the strength of some martin{l˜of type theoriesrathjen/typeohio.pdfthe strength of some martin{l˜of...

The Strength of Some Martin–Lof Type Theories

Michael Rathjen∗†

Abstract

One objective of this paper is the determination of the proof–theoretic strength of Martin–Lof’s type theory with a universe and the type of well–founded trees. It is shown that thistype system comprehends the consistency of a rather strong classical subsystem of second orderarithmetic, namely the one with ∆1

2 comprehension and bar induction. As Martin-Lof intendedto formulate a system of constructive (intuitionistic) mathematics that has a sound philosophicalbasis, this yields a constructive consistency proof of a strong classical theory. Also the proof-theoretic strength of other inductive types like Aczel’s type of iterative sets is investigated invarious contexts.

Further, we study metamathematical relations between type theories and other frameworksfor formalizing constructive mathematics, e.g. Aczel’s set theories and theories of operationsand classes as developed by Feferman.

0 Introduction

The intuitionistic theory of types as developed by Martin-Lof is intended to be a system forformalizing intuitionistic mathematics together with an informal semantics, called “meaning expla-nation”, which enables him to justify the rules of his type theory by showing their validity withrespect to that semantics. Martin-Lof’s theory gives rise to a full scale philosophy of constructivism.It is a typed theory of constructions containing types which themselves depend on the constructionscontained in previously constructed types. These dependent types enable one to express the generalCartesian product of the resulting families of types, as well as the disjoint union of such a family.

Using the Brouwer-Heyting semantics of the logical constants one sees how logical notionsare obtained in this theory. This is done by interpreting propositions as types and proofs asconstructions [ML 84]. In addition to ground types for the finite sets and the set of natural numbersthe theory can be taken to contain types which play the role of successive universes or reflectionsover type construction rules previously introduced. This reflection process can be iterated into thetransfinite in order to obtain successively more strongly closed universes known as Mahlo universes.Finally this theory can be taken to contain a type construction principle which, like the generalCartesian product and general disjoint union, is based on dependent families of types. It is thetype of well-orderings or well-founded trees which can be constructed using a family of branchingtypes indexed by a previously constructed type.

A number of results were known on the proof theoretic strength of Martin–Lof type theories.Many of the earliest results are presented in Beeson [Be 85]. For example, using an embeddingargument it is shown there that intuitionistic type theory without universes has the complexity of

∗Department of Mathematics, The Ohio State University, Columbus, Ohio 43210, USA†The author would like to thank the National Science Foundation of the USA for support by grant DMS–9203443.

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true arithmetic. Aczel in [A 77] showed that intuitionistic type theory with one universe ML1, butwithout well-ordering types, has the same proof theoretic strength as the classical subsystem of anal-ysis Σ1

1–AC. Palmgren later gave in [P 93] a lower bound for the system with well–ordering typesML1W in terms of the fragment of second order arithmetic with ∆1

2 comprehension, ∆12– CA. It

was known by work of Feferman, Buchholz, Pohlers and Sieg (cf. Feferman’s preface to [BFPS 81])that ∆1

2– CA is finitistically reducible to IDi<ε0(O), the intuitionistic theory of Church–Kleene con-

structive tree classes iterated any number < ε0. Palmgren showed that IDi<ε0(O) is interpretable in

ML1W. However, this result was not expected to be a sharp bound. Palmgren conjectured “thatsomething similar to autonomous inductive definitions may be interpreted in ML1W”, [P 93],p.92. In this paper we show that even the theory ∆1

2– CA plus bar induction (∆12– CA + BI for

short) is finitistically reducible to ML1W. ∆12– CA + BI is perceptibly stronger than ∆1

2– CA.∆1

2– CA + BI not only permits one to prove that Π11 inductive definitions can be iterated along

arbitrary well–orderings but (cf. [R 89]) also ∆12 comprehension and Σ1

2 dependent choices1.Key roles in determining a lower bound for the strength of ML1W are played by Fefer-

man’s theory T0 of explicit mathematics, which is known to be proof–theoretically equivalentto ∆1

2– CA + BI (cf. [JP 82],[J 83]), and Aczel’s version of constructive set theory CZF plus theregular extension axiom REA which Aczel interpreted in ML1W(cf [A 86]).

The paper is organized as follows. For the readers convenience and to make the paper more widelyaccessible, we give detailed descriptions of the system T0 in Sect. 1 and of CZF in Sect.2. Forlater use we also gather and prove elementary facts about the latter systems. The fruits are reapedin Sect. 3 where T0 is interpreted in CZF + REA.

Upper bounds for the proof–theoretic strengths of Martin–Lof type theories are obtained byusing recursive realizability models following [Be 82]. Sect. 4 provides a modelling of the typetheory ML1V, which Aczel used in [A 78] to interprete CZF, in Kripke–Platek set theory KP. Asa result, KP, CZF and ML1V have the same strength.

Aczel’s interpretation of CZF+REA in ML1W lends itself to a natural restriction of ML1W,called ML1WV, which has the type of iterated sets V but allows W–formation only for familiesin the universe U. The recursive realizability model for ML1W is extended in Sect. 5 so as toaccomodate the type V and restricted W-formation. This time the modelling is carried out inthe set theory KPi which is an extension of KP that axiomatizes a recursively inaccessible setuniverse. At the end of this section we conclude that the systems ∆1

2– CA + BI, CZF+REA andML1WV are of the same strength.

In connection with the system ML1WV the question arises whether the type V really con-tributes to its strength. V appears to be crucial for the interpretation of CZF+REA in typetheory. To pursue this question we introduce a system of second order IARI that has unrestrictedReplacement but comprehension only for arithmetic properties. In Sect. 6 IARI is interpreted inML1W, that is ML1WV without the type V. In addition, it is shown that IARI suffices for thewell–ordering proof of an ordinal notation system that was used for the analysis of KPi, therebyyielding that ML1W has the strength of ∆1

2– CA + BI too.Finally, in Sect. 7, we investigate the strength of the full system ML1W, where one also has

to deal with W–types at large, i.e. those which do not belong to U. We show that ML1W canbe modelled in a slight extension of KPi. However, we make no attempt to determine the exactbound2 as this theory seems to be of rather limited interest. More interesting are the theories

1These principles do not exhaust the strength of ∆12– CA + BI by far (cf. [R 89]).

