(l+l+)2+(2+m)

2+

+(n+n+n+2)2)+(+o)

2+(+p+12)2))]+[(ic1)2+(i+dM)2+(w+1t)2+(t+Xq)2+(g+1s)2+

+(s+Y+1r)2+(r+Zq)2+((p+l2bi1B)(l+Bp)((biA+Aa+p+l2)2B1)((mK+nmq)2B1)(upr1E)

((prC+Cl+u)2E1)(vps1E)((psD+Dl+v)2E1)(xpt1E)((ptH+Hl+x)2E1)(u+Ev)(q+E+1

v(((vNu)2E1)2+(vRx)2+(uSx)2)(ypz1E)((pzT+Tl+y)2E1)((ERu)2+((EUy)2V1)2+

+(ESv)2))2][mni(mn)(mK+n+r+1mm)(mK+n+r+1mm)(j +ri)(M+r+1j)((f+e2+rbi)2+

+(bis+sa+f+e2)2+(k+h2+tbj)2+(bjW+Wa+k+h2)2+(k+X+1h)2+(f mZ+Ze+F)2+(F+F2G2+gdm)2+

+(F+Yf m)2+(dmB+Bc+F+F2G2)2+(kOR+Rh+G)2+(G+EkO)2+(S+1OIPQ(ef))2+(mO+VmKn)2+

(J+f n)2+(f n+e+J)2+(dn+c+J+J2L2)2+(J+J2L2+dn)2+(dI+c+P+P2Q2)2+(P+P2Q2+dI)2+

+((L G)2 1)2 + (mI+ mK n)2 + (P F)2 + (P J)2 + (P F + J)2 + (Q G)2 + (Q L)2 +(Q G+ L)2)] = 0.

Theorem 7. There exists a real number w such that:

1. if nm

w then I0+ exp proves A(m, n);

2. if n

m > w then ATR0 does not prove A(m, n).

The numberwis the real number introduced by Andreas Weiermann in [25] and is defined as follows: w= 1log

,

where is Otters tree constant (the inverse of the radius of convergence of the generating series for unorderedtrees),w 0.6395781750 . . . .. The number w is of course primitive recursively computable.

This theorem is a new type of result within Andreas Weiermanns Programme of phase transitions betweenprovability and unprovability.

More discussion and commentary will be here.

This story sounded like science fiction in summer 2009 (and we saw disbelief in peoples eyes) but now this isjust another theorem.


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CHAPTER 3. GOING BEYOND PREDICATIVE MATHEMATICS IS NOT THAT DIFFICULT AFTER ALL14

3.3 Graph Minor Theorem

Consider the following polynomial expression G Mwith a quantifier-prefix:

K N ab ij def ghm no xy ABC u klpz Y cqrvw DEF GHI s t L J M X OPQ

RWZ ST UV x1x2x3 y3y4 x4x5x6x7x8x9 y1y2

[(i+u+1N)2+((k+k2l2+k3l3p3bib1D)((biY+bY+Ya+k+k2l2+k3l3p3)2D1)(K+i+Dp)((D+2v)2+

+(((wc)2E1)2+((wq)2F1)2+((wr)2G1)2)((w+E+1p)2+(s+Flwl)2+(lwG+lG+Gk+s)2+

+(vHs)2)+((cq)2I1)2+((qr)2J1)2+((rc)2M1)2)2][(i+u+1j)2+(d+d2e2+d3e3f3+bib)2+

+(j++1N)2+(bi+b+a+d2e2+d3e3f3)2+(g+g2h2+g3h3m3+bjb)2+(bj+b+a+g+g2h2+g3h3m3)2+

+((p1)p((zk)2(zl)2+((f++1z)(qeze1)((ez+e+d+q)21)((q)21))2)((+2r)2+

+(v+yky)2+(yk+y+x+w)2+(w+yly)2+(yl+y+x+w)2+(((sv)21)2+((sw)21)2)

((s + + 1 f)2 + (r t)2 + (t + os s)2 + (os+ o+ n + t)2))2 + ((C+ k) ((BL + B + A + J)2+

+(J+BLB)2+(((Lk)21)(D+D2E2+D3E3F3J))2+(((Lk1)21)(G+G2H2+G3H3I3J))2+

+(((L1)21)(g+g2h2+g3h3m3J))2+(((LG)21)(n+n2o2+n3o3f3J))2+(+1M)2+(P+EM)2+

+(EM+D+P)2+(U+HR)2+(HR+G+U)2+(S+ER)2+(ER+D+S)2+(T+ERE)2+

+(ER + D + T)2 + (R (I+ x1+ 1R) ((x1+ 2O)2 + (M+ x2+ 1X)

2 + (X+ x3F)2 + (Q + x4EX)

2+

+(EX x5+x5D +Q)2+(((IF+1)2+(Ox6P)

2+(Ox7Q)2+(((RM)2+(UP Q)2)((X+x8R)

2+(UT)2)

((R+x8X)2+(US)2+((RM)2x91)

2))((IF)2+(Ox6P)2+(Ox7Q)

2+(OP x8+Qx9)2+((RM)2+

+(UV)2 + (OV Q)2) ((RX)2 +(UV)2 + (OV Q)2) (((RX)2y11)2 + ((RM)2y21)

2 + (US)2)))

((M+x1F)2+(IF+1)2+(+x2EW)

2+(EW x3+x3D+)2+(((Zy3P)

2x41)((Zy4)2x41)(MW)

(Ix4+1W)W(Z1))2+((R+x5+1M)

2+(US)2)((M+x5R)2+(UT)2))((IF)2+(US)2)2)2] = 0.

Theorem 8. The statement GM is equivalent to the finite Graph Minor Theorem, and hence is unprovablein at least 11-CA0.

This rough polynomial expression has size1067.

Discussion and commentary will be here.

3.4 Phase transition polynomial equivalent to the planar graph mi-nor threshold

Will be added here.

This is a variation of the above polynomial for graph minors, based on the article [2].

There is no exact phase transition result for full Graph Minor Theorem, but there is a neat threshold resultfor the much smaller (exponential) class of planar graphs, with the constant 1

log2 separating provable and

unprovable instances of the Graph Minor Theorem restricted to planar graphs. Here is a classical constant,the planar graph constant from the graph enumeration theory, 29.06<


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Chapter 4

Magic polynomials

4.1 Pell equation as a rudimentary example4.2 Superexponentiation

4.3 Ackermannianness

15


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Chapter 5

Universality and hopping betweenequivalence classes

5.1 Jones 1978 theorem

5.2 Dyson-Jones-Shepherdson results

5.3 More hopping

16


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Chapter 6

Polynomials for Nash-Williams theoryand Banach spaces

This section will be filled during the spring and summer of 2010.

Galvins Lemma and Nash-Williams Theorem

Formulations and explanations of the new Nash-Williams-style unprovable statements, which we got from newindiscernibles, will be here.

Galvin-Prikry theorem

Gowers Theorem and its polynomialAbout the Polynomial Footprint of all infinitary statements

Capturing the meaning and the strength of an infinitary statement via its first-order version (the core)and then finding the polynomial expression for this first-order version to act as the invariant signature orfootprint of the original second-order statement. This will be the story of the polynomial DNA of everyinfinitary statement.

17


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Chapter 7

Values of polynomials and BAF-terms

7.1 A prefixed polynomial equation that knows values of all poly-nomials on all inputs

We will represent a polynomial by a four-tuple of numbers a, b, c, d, where a, b, c code, by Godel-coding, asequence of length c of some triples, each of which codes a finite sequence that represents a monomial of ourpolynomial. The first element of this sequence will be the coefficient of the monomial. The rest of the elementsdetermine which variables occur in the monomial. For example, 4x31x2x

27= 4x1x1x1x2x7x7 can be represented

by (4, 1, 1, 1, 2, 7, 7), or by (4, 7, 1, 2, 7, 1, 1), or by any other permutation that starts with the 4. The numberc is the total number of monomials and d is the number of positive monomials among them, so 0 d c.The input of each polynomial will be a sequence (x1, x2, . . . . . .) Godel-coded by a pair x, y (there is no needto fix the length of the input).

We are interested in the relation the polynomial coded by a, b, c, d, when fed the input coded by x, yreturns the value w. This relation will allow us to quantify over all polynomials and to quantify over all

computably enumerable sets (since they are sets of values of polynomials).

Consider the following polynomial equation (a, b, c, d, x, y, w), with free variables a, b, c, d, x, y, w, whichwe showed in bold:

st uv i efg qr h j klmnop ABCDEFGHIJMN OPQRSTU V WX

[i (c + A + 1 i) j (g + Aj) ((e + e2f2 + e3f3g3 + Abi)2 + (biB + B a + e + e2f2 + e3f3g3)2 + (h + C vi)2+

+(viD + Du+ h)2 +(k + Erj)2+(rjF+ Fq+ k)2 +(l + Grj r)2 +(rjH+ rH+ Hq+ l)2 +(m+ If jf)2+

+(f jJ+f J+Je+m)2+(n+Mymy)2+(ymN+yN+Nx+n)2+(lkn)2+((j1)2O1)2 ((k +Of)2+

+(f P+Pe+k)2)2+((jg+1)2Q1)2(lh)2+(o+Rti)2+(tiS+Ss+o)2+(p+Tti+t)2+(tiUtU+Us+p)2+

+((i1)2+h2)2((2+Vi)2+(i+Wd)2+(oph)2)2((d+V+1i)2+(op+h)2)2+((ic)2X1)2(ow)2)] = 0.

Theorem 9.For any a, b, c, d, x, y, w, the polynomial coded by a, b, c, d, on input coded by x, y assumes the value w ifand only if(a, b, c, d, x, y, w).