2The exact strength of ML1W is investigated in [S 93].

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MLnW with n universes. The methods of this paper can also be applied to those theories so asto yield their proof–theoretic strength. As a rule of thumb, the tower of n universes in MLnWcorresponds to n recursively inaccessible universes in classical Kripke–Platek set theory.

1 Some Background on Feferman’s T0

1.1 Feferman’s T0

T0 is the formal system for constructive mathematics due to Feferman. In the literature it hasalready been shown that within T0 one can define notions like arithmetic truth, the ramifiedanalytic hierarchy and the constructive tree classes. Since T0 is one of the tools we use to estimatethe strength of intuitionistic type theory with universes closed under well-ordering types, we includea brief discussion and summary.

The language of T0, L(T0), has two sorts of variables. The free and bound variables (a, b, c, . . .and x, y, z . . . ) are conceived to range over the whole constructive universe which comprises opera-tions and classifications among other kinds of entities; while upper-case versions of these A,B, C,... and X,Y, Z, ... are used to represent free and bound classification variables.

The intended objects of the constructive universe are in general infinite, seen classically. Sincethe underlying logic of T0 is intuitionistic the objects of this universe are presented in a finitarymanner and, without loss of generality, by natural numbers. As far as algorithmic procedures onnatural numbers are concerned there are a variety of mathematical formalizations each of whichgives a way of presenting the derivations of algorithms using fixed rules for computing as finiteobjects. To be treated as objects these are then coded by natural numbers. Examples are Godelnumbers of formulae defining these functions in a formal theory, of formal derivations using schemesand codes of Turing machines. Yet another alternative, and the one adopted here, is Godel num-bers of application terms built up from variables and the two basic combinators k and s using abinary application operation to represent effectively computable number theoretic functions. Thisis the sense in which these objects are elements of the constructive universe. In a similar fashionone obtains natural numbers coding the formulae defining properties which give classifications aselements of the constructive universe. Which of these objects actually populate this universe willof course depend on the axioms of T0. To facilitate the formulation of these axioms, the languageL(T0) is equipped with defined symbols for term application and strong or complete equality.

N is a classification constant taken to define the class of natural numbers. 0 , sN and pN areoperation constants whose intended interpretations are the natural number 0 and the successor andpredecessor operations. Additional operation constants are k, s, d, p, p0 and p1 for the two basiccombinators, definition by cases on N, pairing and the corresponding two projections. Additionalclassification constants are generated using the axioms and the constants j, i and cn(n < ω) forjoin, induction and comprehension.

There is no arity associated with the various constants. The terms of T0 are just the variablesand constants of the two sorts. The atomic formulae of T0 are built up using the terms andthree primitive relation symbols =, App and ε as follows. If q, r, r1, r2 are terms, then q = r,App(q, r1, r2), and q ε r (where r has to be a classification variable or constant) are atomic formulae.App(q, r1, r2) expresses that the operation q applied to r1 yields the value r2; q ε r asserts3 that qis in r or that q is classified under r.

3It should be pointed out that we use the symbol “ ε ” instead of “∈”, the latter being reserved for the set–theoreticelementhood relation.

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We write t1t2 ' t3 for App(t1, t2, t3).The set of formulae are then obtained from these using the propositional connectives and the

two quantifiers of each sort.In order to facilitate the formulation of the axioms, the language of T0 is expanded definitionally

with the symbol ' and the auxilliary notion of an application term is introduced. The set ofapplication terms is given by two clauses:

1. all terms of T0 are application terms; and

2. if s and t are application terms, then (st) is an application term.

For s and t application terms, we have auxilliary, defined formulae of the form:

s ' t := ∀y(s ' y ↔ t ' y)

where the occurances of y are instances of a bound operation variable. We then define s ' a (fora a free variable) inductively by:

s ' a is

s = a, if s is a variable or a constant,∃x, y[s1 ' x ∧ s2 ' y ∧ App(x, y, a] if s is an application term (s1s2)

Some abbreviations are t1 . . . tn for ((...(t1t2)...)tn); t ↓ for ∃y(t ' y) and φ(t) for ∃y(t ' y ∧ φ(y)).Godel numbers for formulae play a key role in the axioms introducing the classification constants

cn. A formula is said to be elementary if it contains only free occurences of classification variablesA (i.e., only as parameters), and even those free occurrences of A are restricted: A must occur onlyto the right of ε in atomic formulas. The Godel number cn above is is the Godel number of anelementary formula. We assume that a standard Godel numbering numbering has been chosen forL(T0); if φ is an elementary formula and a, b1, . . . , bm, A1, . . . , An is a list of variables which includesall parameters of φ, then x : φ(x, b1, . . . , bn, A1, . . . , An) stands for cn(b1, . . . ,bn,A1, . . . ,An);n is the code of the pair of Godel numbers 〈dφe, d(a, b1, . . . , bm, A1, . . . , An)e〉 and is called the‘index’ of φ and the list of variables.

Some further conventions are useful. Systematic notation for n-tuples is introduced as follows:(t) is t, (s, t) is pst, and (t1, . . . , tn) is defined by ((t1, . . . , tn− 1), tn). Finally, t′ is written for theterm sNt, and ⊥ is the atomic formula 0 ' 0′.

T0’s logic is intuitionistic two-sorted predicate logic with identity. Its non-logical axioms are:

I. Basic Axioms

1. ∀X∃x(X = x)

2. App(a, b, c1) ∧ App(a, b, c2) → c1 = c2

II. App Axioms

1. (kab) ↓ ∧kab ' a,

2. (sab) ↓ ∧ sabc ' ac(bc),

3. (pa1a2) ↓ ∧ (p1a) ∧ (p2a) ↓ ∧pi(pa1a2) ' ai for i = 0, 1,

4. (c1 = c2 ∨ c1 6= c2) ∧ (dabc1c2) ↓ ∧ (c1 = c2 → dabc1c2 ' a) ∧ (c1 6= c2 → dabc1c2 ' b),

5. a εN ∧ b εN → [a′ ↓ ∧pN(a′) ' a ∧ ¬(a′ = 0) ∧ (a′ ' b′ → a ' b)].