Let us introduce a simple notation for the block of quantifiers of and for the polynomial of . LetPQuantifiers(x, Y) be the block of quantifiers in , where

x= stuviefgqrhjklmnopABCDEFGHIJ MN OP QRST UV W X

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CHAPTER 7. VALUES OF POLYNOMIALS AND BAF-TERMS 19

is the list of all the bound variables of in order of appearance in the quantifier-prefix, and with a new variableYadded at the end, i.e.

PQuantifiers(x, Y) = st uv i efg qr h j klmnop ABCDEFGHIJMN OPQRSTU V WXY.

Now, letP(x,Y, a, b, c, d, x, y, w) be the following polynomial:

i(c+A+1i)j (g+Aj)((e+e2f2+e3f3g3+Abi)2+(biB+Ba+e+e2f2+e3f3g3)2+(h+Cvi)2+(k+Erj)2+

+(viD+Du+h)2+(rjF+Fq+k)2+(l+Grjr)2+(rjH+rH+Hq+l)2+(m+If jf)2+(f jJ+f J+Je+m)2+

+(n+Mymy)2+(ymN+yN+Nx+n)2+(lkn)2+((j1)2O1)2 ((k+Of)2+(f P+Pe+k)2)2+(lh)2

((jg+1)2Q1)2+(o+Rti)2+(tiS+Ss+o)2+(p+Tti+t)2+(tiUtU+Us+p)2+((i1)2+h2)2((2+Vi)2+

+(i + W d)2 + (o p h)2)2 ((d + V + 1 i)2 + (o p + h)2)2 + ((i c)2 X 1)2 ((o w)2 Y)2).

Recall that x covers all variables that occur in PQuantifiers(x, Y), apart from Y. This polynomial differsfrom the polynomial in(a, b, c, d, x, y, w) by only one term at the very end of the expression. This is a switch

that we introduce so that the polynomialP(x,Y,

a,

b,

c,

d,

x,

y,

w)would cover both

(a,

b,

c,

d,

x,

y,

w) andthe opposite of(a, b, c, d, x, y, w), as follows.

For any a, b, c, d, x, y, w, the polynomial coded by a, b, c, d, on input coded by x, y does not assume thevaluew if and only ifP Quantif iers(x, Y)[P(x, Y + 1, a, b, c, d, x, y, w) = 0].

The other equivalence is our Theorem 7 above:

(a, b, c, d, x, y, w) P Quantifiers(x, Y)[P(x, 0, a, b, c, d, x, y, w) = 0].

In this sense knows values of all polynomials on all inputs.

The mention of the universal polynomial from [11] will be here.

The mention that actually knows the actual values (not just existence of the solution as in universalpolynomials in the sense of [11]) - will be here.

7.2 An exp-polynomial expression that knows values of all polyno-mials on all inputs

will be here.

Target length: 1-2 lines.

7.3 The polynomial expression for values of BAF-terms

Let us first introduce BAF-terms, as in [7]. We shall use the symbols + (addition),.

(cut-off subtraction, i.e.x y if x > y and 0 otherwise), (multiplication), exp (base-2 exponentiation) and log (the integer part ofthe binary logarithm, with log(0) = 0).

Definition.

BAF-terms (basic arithmetic functions, BAF-functions) are functions built from 0 , 1, +,.

, , exp, log andvariables v1, v2, v3, . . ..

Examples of BAF-terms: examples here.


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What is the difference with poly-exp-log functions: explanation here.

Consider a BAF-termfof arityk, i.e. no more than k different variablesv1, . . . , vk occur in it. We will encode

tas a sequence (a1, a2, . . . , as) of triples. The last element of the triple will represent an operation as follows:

symbol translation

+ 0.

1

2

exp 3

log 4

The first two components of the triple will represent the input as follows:

symbol translation

0 1

1 2

v1 3...

vk k+ 2

result after step i i + k+ 2.

By the last line we mean once we are computing the value of the BAF-term, given a certain input, take theresult you have obtained so far after i steps. A binary operation coded by the third component acts on thetwo other components, taking the first one as the leftmost. If the operation is unary, it works only on the firstelement.

The sequence (a1, a2, . . . , as) of length s will be coded by (a,b,s). The only other piece of informationneeded is the arityk , so a BAF-term is completely determined by ( a,b,s,k). Using the translation above, thestatement (a,b,s,k) determines a BAF-term is equivalent to

i efg hjlmnop

(s+hi)((e+e2f2+e3f3g3+hbib)2+(bij+bj+ja+e+e2f2+e3f3g3)2+(g+l4)2+(m+1e)2+(e+nik2)2+

+(o + 1 f)2 + (f+ p i k2)2) = 0.

We would like to express wis the value of the BAF-term coded by (a, b, s, k) on the input-sequence codedby (x, y).

Consider the following prefixed polynomial equation (a, b, s, k, x, y, w), with free variables a, b, s, k, x, y, w,which we show in bold:

cd i efg hjl mnopqrtuvz ABC D EFG HI JK LM NO PQ RS

[(hmdid)2+(din+dn+nc+h)2+(o+1i)2h2+((i1)2p1)2(h1)2+(i+q1)2(k+2+qi)2((h+qyi+y)2+

+(yiryr + r + x + h)2)2 + (i + tk2)2 (k + 2 + s + t i)2 ((e + e2f2 + e3f3g3 + tbi + bk + b)2 + (k + v de)2+

+(dez+zc+k)2+(biubkubu+ua+e+e2f2+e3f3g3)2+(l+Adf)2+(df B+Bc+l)2+(g2+(hkl)2)2((g1)2+

+(k + E+ 1 l)2 (hk + l)2 + (l + F k)2 h2)2 ((g 2)2 + (hkl)2) ((g 3)2 +(1+ JF)2 + (F K+ KE+ 1)2+

+(h+LFkF)2+(FkM+F M+ME+h)2+(k+N+1G)2((H+NF G)2+(F GO+OE+H)2+(I+PF GF)2+


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+(F GQ+F Q+QE+I)2+(I2H)2)2)2 ((g4)2+((k2+h2)(((1+JF)2+(F K+KE+1)2+(D+LFyF)2+

+(F hM+ F M+ ME+ D)2 + (h + N+ 1 i)2 ((H+ NF G)2 + (F GO + OE+ H)2 + (F GQ + F Q + QE+ I)2+

+(I+ P F GF)2

+ (I2H)2

)2

)2

+ (D + Rx)2

+ (k + S+ 12D)2

))2

)2

+ ((k + 1 + s i)2

C1)2

(hw)2

] = 0.

Theorem 10. For any a, b, s, k, x, y, w, the BAF-term coded by a, b, s, k, on input coded by x, y assumesthe value w if and only if (a, b, s, k, x, y, w).

As we did for and P in the previous section, let us introduce notation for the block of quantifiers of and the polynomial of . LetQQuantifiers(x, T) be the following block of quantifiers (where x is the listcdiefghjlABCDEFGHIJKLMNOPQRSTof all bound variables of , and with Tadded at the end):

QQuantifiers(x, T) = cd i efg hjl mnopqrtuvz ABC D EFG HI JK LM NO PQ RS T.

Now defineQ(x,T, a, b, s, k, x, y, w) to be the following polynomial:

(hmdid)

2

+(din+dn+nc+h)

2

+(o+1i)

2

h

2

+((i1)

2

p1)

2

(h1)

2

+(i+q1)

2

(k+2+qi)

2

((h+qyi+y)

2

++(yiryr + r + x+ h)2)2 +(i + tk2)2 (k+ 2+s + ti)2 ((e + e2f2 + e3f3g3 + tbi+bk +b)2+ (dez+ zc+ k)2+

+(k+vde)2+(biubkubu+ua+e+e2f2+e3f3g3)2+(l+Adf)2+(df B+Bc+l)2+(g2+(hkl)2)2((g1)2+

+(k + E+ 1 l)2 (hk + l)2 + (l + F k)2 h2)2 ((g 2)2 + (hkl)2) ((g 3)2 +(1+ JF)2 + (F K+ KE+ 1)2+

+(h+LFkF)2+(FkM+F M+ME+h)2+(k+N+1G)2((H+NF G)2+(F GO+OE+H)2+(I+PF GF)2+

+(F GQ+F Q+QE+I)2+(I2H)2)2)2 ((g4)2+((k2+h2)(((1+JF)2+(F K+KE+1)2+(D+LFyF)2+

+(F hM+ F M+ ME+ D)2 + (h + N+ 1 i)2 ((H+ NF G)2 + (F GO + OE+ H)2 + (F GQ + F Q + QE+ I)2+

+(I+ PF GF)2 + (I2H)2)2)2 + (D + Rx)2 +(k + S+ 12D)2))2)2 + ((k +1 + s i)2C1)2 ((hw)2T)2.

Recall that xcovers all variables occurring in QQuantifiers(x, T), apart from T. This polynomial differsfrom the polynomial of (a, b, s, k, x, y, w) by only one term at the very end of the expression. This is a switch

that we introduce so that this polynomial Q(x,Y, a, b, s, k, x, y, w) would cover both (a, b, s, k, x, y, w) andthe opposite of (a, b, s, k, x, y, w), as follows.

For any a, b, s, k, x, y, w, the BAF-term coded by a, b, s, k, on input coded by x, y does not assume thevaluew if and only ifQQuantifiers(x, T)[Q(x, T+ 1, a, b, s, k, x, y, w)= 0]. The other equivalence is ourTheorem 8:

(a, b, s, k, x, y, w) QQuantifiers(x, T)[Q(x, 0, a, b, s, k, x, y, w) = 0].

7.4 A poly-exp-log expression that knows values of all BAF-termson all inputs

will be here.

Target length: 2-3 lines.

7.5 Arbitrary programming languages

Not to forget the story of any programming language. Indeed a polynomial expression that knows the HaltingProblem is just a minor modification of .

We dont have to do it for Turing machines but write a more conventient Turing-complete programminglanguage instead.


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7.6 About Katies project

An explanation of how much more we can now express will be here.