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III. Classification Axioms

Elementary Comprehension Axiom (ECA)

∃X[X ' x : ψ(x) ∧ ∀x(x ε X ↔ ψ(x))]

for each elementary formula ψa, which may contain additional parameters.

Natural Numbers

(i) 0 ∈ N ∧ ∀x(x εN → x′ εN)

(ii) φ(0) ∧ ∀x(φ(x) → φ(x′)) → (∀x εN)φ(x) for each formula φ of L(T0).

Join (J)

∀x∃Y fx ' Y → ∃X[X ' j(A, f) ∧ ∀z(z εX ↔ ∃x ε A∃y(z ' (x, y) ∧ y ε fx))]

Inductive Generation (IG)

∃X[X ' i(A,B) ∧ ∀x ε A[∀y[(y, x) εB → y ε X] → x ε X]

∧ ∀x εA [∀y ((y, x) εB → φ(y)) → φ(x)] → ∀x ε X φ(x)]]

where φ is an arbitrary formula of T0.

1.2 A Finite Axiomitization of ECA

Later on in section 3 we are going to interprete T0 in intuitionistic set theory. For technical reasonswhich will become apparent then, we show that ECA is already captured by finitely many of itsinstances.

Definition 1.1 For each elementary φ(x, y1, . . . , yn, A1, . . . , Am) write

x : φ(x, y1, . . . , yn, A1, . . . , An) for cφ(y1, . . . , yn, A1, . . . , Am).

We may then make the following abbreviations:

V for x : x = x;∅ for x : ⊥;

a1, . . . , an for x : x = a1 ∨ · · · ∨ x = an;A2B for x : x ε A2x εB, (2 ∈ ∧,∨,→)

AB for f : x ε B → fx εA;Diag=(A,B, s, t) for f : f ε AB ∧ fs ' ft ∧ s, t ε B

DiagApp(A,B, s, t, t′) for f : f ε AB ∧ App(fs, ft, ft′) ∧ s, t, t′ εBProj(A,B, C) for f : f ε AB ∧ ∃g ε C ∀x ε B(gx ' fx)

Ext(A,B, C, s) for f : f ε AB ∧ ∀z ∃g ε C[gs ' z ∧ ∀x εB(gx ' fx)];f−1a for x : fx ' a.

In the next definition, to facilitate the writing, we will occasionally write u ∈ x : φ(x) instead ofφ(u). Notice that we use the set–theoretic elementhood sign here.

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Definition 1.2 For the finite axiomatization of ECA we use the universal closures of the followingformulae:

(A1) ∃X [X ' (A2B) ∧ ∀u(u ε X ↔ u ε A2u ε A)]where 2 is one of the connectives ∧,∨,→

∃X [X ' ∅ ∧ ∀u(u ε X ↔ u ∈ ∅)](A2) ∃X [X ' f−1a ∧ ∀u(u ε X ↔ fu ' a)](A3) ∃X [X ' a ∧ ∀u(u ε X ↔ u = a)](A4) ∃X [X ' Diag=(A,B, a, b)) ∧ ∀u(u ε X ↔ u ∈ Diag=(A,B, a, b))](A5) ∃X [X ' DiagApp(A, B, a, b, b′) ∧ ∀u(u εX ↔ u ∈ DiagApp(A, B, a, b, b′))](A6) ∃X [X ' Proj(A,B, C) ∧ ∀u(u ε X ↔ u ∈ Proj(A,B, C))](A7) ∃X [X ' Ext(A,B, C, a) ∧ ∀u(u εX ↔ u ∈ Ext(A,B, C, a))](A8) ∃X [X ' AB ∧ ∀u(u εX ↔ u ∈ AB)](A9) ∃X [X ' u : ∃f(f ε A ∧ f0 ' u ∧ ∀u(u ε X ↔ ∃f(f ε A ∧ f0 ' u))]

Let Tf0 arise from T0 by restricting the schema ECA to the finitely many instances from

(A1),. . . ,(A9) and only retaining those constants cn that enter these axioms.If ψ(b) is an elementary formula of Tf

0 and tψ is an application term of Tf0 such that

Tf0 ` ∃X[X ' t ∧ ∀u (u εX ↔ ψ(u))]

and the free variables of tψ are among the free variables of ψ(0) we will say that u : ψ(u) is aclassification (provably) in Tf

0.

Lemma 1.3 If φ(a0, . . . , an−1) is an elementary formula of Tf0 with free non–classification vari-

ables among a0, . . . , an−1, then we can construct an application term tφ which witnesses that

u : u ε A0,... ,n−1 ∧ φ(u0, . . . , u(n− 1))]

is a classification in Tf0.

Proof : Write An for A0,... ,n−1 and ~a for a0, . . . , an−1. Note that An is a classification inTf

0 using (A3), (A1), and (A8). We shall denote the term sought after by f ε A0,... ,n−1 :φ(f0, . . . , f(n − 1))]. Set Hφ := f ε An : φ(f0, . . . , f(n − 1)). We want to prove the lemmaby induction on the generation of elementary formulas.

1. If φ(~a) is ai = aj , then Hφ = Diag=(A, 0, . . . , n− 1, i, j). Since 0, . . . , n− 1 is a classifi-cation using (A3) and (A1), so is Hφ by (A4).

2. φ(~a) is ai = c, where c is a constant. Let ψ(~a) be the formula ai = an. By (A3), (A2), and(A1), C := Hψ ∩ f ε An+1 : fn ' c is a classification. Hence Hφ is a classification by (A6)and (1) as Hφ = Proj(A, 0, . . . , n− 1, C).

3. φ(~a) is c = d where c and d are both constants. Let ψ(~a) be an = an+1 and put

C := Hψ ∩ f εAn+1 : fn ' c ∩ g ε An+2 : g(n + 1) ' d.As C is a classification, so is Hφ = Proj(A, 0, . . . , n− 1, C).

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4. If φ(~a) is App(ai, aj , ak), then Hφ = DiagApp(A, 0, . . . , n − 1, i, j, k) is a classification by(A5).

5. Suppose φ(~a) is App(f, s, t) where some of the terms f, s, t are constants. Define ψ(a0 . . . , an+2)by App(an, an+1, an+2). ThenHψ is a classification by (4). If for instance φ(~a) is App(f, ai, aj)with f a constant, then Hφ = Proj(A, 0, . . . , n− 1, C) with C being

Hψ ∩ g ε An+3 : gn ' f ∩ g ε An+3 : g(n + 1) ' ai ∩ g ε An+3 : g(n + 2) ' aj.

Similarly, Hφ can be shown to be a classification for the remaining possibilities.

6. If φ(~a) is of the form φ0(~a)2φ1(~a), where 2 is one of the connectives ∧,∨,→, then theassertion follows inductively using (A1).

7. If φ(~a) is ∃z ψ(~a, z), then Hθ = Proj(A, 0, . . . , n− 1,H) with θ(a0, . . . , an) being ψ(~a, an).So Hφ is a classification by (A6) and the inductive assumption.

8. If φ(~a) is ∀z ψ(~a, z), thenHφ = Ext(A, 0, . . . , n−1,Hθ, n) with θ(a0, . . . , an) being ψ(~a, an).Thus Hφ is a classification using (A7) and the inductive assumption.

9. If φ(~a) is ai εB, then Hφ = f εAn : fi ε B = f ε An : f εBi = An ∩Bi. Thus Hφ is aclassification employing (A3), (A8), and (A1).

10. If φ(~a) is c ε B where c is a constant, then Hφ = Proj(A, 0, . . . , n − 1, C) with C beingHψ ∩ g ε An+1 : gn ' c, where ψ(a0, . . . an) is an ε B. 2

Proposition 1.4 Tf0`ECA, that is to say u : ψ(u) is a classification in Tf

0 for every elementaryformula ψ of Tf

0.

Proof : Let φ(a0, . . . , an−1) be elementary. Then

C := f ε An : φ(f0, . . . , f(n− 1)) ∧ f1 ' a1 ∧ · · · f(n− 1) ' an−1

is a classification using Lemma 1.3, (A1), (A2), (A3). Now

x : φ(x, a1, . . . , an−1) = x : ∃f ε C(f0 ' x),

and hence x : φ(x, a1, . . . , an−1) is a classification by (A9). 2

To conclude from the previous result that T0 can be interpreted in Tf0, we need the following

lemma.

Lemma 1.5 (cf. [FS 81]) (Abstraction Lemma) For each application term t(c1, . . . , cn) of Tf0

there is a new application term t∗ such that the parameters of t∗ are among the parameters oft(c1, . . . , cn) minus c1, . . . , cn and such that

Tf0 ` t∗ ↓ ∧ t∗(c1, . . . , cn) ' t.

λ(c1, . . . , cn).t is written for t∗.

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Proof : To begin with, we show this for one parameter. Define λc.t(c) by the complexity ofthe term t as follows. λc.c is skk; λc.t is kt for t a constant or a variable other than c; λc.t0t1 iss(λc.t0)(λc.t1).

For two parameters c1, c2, define λ(c1, c2).t(c1, c2) := λc.t(p0(c),p1(c)); etc.Notice that if the list c1, . . . , cn comprises all the parameters of t, then λ(c1, . . . , cn).t is a closed

application term without any classification variables. 2

For later use we state a consequence of the Abstraction Lemma which can be derived in the sameway as for the λ–calculus (cf. [F 75],[Be 85],VI2.7).

Corollary 1.6 (Recursion Theorem)

∀f∃g∀x1 . . .∀xn g(x1, . . . , xn) ' f(g, x1, . . . , xn).

Corollary 1.7 T0 can be interpreted in Tf0.

Proof : The interpretation consists in successively replacing all those comprehension constants cφ

which do not belong to Tf0 with a suitable closed application term c¦φ which arises from abstracting

an application term as provided by Proposition 1.4.The construction of c¦φ proceeds along the generation of cφ. So assume that cφ is the index of an

elementary formula φ(b, a1, . . . , an, B1, . . . , Bm) and a list of variables b, a1, . . . , an,B1, . . . , Bm thatincludes all parameters of φ. Assume that we have already translated φ(b, a1, . . . , an, B1, . . . , Bm)into an elementary formula

φ(b, a1, . . . , an, B1, . . . , Bm)¦

of Tf0. Then let t(a1, . . . , an, B1, . . . , Bm) be the term constructed in Proposition 1.4 which wit-

nesses that u : φ(u, a1, . . . , an, B1, . . . , Bm)¦ is a classification in Tf0.

Let c¦φ := λ(a1, . . . ,an,B1, . . . ,Bm).t(a1, . . . ,an,B1, . . . ,Bm). If we now replace cφ in anelementary formula with c¦φ, then the result4 will also be an elementary formula. The faithfulnessof this interpretation is immediate by 1.4 and 1.5 2

In the sequel we shall identify T0 with Tf0.

2 Constructive set theories

2.1 The System CZF

In this subsection we will summarize the language and axioms for Aczel’s constructive set theoryor CZF. The language of CZF is the first order language of ZF whose only non-logical symbolis ∈. The logic of CZF is intuitionistic first order logic with equality. Its non-logical axiomsare extensionality, pairing, union, and infinity in their usual forms. CZF has additionally axiomschemata which we will now proceed to summarize. Each is followed with some comments on theintuition involved and questions of constructivity.

4More precisely, that which is obtained after rewriting the result as a formula of T0.

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Set Induction∀x[∀y ∈ xφ(y) → φ(x)] → ∀xφ(x)

for all formulae φ. Set induction replaces primarily the foundation axiom and corresponds to theprinciple of ∈-induction which is a theorem of ZF. It should be compared, as far as constructivitygoes, with the induction principle associated with any generalized inductive definition. Examplesare the inductive definitions giving the higher number classes.

Bounded Separation∀a∃b∀x[x ∈ b ↔ x ∈ a ∧ φ(x)]

for all bounded formulae φ, where φ is said to be bounded if all quantifiers in φ are bounded, i.e.,having constructed the set a as well as those giving explicit bounds on all of the quantifiers in φ,we can construct the set of all elements of a which have the property φ.