Example: we can now feasibly quantify over all computable functions, because we have written (in thischapter above) a short polynomial expression that knows values of all polynomials on all inputs. Indeed, sinceevery computably enumerable set (including graphs of computable functions) is the set of all values of somepolynomial of several variables, we can easily write for every computably enumerable set A . . .

Katie Pearces MSci project [19] was devoted to this kind of questions and includes a polynomial expressionthat is equivalent to Harvey Friedmans recent unprovable statement for all computable functionsf: Nk Nk

there are distinct natural numbers x1, x2, . . . , xk+1 such that f(x1, x2, . . . , xk) is coordinate-wise smaller orequal to f(x2, x3, . . . , xk+1).


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Chapter 8

Polynomial expressions that cant betackled by ZFC and stronger theories

The coarse ZFC-unprovable polynomial expression you will find below is still a bit long. We only did the mostrough raw version, without fiddling with the initial combinatorial statement beforehand. We safely predictthat by the end of spring 2011, the polynomial will shrink to a reasonable size and the poly-exp-log expressionwill be quite simple. (See discussion in the end of this chapter.)

Instead of carefully thinking and spending some time on playing with Friedmans Boolean Relation Theoryand careful efficient coding, we really couldnt resist the temptation and wrote down the reckless attemptstraight away. This is the first time that a polynomial expression of such high strength has been explicitlywritten down. The length is far from optimal and the readers shouldnt think that the final clever answer willbe anywhere near that long.

Recall the story how since the early 1970s solution of Hilberts Tenth Problem, some people dreamt of findinga polynomial expression (usually imagined in the shape of the statement of non-solvability of some Diophantineequation, i.e. in explicitly 01 form) having consistency strength of, say, ZFC.

It was around that time that the opinion was expressed that short polynomial equations of this strength cantbe written. There is a historical explanation for this opinion. Unprovability Theory, as we know it now in 2010,did not yet exist in the 1970s. At that time people only knew one kind of unprovable first-order arithmeticalstatement, namely ConT, that is why they could only think in terms of writing down ConTby expressing someprovability predicate PrT for some proof system and then writing down PrT(0 = 1) for it. Of course inthis form this is not really a feasible task, but luckily we dont have to think in terms of ConT any longer.Nowadays, with all the sophisticated unprovability-machinery available, polynomial equations are ready tospring from different corners of the subject: Ramsey theory, well-partial-order theory, Nash-Williams theoryand from Friedmans Boolean Relation Theory (as well as polynomial expressions to be obtained from thetheory of dags, greedy chains, set series, etc). We hope that this paper demonstrates it well.

We chose Proposition E, which is ACA0-equivalent to 1-Con(ZFC+ {there exist n-Mahlo cardinals}n), herebecause it has recently been studied and carefully checked during the Bristol Boolean Relation Theory researchmeeting.

8.1 Friedmans Proposition E

For x a k -tuple (x1, . . . , xk) of natural numbers we write |x| for max{x1, . . . , xk}. The following statement isour variation of Friedmans Proposition E (see [7]):

23


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CHAPTER 8. POLYNOMIAL EXPRESSIONS THAT CANT BE TACKLED BY ZFC AND STRONGER THEOR

For any two terms f , g BAF of several variables, such that there exist rational numbers a,b,c,d >1witha|x| f(x) b|x| and c|y| g(y) d|y| for all but finitely many x and y,

there exist computably enumerable sets A B C N,each containing infinitely many powers of 2, such that

f A B gB and f B C gCand B gB = and C gC=.

As expected, f A stands for the image under fof all k-tuples of elements ofA, where fhas arity k .

This will be the statement we translate into a polynomial equation. Using BRT, it is easy to show that thisvariation is equivalent to Friedmans original Proposition E so the strength is preserved.

Recall the definitions ofP andQ and PQuantifiers(x, y) and QQuantifiers(x, y) in Sections 7.1 and7.3. The polynomials P and Q contain the information stored in and , respectively, in the following way:

(a, b, c, d, x, y, w) P Quantif iers(x, y)[P(x, 0, a, b, c, d, x, y, w) = 0],

and

(a, b, s, k, x, y, w) QQuantifiers(x, y)[Q(x, 0, a, b, s, k, x, y, w) = 0].Let us first write an intermediate polynomial expression which contains the polynomials P and Q. Let Ebethe following polynomial expression:

abcd efgh tuvw klm xy ABCD EFGH IJKL Y z nopq MNOP QRST UX j

i V WZa1b1 r s i1 c1d1 e1f1 g1h1 k1l1 m1n1 a2a3a4

QQuantifiers(x, x0) PQuantifiers(y, y0) QQuantifiers(z, z0) PQuantifiers(r, r0) QQuantifiers(s, s0)

PQuantifiers(t, t0) PQuantifiers(u, u0) QQuantifiers(v, v0) QQuantifiers(w, w0) QQuantifiers(q, q0)