Strong Collection∀a[∀x ∈ a∃yφ(x, y) → ∃bφ′(a, b)]

for all formulae φ(x, y), where φ′(a, b) is the formula

∀x ∈ a∃y ∈ b φ(x, y) ∧ ∀y ∈ b∃x ∈ aφ(x, y)

This axiom allows one to construct a set corresponding exactly to the image of a previously definedrelation between sets one has already constructed.

Subset Collection

∀a∀a′∃c∀u(∀x ∈ a∃y ∈ a′φ(x, y) → ∃b ∈ c φ′(a, b))

for all formulae φ(x, y). Note that u may occur free in φ(x, y), i.e., this is subset collection withparameters.

Dependent Choices (DC)

∀x(θ(x) → ∃y(θ(y) ∧ φ(x, y))) → ∀x(θ(x) → ∃zψ(x, z))

for all formulae θ(x) and φ(x, y), where ψ(x, z) is the formula which asserts that z is a functionwith domain the natural numbers such that z(0) = x and for every natural number n, θ(z(n)) ∧φ(z(n), z(n + 1)) holds.

The formula in the language of CZF defining the property of a set A that it is regular states thatA is transitive, and for every a ∈ A and set R ⊆ a×A if ∀x ∈ a∃y (〈x, y〉 ∈ R), then there is a setb ∈ A such that

∀x ∈ a ∃y ∈ b (〈x, y〉 ∈ R) ∧ ∀y ∈ b∃x ∈ a (〈x, y〉 ∈ R).

In particular, if R : a → A is a function, then the image of R is an an element of A. Let Reg(A)denote this assertion. With this auxilliary definition we can state:

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Regular Extension Axiom (REA)

∀x∃y[x ⊆ y ∧ Reg(y)]

Aczel in [A 86] interpreted CZF plus the regular extensions axiom in the intuitionistic theory oftypes due to Martin-Lof using a single universe closed under the type of well-orderings. Alter-natively he showed that, by introducing rules in type theory for the collection of iterative sets(in anology with Russell’s ramified theory of types), one can model the sets and formulae of thelanguage of CZF using the types in this extension of intuitionistic type theory.

2.2 Some Properties of CZF

The mathematical power of CZF resides in the possibility of defining class functions by ∈–recursionand the fact that the scheme of Bounded Separation allows for an extension with provable classfunctions occurring in otherwise bounded formulae.

We shall use the informal notion of class as in classical set theory.

Proposition 2.1 (Definition of Recursion in CZF) Let G be a total (n + 2)–ary class function ofCZF, i.e. CZF ` ∀~xyz∃!uG(~x, y, z) = u. Then there is a total (n + 1)–ary class function F ofCZF such that5

CZF ` ∀~xy[F (~x, y) = G(~x, y, (F (~x, z)|z ∈ y))].

Proof : Let Φ(f, ~x) be the formula

[f is a function] ∧ [dom(f) is transitive] ∧ [∀y ∈ dom(f) (f(y) = G(~x, y, f ¹ y))].

Set ψ(~x, y, f) := [Φ(f, ~x) ∧ y ∈ dom(f)]. 2

Claim CZF ` ∀~x, y∃!fψ(~x, y, f).

Proof of Claim: By ∈–induction on y. Suppose ∀u∈ y ∃g ψ(~x, u, g). By Strong Collection we find aset A such that ∀u∈ y ∃g ∈Aψ(~x, u, g) and ∀g ∈A∃u∈ y ψ(~x, u, g). Let f0 =

⋃g : g ∈ A. By ourgeneral assumption there exists a u0 such that G(~x, y, (f0(u)|u ∈ y)) = u0. Set f = f0 ∪ 〈y, u0〉.Since for all g ∈ A, dom(g) is transitive we have that dom(f0) is transitive. If u ∈ y, thenu ∈ dom(f0). Thus dom(f) is transitive and y ∈ dom(f). We have to show that f is a function.But it is readily shown that if g0, g1 ∈ A, then ∀x ∈ dom(g0) ∩ dom(g1)[g0(x) = g1(x)]. Thereforef is a function. This also shows that ∀w∈dom(f)[f(w) = G(~x,w, f ¹ w)], confirming the claim(using ∈–induction).

Now define F byF (~x, y) = w := ∃f [ψ(~x, y, f) ∧ f(y) = w]. 2

Corollary 2.2 There is a class function TC such that

CZF ` ∀a[TC(a) = a ∪⋃TC(x) : x ∈ a].

5(F (~x, z)|z ∈ y) := 〈z, F (~x, z)〉 : z ∈ y

10

Proposition 2.3 (Definition by TC–Recursion) Under the assumptions of Proposition 2.1 thereis an (n + 1)–ary class function F of CZF such that

CZF ` ∀~xy[F (~x, y) = G(~x, y, (F (~x, z)|z ∈ TC(y)))].

Proof : Let θ(f, ~x, y) be the formula

[f is a function] ∧ [dom(f) = TC(y)] ∧ [∀u∈dom(f)[f(u) = G(~x, u, f ¹ TC(u))]].

Prove by ∈–induction that ∀y∃!f θ(f, ~x, y). Suppose ∀u∈ y ∃!g θ(g, ~x, u). By Strong Collection wefind a set A such that ∀u∈ y ∃g ∈A θ(g, ~x, u) and ∀g ∈A∃u∈ y θ(g, ~x, u). By assumption, we alsohave

∀u∈ y∃!z∃g∃a[θ(g, ~x, u) ∧G(~x, u, g) = a ∧ z = 〈u, a〉].Again by Strong Collection there is a function h such that dom(h) = y and

∀u∈ y ∃g [θ(g, ~x, u) ∧G(~x, u, g) = h(u)] .

Now let f = (⋃g : g ∈ A) ∪ h. Then θ(f, ~x, y). 2

The next result was proved by Myhill (cf. [My 75]) for the theory CST.

Proposition 2.4 (Extension by Defined Function Symbols) Suppose CZF ` ∀~x∃!y Φ(~x, y). LetCZF′ be obtained by adjoining a function symbol FΦ to the language6, extending the schemata tothe enriched language, and adding the axiom ∀~x Φ(~x, FΦ(~x )). Then CZF′ is conservative overCZF.

Proof : We define the following translation ∗ for formulas of CZF′:

φ∗ ≡ φ if FΦ does not occur in φ

(FΦ(~x ) = y)∗ ≡ Φ(~x, y)

If φ is of the form t = x with t ≡ G(t1, . . . , tk) such that one of the terms t1, . . . , tk is not a variable,then let

(t = x)∗ ≡ ∃x1 . . .∃xk [(t1 = x1)∗ ∧ · · · ∧ (tk = xk)∗ ∧ (G(x1, . . . , xk) = x)∗] .

The latter provides a definition of (t = x)∗ by induction on t. If either t or s contains FΦ, then let

(t ∈ s)∗ ≡ ∃x∃y[(t = x)∗ ∧ (s = y)∗ ∧ x ∈ y],(t = s)∗ ≡ ∃x∃y[(t = x)∗ ∧ (s = y)∗ ∧ x = y],

(¬φ)∗ ≡ ¬φ∗

(φ02φ1)∗ ≡ φ∗02φ∗1, if 2 is ∧,∨, or →(∃xφ)∗ ≡ ∃xφ∗

(∀xφ)∗ ≡ ∀φ∗.Let CZF0 be the restriction of CZF′, where FΦ is not allowed to occur in the Bounded Sep-aration Scheme. Then it is obvious that CZF0 ` φ implies CZF ` φ∗. So it remains toshow that CZF′ proves the same theorems as CZF0. In actuality, we have to prove CZF0 `∃x∀y [y ∈ x ↔ y ∈ a ∧ φ(a)] for any bounded formula φ of CZF′. We proceed by induction on φ.

6This changes the notion of bounded formula.

11

1. φ(y) ≡ t(y) ∈ s(y). Now

CZF0 ` ∀y ∈ a∃!z[(z = t(y)) ∧ ∀y ∈ a∃!u(u = s(y))].

Using Replacement we find functions f and g such that

dom(f) = dom(g) = a and ∀y ∈ a [f(y) = t(y) ∧ g(y) = s(y)] .

Therefore y ∈ a : φ(y) = y ∈ a : f(y) ∈ g(y) exists by Bounded Separation in CZF0.

2. φ(y) ≡ t(y) = s(y). Similar.

3. φ(y) ≡ φ0(y)2φ1(y), where 2 is any of ∧,∨,→. This is immediate by induction hypothesis.

4. φ(y) ≡ ∀u∈ t(y) φ0(u, y). We find a function f such that dom(f) = a and ∀y ∈ a f(y) = t(y).Inductively, for all b ∈ a, u ∈ ⋃

ran(f) : φ0(u, b) is a set. Hence there is a function g withdom(g) = a and ∀b ∈ a g(b) = u ∈ ⋃

ran(f) : φ0(u, b). Then y ∈ a : φ(y) = y ∈ a : ∀u ∈f(y)(u ∈ g(y)).

5. φ(y) ≡ ∃u∈ t(y)φ0(u, y). With f and g as above, y ∈ a : φ(y) = y ∈ a : ∃u∈ f(y)(u ∈g(y)). 2

Remark 2.5 The proof of Proposition 2.4 shows that the process of adding defined function symbolsto CZF can be iterated. So if e.g. CZF′ ` ∀~x∃y ψ(~x, y), then also CZF′+ ∀~x∃y ψ(~x, Fψ(~x)) willbe conservative over CZF.

3 A Lower Bound for the Strength of ML1W

Employing an interpretation of intuitionistic theories of iterated strictly positive inductive defini-tions in Martin–Lof’s type theory, Palmgren showed in [P 93] that ML1W has at least the strengthof ∆1

2– CA. In this section we will establish that ML1W is much stronger than ∆12– CA in that

we interpret Feferman’s theory T0 in ML1W. By [FS 81], [JP 82], and [J 83] it is known that T0

has the same strength as ∆12-comprehension plus bar induction. However, rather than modelling

T0 directly in ML1W, our line of attack will be to model T0 in Aczel’s constructive set theoryCZF + REA, for which Aczel in [A 86] provided an interpretation in ML1W. Roughly, our inter-pretation of T0 in CZF + REA proceeds by showing that Feferman’s set–theoretical interpretationof T0 (cf. [F 75]) can be carried out in CZF + REA.

3.1 The Interpretation of T0 in CZF + REA

For later use we show that, provably in CZF + REA the well-founded part of a binary set relationis a set.

For any class Φ the class X is said to be Φ–closed if A ⊆ X implies a ∈ X for every pair〈a,A〉 ∈ Φ.

Lemma 3.1 (CZF) (Aczel, cf. [A 82],4.2) For any class Φ there is a smallest Φ–closed class I(Φ).

12

Proof : Call a set relation G good if whenever 〈x, y〉 ∈ G there is a set A such that 〈y,A〉 ∈ Φ and

∀u ∈ A∃v ∈ x 〈v, u〉 ∈ G.

Call a set Φ–generated if it is in the range of some good relation. To see that the class of Φ–generatedsets is Φ–closed, let A be a set of Φ–generated sets, where 〈a,A〉 ∈ Φ. Then

∀y ∈ A∃G [G is good ∧ ∃x (〈x, y〉 ∈ G)].

By Strong Collection there is a set C of good sets such that

∀y ∈ A∃G ∈ C ∃x (〈x, y〉 ∈ G).

Letting G0 =⋃

C ∪ 〈b, a〉, where b = u : ∃y 〈u, y〉 ∈ ⋃C, G0 is good and 〈b, a〉 ∈ G0. Thus

a is Φ–generated. Whence I(Φ) is Φ–closed. Now if X is another Φ–closed class and G is good,then by set induction on x it follows 〈x, y〉 ∈ G → y ∈ X, so that I(Φ) ⊆ X. 2

Lemma 3.2 (Aczel, [A 86], Corollary 5.3) For R a set let Ri = j : 〈j, i〉 ∈ R. Let WF(A,R)be the smallest class X such that for i ∈ A

Ri ⊆ X implies i ∈ X.

Then WF(A,R) = I(Φ) where Φ = 〈i, Ri〉 : i ∈ A. CZF + REA proves that, for sets A and R,WF(A,R) is a set.