[((ta)2+(ub)2+(vc)2+(wd)2)((te)2+(uf)2+(vg)2+(wh)2)+((Y++1v)2+(+22+333uYu1)2

((uY+ u+ t++22+333)21)2 (5+)2 2 (Y+w+3+)2 2 (Y+w+3+)2 (nypy1)2

(Q(x, x0,t,u,v,w,x,y,z ))2

((yp+ y+ x+n)21)2 (oyq y 1)2 ((yq+ y+ x+o)21)2 (w+ p)2

(w+q)2

(ryiy1)2

((yi+ y+ x+r)2

1)2

((i+1w)2

+(n++1r)2

(r++1o)2

)((m++1n)2

++(z ++1 kn)2 (ln+ +1 z)2) (z ++1 n)2] [((((MA)21) ((NB)2 1) ((O C)21) ((PD)21)

((QE)21)((RF)21)((SG)21)((TH)21))2+(((ME)21)((NF)21)((OG)21)

((PH)21)((QI)21)((RJ)21)((SK)21)((TL)21))2)((1+e1b1)2+(b1f1+f1a1+1)

2+

+(W+ g1 b1V b1)2 + (b1V h1 + b1h1 + h1 a1 + W)

2 + ((V + k1 + 1 i1) ((c1 + k1 b1i1)2 + (b1i1n1 + b1n1 + n1 a1 + d1)

2+

+(b1i1l1+ l1 a1+ c1)2 + (d1+ m1 b1i1 b1)

2 + (d1 2c1)2))2 + (j+ + 1 W)2 + ((P(y, 0, M , N , O , P , , , W ))

2+

+((Zj 1)((j +++Z)21)Q(z, z0, e , f , g , h , , , U )((j+1+h)2+((P(r, r0, Q , R , S , T , , , Z ))2

)

((U X)2 1))2 + (Q(s, s0,a,b,c,d,,,U) ( j 1 a4) ((j + + + )2 a4 1) (((j + a4 + 1 d)2+

+(P(t, t0, M , N , O , P , , , ))2

) (((P(u, 0, Q , R , S , T , , , ))2

+ (P(v, 0, Q , R , S , T , , , U ) (((sa2+ a2+ a2 + )2+

+(Q(w, 0, e , f , g , h , , , U ))2

+ (h + a3 s)2 ((P(q, 0,Q,R,S,T,,,))2

+ (+ a4 s )2)))2)] = 0.

Theorem 11. The polynomial expression E is (I0+ exp)-equivalent to Friedmans Proposition E, which isACA-equivalent to 1-Con(ZFC + {there exists an n-Mahlo cardinal}n).

8.2 An intermediate poly-exp-log-expression equivalent to 1-Con(ZFC+Mahlo cardinals)

will be here.


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CHAPTER 8. POLYNOMIAL EXPRESSIONS THAT CANT BE TACKLED BY ZFC AND STRONGER THEOR

Theorem 12. is ACA0-equivalent to1-Con(ZFC + {there exists an n-Mahlo cardinal}n).

Corollary 13. The statement above is a sinlge axiom, whose 02 consequences over ACA are exactly the

02 theorems of ZFC + {there exists an n-Mahlo cardinal}n).

In the sense of Corollary 13, theoretically you dont have to use ZFC+ {there exists ann-Mahlo cardinal}nforany 2-quantifier arithmetical purposes (for example if you object to that particular theory on some philosophicalgrounds). Instead, you can use the purely arithmetical single axiom . The authors actually know how topractically use Axiom in proving arithmetical sentences, following the monograph [7].

The authors have no philosophical objections to the theory ZFC+ {there exists ann-Mahlo cardinal}n or toNF or to any other theory. We find it absolutely fascinating to study how one can axiomatize large portions ofthese theories arithmetical fragments by a single, pure arithmetical axiom about one polynomial. (It would beeven more fascinating to discover that these theories, and a billion others, contradict each other on first-orderarithmetical sentences: see a discussion of Arithmetical Splitting in [4].)

8.4 What will be the final length after some optimization

Boolean Relation Theory has a lot of flexibility, so we are planning to fiddle with Proposition E to decreasethe number of appeals to P andQ and re-use these appeals in a couple of places. Also,P andQ will getsimplified (especially with the use of exp and log).

With the current 4 appeals to Q and 6 appeals to P, the size of the answer is approximately 4|Q|+6 |P|+9lines (subscripts increased it from 72 to 80 lines above).

We expect to get down to 3 uses ofQ and 4 uses ofP, so if we also save one line in P and two lines in Q,the answer will be shorter than 3 6 + 4 4 + 9 = 43 lines.

In poly-exp-log format, we safely predict an answer that fits in fewer than 3 2 + 4 1 + 3 = 13 lines.

8.5 A short poly-exp-log expression equivalent to 1-Con(ZFC+ Mahlocardinals)

will be here.


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Chapter 9

Subtle cardinals

28


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Chapter 10

Seeds

About classification of short polynomial equations with quantifiers: seeing the full distribution of EFA-provableequivalence classes of polynomial equations with quantifiers.