Proof : We argue in CZF + REA. For each set x let

Γ(x) = ı : i ∈ A ∧ Ri ⊆ x. (1)

Note that Γ(x) is a set by Bounded Separation. Let X be the smallest class such that if

∀y ∈ a ∃z ∈ b 〈y, z〉 ∈ X ∧ ∀z ∈ b ∃y ∈ a 〈y, z〉 ∈ X

then〈a,Γ(

⋃b)〉 ∈ X.

This definition can be couched in terms of an inductive definition falling under the scope of Lemma3.1. Using ∈–induction it is easily verified that for each set a there is a unique set y such that〈a, y〉 ∈ X, and if this unique y is written Γa then

Γa = Γ(⋃Γy : y ∈ a). (2)

Next we want to show

I (Φ) =⋃Γu : u ∈ V. (3)

First note that if Γy ⊆ I (Φ) for all y ∈ a, then⋃Γy : y ∈ a ⊆ I (Φ), whence Γa ⊆ I (Φ) since I (Φ)

is Φ–closed. Thus by set induction, Γa ⊆ I (Φ) for all sets a, confirming “⊆” of (3). For the converseinclusion it suffices to show that

⋃Γu : u ∈ V is Φ–closed. So suppose x ⊆ ⋃Γu : u ∈ V. Then

∀y ∈ x ∃a (y ∈ Γa).

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Employing Strong Collection there is a set b such that

∀y ∈ x ∃a ∈ b (y ∈ Γa).

Therefore x ⊆ ⋃Γu : u ∈ b and hence Γ(x) ⊆ Γb ⊆ ⋃Γu : u ∈ V.Finally we must show that I (Φ) is a set. By the Regular Extension Axiom we can pick a regular

set B such that ∀i ∈ A Ri ∈ B. Let

I =⋃Γu : u ∈ B.

Strong Collection and Union guarantee that I is a set. By (3) we have I ⊆ I (Φ). So it remainsto show that I is Φ–closed, since then I (Φ) ⊆ I. So assume Ri ⊆ I for some i ∈ A. Then∀j ∈ Ri ∃u ∈ B j ∈ Γu. By the regularity of B there is b ∈ B such that ∀j ∈ Ri ∃u ∈ b j ∈ Γu. Itfollows that Ri ⊆

⋃Γu : u ∈ b, yielding, by (2), that i ∈ Γb ⊆ I. 2

Corollary 3.3 CZF + REA minus Subset Collection proves that, for sets A and R, WF(A,R)is a set.

Proof : This is because nowhere in the proofs of 3.1 and 3.2 we had to use Subset Collection. 2

Since I (Φ) is the smallest Φ–closed class we get the following induction principle for WF(A,R).

Corollary 3.4 (CZF + REA) For an arbitrary formula of set theory ψ(x) and sets A,R,

∀i∈A[∀j ∈Ri ψ(j) → ψ(i)] → ∀i∈WF(A,R) ψ(i),

where Ri = j ∈ A : 〈j, i〉 ∈ R. 2

For the interpretation of T0 it is convenient to work in a definitional extension of CZF which wedenote again by CZF. Drawing on Proposition 2.4, we may assume that CZF has a constant ωfor the unique set whose existence is asserted in the axiom of infinity7. By the same token we mayassume that CZF has a function symbol S for the successor function given by S(x) = x∪x, anda constant 0 denoting the empty set. CZF proves ω–induction (cf. [TD 88], Ch. 11, 8.5)

φ(0) ∧ ∀x [φ(x) → φ(S(x))] → ∀x∈ω φ(x).

The Peano Axioms for zero and successor can be shown in CZF to hold for 0 and S on ω. EmployingProposition 2.1 and Proposition 2.4 we may assume that CZF has as many function symbols forprimitive recursive functions as we like and their defining axioms being among the axioms of CZF.In particular we will assume that all those primitive recursive functions (and their defining axioms)required for introducing Kleene’s T–predicate and the pertinent evaluation function U are part ofCZF. As usual we define

x(y) ' z := x, y, z ∈ω ∧ ∃v ∈ω[T (x, y, v) ∧ U(v) = z].7Uniqueness is proved in [TD 88] Ch. 11, 8.3.

14

We shall assume that CZF has a constant n for each (meta) natural number n which names the nth

numeral, i.e. the nth element of ω. We are going to translate App(x, y, z) as x(y) ' z. Particularnumerals sN, pN, k, s, d, p0, p1 can be determined to satisfy (provably in CZF) the translation ofthe applicative axioms pertaining to the constants sN,pN,k, s,d,p0,p1 of T0. N is translated asN := ω. Select cn, j, i in such a way so that

CZF ` ∀x, y ∈ω[cn(x) ' (1, n, x) ∧ j(x, y) ' (2, x, y) ∧ i(x, y) ' (3, x, y)

],

where (x, y) := p(x) (y) and (x, y, z) := ((x, y), z). By Proposition 1.4, ECA can be finitelyaxiomatized, say via the instances

∀A,B, C ∀x, y, z ∃X[cφi

(x,y, z,A,B,C) ' X ∧ ∀u(u εX ↔ φi(u,x,y, z,A,B,C))],

where 1 ≤ i ≤ M for some M .We are now in a position to define a class model for T0 in CZF + REA. Since in constructive

set theory the ordinals are not at all as well behaved as they are classically, we cannot define thismodel by recursion along the ordinals as in [F 75], Theorem 1.1. Instead, we shall define it byrecursion on the transitive closure along all sets, thereby employing Proposition 2.3.