The idea is to go through all polynomial equations, starting from length n = 1 and generate all possiblepolynomial equations and use the automatical polynomial transformation software or some kind of prover toeliminate all quickly-provable/refutable junk and isolate and extract the candidates for the unprovable ones,and then use automatic transformation programs, to compare them with the PH2 polynomial of Theorem 2and other strong ones: below I1 and above I1 alike. (Or just do those hard cases by hands!)

We can actually find some amazing polynomials, the seeds of strength in very short lengths, something thatpeople absolutely dont expect.

At the moment, we have an example of length 131 (Theorem 2) and lots above it, but we might bump intosome absolutely unexpected examples that have size, say 23 or 37.

It is also a good idea to try this in the poly-exp or poly-exp-log set-up to obtain very impressive examples.Here is how we count the size of a polynomial expression (the same definition as on page 1). In the

polynomial equation with quantifiers:

- every occurence of a variable inside the polynomial adds 1 to the total size;

- every operation symbol, + or between variables adds 1 to the total size;

- for a natural number n, +n or n adds n to the total size;

- for a natural number n, n or n adds (n 1) to the size;

- the power n adds (n 1) to the total size;

- brackets are disregarded;

- the quantifier-prefix is disregarded.

It is not final that we will use this size measure in the future, but for a while it will be. What we want ofa measure is that it is monotone and somehow reflects the simplicity of the expression. We dont want tocount the size of squaring (t(x))2 by doubling the size oft(x) for exactly this reason: in the computation ofit, we grab t(x) and square it in a single act of squaring, instead of computing it again and then multiplyingit by itself.

With this measure, the raw coarse polynomials we have so far had the following sizes:

29


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CHAPTER 10. SEEDS 30

our name unprovable in theory current size (6.12.2page 9: Theorem 2 1-Con(I1) 1 I1 131

page 10: Theorem 3 1-Con(I2) 2 I2 174

page 10: Theorem 4 1-Con(PA) PA 352

page 12: Theorem 6 1-Con(ATR0) K ATR0 648page 14: Theorem 8 GMT GM 11-CA0 1067page 27: Theorem 12 1-Con(ZFC + {n-Mahlo}n ZFC + {n-Mahlo}n 4620

10.1 Provable and refutable cases

How to extract and dispose of all provable and refutable rubbish.

10.2 Some short known open problems

Some open problems may be seeds, or will follow from seeds.

10.3 What are seeds

Clearly, every equivalence class of a prefixed polynomial equation thatis present in size n, is also present inthe sizes larger than n (add 1 and subtract 1 from some variable). The same expression is also spread aroundby polynomial rewriting and EFA-proofs.

For every polynomial equation with quantifiers, consider the set of all polynomial equations that are EFA-equivalent to it. A seed, or a purely polynomial seed is a polynomial equation of smallest size in its equivalenceclass. If a seed belongs to the EFA-equivalence class of an arithmetical formula, we shall say it is a seed of.We are not saying the seed of because there may be several seeds of of the same size.

We shall not count the two trivial seeds: 0 = 0) (the seed of truth, the seed belonging to the equivalenceclass of all provable rubbish) and 1 = 0) (the seed of lies, the seed of all refutable rubbish) as seeds.

We also have have slightly lower interest in the intermediate seeds (provable in I1 but unprovable in EFA),for example in the seed for the Finite Ramsey Theorem (see discussion on page 46), which is provable inI0+ tower but unprovable in I0+ exp. For us, the stronger the theory - the more precious the seed.

Clearly, we can expect seeds of quite reasonable sizes, for example, by Theorem 2 there is a seed of 1-Con( I1)of size


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CHAPTER 10. SEEDS 31

Conjecture 2: there is a seed of 1-consistency ofI1 of size


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Chapter 11

What use could a fragment of theAtlas be

This is just a copy-paste of my email from April 2010. The text will be severly edited in the future.The Atlas of Truth and Strength is an exciting possibility that follows from this polynomial project.

11.1 The Atlas as a knowledge database

Mathematicians keep re-proving each others results again and again, in different languages. The results mayseem to talk or prove lemmas about p-adics, or complex numbers or finite groups or about graph theory buthardly ever know when they re-do the work that has been already done many times, possibly in anothersubject, possibly cast in absolutely different terms. There is a lot of wasted time spent re-proving the samebasic I0+ exp combinatorial lemmas.

However, occasionally mathematics stumbles upon lemmas with a bit of strength and people get puzzled:they sense the difference but dont know how to explain why Ramsey Theorem is not the same as the primenumber theorem. (We, logicians, metamathematicains, that this intuitive reason manifests itself in the factthat Ramsey Thoerem is not EFA-provable, but the prime number theorem isnt. In particular, these twostatements are not equivalent.)

The of course there is the Paris-Harrington Principle and the amasing world of logical strength and un-provability, giving us other statements that are not equivalent to each other.

What if we produce the Atlas (or Encyclopedia, or Dictionary) of (suitably chosen normal forms of) allpolynomial expressions with quantifiers, of lengths, say, < 500 and teach people how to translate most of theirquestions, in their subjects, into that language like it is done in this paper, and to routinely check in the Atlasfor the strength or provability or refutability (the metamathematical status) of every small statement theyever encounter? (more precisely: to look up the EFA-provable equivalence class of their statement).

How would the teaching process go?

1. Example 1: Paris-Harrington.

Teaching how the translation goes, checking the polynomial in the Atlas. Understanding the (quite big)strength. Being directed to see the small and cute seed of this statement.

2. Example 2: Some Sylows theorem in group theory and PNT.

Translating Sylows theorem into its polynomial, checking in the Atlas. Translate the Prime NumberTheorem, Check the Atlas. Indeed both statements are EFA-provable. No need to spend time provingthem.

3. Example 3: Finite Ramsey Theorem.

32


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CHAPTER 11. WHAT USE COULD A FRAGMENT OF THE ATLAS BE 33

Translating the Ramsey Theorem, ending up with a polynomial statement unprovable in I0+ exp,checking it in the Atlas and seeing this fact and the strength clearly marked in the Atlas

4. Example 4: Gowerss theorem about Banach spaces.This is an adventurous and creative bit (unlike the other, mechanical exercises!) Showing how to minia-turise such second-order statements, (find a first-order fragment preserving a lot of strength), thentranslate into a polynomial expression, that is marked in the Atlas as, say, unprovable in 11-CA0 (con-

jecture).

The next step is finding the seed of this statement.

How to communicate this second-order statement to hypothetical aliens on other planets? Send themthe seed of this statement! Alien mathematicians will understand the language of polynomials. Thealiens on other planets will recognize this seed as the unique code of this mathematical phenomenon.

5. Example 5: Classical Open Problems.

Translating twin prime, Hypothesis H, P=NP etc, etc... These equivalence classes will produce lots of

equivalent polynomial expressions. Mathematicians will always be able to check whether their problemin whatever subject happens to be equivalent to a known open problem! Just check the Atlas!

6. Example 6: Graph minor theorem.

Showing the translation from Chapter 3 above, clearly marking what are the known lower and upperbounds on the stregth. GMT itself is not an open problem but the Atlas of Truth and Strength willclearly state what IS the problem (exact strenghth yet unknown).

All possible logical strengths will be clearly marked! All open problems will be clearly marked! Allpolynomial seeds will be clearly marked as seeds!

Polynomials we get from Boolean Relation Theory will amaze people, and well genarate very many exam-ples in the reasonable lengths.

Logic will be crucial there, and they will see a lot of important equivalence classes throughout the placeapart from the commonly encountered classes (equivalents of 0 = 0 and equivalents ofx x= 0).

The Atlas will indeed educate the world about logic: they will see that mathematical statements are nolonger true or false but fall into so many important and interesting metamathematical classes.

At an early size there will be a polynomial for, say, 1-Con(NF). Suddenly NF will be not a weird settheory, but one of the canoical metamathematical equivalence classes.

As an Appendix to the Atlas, we can list all discovered and suspected seeds.Because seeds and suspected seeds are so rare, and EFA-equivalence classes are so few, the list of seeds

will be a short, 5-page table.Probably this whole project should first be tried for a very small maximal length, up to, say, 132, to see

how many polynomial expressions we really want to list in the ATLAS (the EFA-equivalence classes are huge,but the number of classes, I guess, is quite small).

Of course there are lots of seeds already below 132, for example the seed of Paris-Harrington for pairs.

This sounds almost like a computer science exercise, but all messages the project sends to the world ofmathematicians are about unprovability, templates and the true nature of various arithmetizable mathematicalsubjects.

Is Isabelle the right package to learn?Later, the sister-Atlases can be produced: fine-tuned for the rationals or allowing exponentiation and

logarithm, but, most importatnly, the Atlas in the Rich Language of finite set theory, very close to the actuallanguage that mathematicians are using.


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CHAPTER 11. WHAT USE COULD A FRAGMENT OF THE ATLAS BE 34

11.2 Building some fragment of the Atlas

First notice that the equivalence of two arithmetical statements is undecidable. However, we can try togenerate

the equivalence classes, see the gaps and try to fill in the thousands of gaps by hands.Isabelle or some other automatic deduction software, can be used to generate polynomials with quantifiers

and do automatic proofs to carve the equivalence classes.I guess we can generate the first LaTeX file for the Atlas automatically, and then automatically mark the

equivalence classes. We already have names for several amazing equivalence classes (PH, RT, E).The ATLAS can be arranged like a dictionary, so that any polynomial expression could be quickly found.You see: there are very many polynomial expressions with quantifiers, but theres always a short dictionary

route to locate each of them!!! We dont even have to produce a paper version of this Atlas - just the searchabledatabase-file or a pdf file, or both.

Well... maybe distributing a few hard copies will be cool too.Now imagine the impact this Atlas (a huge volume in large A3 format) will have. It is fine it it takes one

century to build, as long as it is eventually built, and all answers recorded.All small-sized questions in everyones own mathematical subject that may ever come up will have the

answer (its metamathematical status) written there.(And, perhaps, the majority of questions is, or can be

studying the atlas of all possible polinomial equations with quantifier-prefixes

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