Definition 3.5 For each set s, sets Es and CLs are defined by TC–recursion as follows:

CL<s =⋃CLu : u ∈ TC(s)

E<s =⋃Eu : u ∈ TC(s)

CLs = CL<s ∪ N ∪cφi(x, y, z, a, b, c) : a, b, c ∈ CL<s; x, y, z ∈ ω; 1 ≤ i ≤ M ∪j(a, f) : a ∈ CL<s ∧ f ∈ ω ∧ ∀x ∈ ω[〈x, a〉 ∈ E<s → f(x) ∈ CL<s]

∪

i(a, b) : a, b ∈ CL<s

;

〈u, a〉 ∈ Es iff either of the following holds:

1. 〈u, a〉 ∈ E<s;

2. a = N ∧ u ∈ ω;

3. a ' cφi(x, y, z, b, c, d) with b, c, d ∈ CL<s, x, y, z ∈ ω and φi(u, x, y, z, b, c, d) is true underthe interpretation which takes v ε w to mean 〈v, w〉 ∈ E<s and interprets App and the constantsof T0 as stated above;

4. a ' j(b, f) with b ∈ CL<s, f ∈ ω such that ∀x∈ω[〈x, b〉 ∈ E<s → f(x) ∈ CL<s] andu = (u0, u1) satisfies 〈u1, f(uo)〉 ∈ E<s;

5. a ' i(b, c) with b, c ∈ CL<s and u ∈ WF(b<s, c<s), where b<s = x : 〈x, b〉 ∈ E<s andc<s = 〈x, y〉 : 〈(x, y), c〉 ∈ E<s.

We have to verify that the preceding definition falls under the scope of Proposition 2.3. Butthis follows once we see that the passage from (〈CLu,Eu〉|u ∈ s) to 〈CLs,Es〉 determines a classfunction of CZF + REA. It is obvious that CLs is a set again. Regarding Es this follows from the

15

fact that Es can be obtained by Bounded Separation from (〈CLu,Eu〉|u ∈ s) and WF(b<s, c<s) :b, c ∈ CLs, the latter being a set by Lemma 3.2 and Strong Collection.

To complete the interpretation of T0 in CZF + REA we are going to let classification variablesrange over CL =

⋃CLu : u ∈ V and translate x ε s as 〈x, s〉 ∈ E, where E =⋃Eu : u ∈ V.

We shall write uEsx and uE<sx instead of 〈u, x〉 ∈ Es and 〈u, x〉 ∈ E<s, respectively. Verifyingthat the classification axioms of T0 are preserved under the above interpretation requires a littlelemma to the extent that all E–elements of a member x of CL are added at the same time as x isdeclared to be in CL.

Lemma 3.6 If a ∈ CLs ∩CLt, then u : uEsa = u : uEta.The proof of 3.6 requires another lemma.

Lemma 3.7 Let e ∈ CLz. Then

1. e ' cφi(x, y, z, a, b, c) → a, b, c ∈ CL<z,

2. e ' j(a, f) → a ∈ CL<z,

3. e ' i(a, b) → a, b,∈ CL<z.

Proof : Use ∈-induction on z. 2

Proof of 3.6: We use ∈-induction on s. Set Ez(a) = u : uEza. If a ∈ CL<s, then a ∈ CLz forsome z ∈ s and inductively Ez(a) = Es(a) and Ez(a) = Et(a), thus Es(a) = Et(a). If a ' i(c, d)with c, d ∈ CL<s, then c, d ∈ CL<t by 3.7. Hence E<s(c) = E<t(c) and E<s(d) = E<t(d), usingthe inductive assumption. Thus Es(a) = Et(a) by 2.3(5). The argument proceeds along the sameway in the remaining cases. 2

Corollary 3.8 If a ∈ CLs, then u : uEsa = u : uEa. 2

In view of 3.8 and 3.5(3) it is clear that the interpretation renders the elementary comprehensionaxioms true. As to Join, suppose a ∈ CL and ∀x[xEa → f(x) ∈ CL]. Then there exists s sothat a ∈ CLs and ∀x∈Es(a)∃z [f(x) ∈ CLz]. Using Strong Collection we find a t such that∀x∈Es(a)∃z ∈ t [f(x) ∈ CLz], and hence ∀x∈Es(a) f(x) ∈ CL<t. For s′ = s, t it follows∀x[xE<s′a → f(x) ∈ CL<s′ ] using 3.6. Hence j(a, f) ∈ CLs′ and

∀u[uEs′ j(a, f) ⇐⇒ ∃u0∃u1[u ' (u0,u1) ∧ u1E<s′f(u0)]

].

Employing 3.6, this yields

∀u[uEj(a, f) ⇐⇒ ∃u0∃u1(u ' (u0,u1) ∧ u1Ef(u0))

],

confirming the faithfulness of our translation with regard to Join. For Inductive Generation supposea, b ∈ CL. Then a, b ∈ CL<s for some s. With a<s = x : xE<sa and b<s = 〈x, y〉 : (x, y)E<sbwe obtain

∀u[uE<si(a,b) ⇐⇒ u ∈ WF(a<s,b<s)

],

16

according to 3.5.5. Therefore, using 3.6 and 3.8, we have

∀u[uEi(a,b) ⇐⇒ u ∈ WF(aE,bE)

],

where aE = x : xEa and bE = 〈x, y〉 : (x, y)Eb. Consequently, by 3.4, we come to see that theinterpretation also preserves the axiom of Inductive Generation.

Theorem 3.9 T0 can be interpreted in CZF + REA. 2

4 Interpretation of ML1V in KP

ML1V is the extension of ML1 with Aczel’s set of iterative sets V (cf. [A 78]). The rules pertainingto V are:8

(V–formation) V set

(V–introduction)A ∈ U f ∈ A → V

sup(A, f) ∈ V

(V–elimination)c ∈ V

[A ∈ U, f ∈ A → V][z ∈ (Πv ∈ A)C(f(v))]

d(A, f, z) ∈ C(sup(A, f))TV(c, (A, f , z)d) ∈ C(c)

(V–equality)B ∈ U g ∈ B → V

[A ∈ U, f ∈ A → V][z ∈ (Πv ∈ A)C(f(v))]

d(A, f, z) ∈ C(sup(A, f))TV(sup(B,g), (A, f , z)d) = d(B,g, (λv)TV(g(v), (A, f , z)d)) ∈ C(sup(B,g))

.

KP stands for Kripke–Platek set theory which we shall assume to contain the axiom of infinity.As usual in set theory we identify the natural numbers with the finite ordinals, i.e. N := ω. Tosimplify the treatment we will assume that KP has names for all (meta) natural numbers. Letn be the constant designating the nth natural number. We also assume that KP has functionsymbols for addition and multiplication on N as well as for a primitive recursive bijective pairingfunction 〈, 〉N : N2 → N and its primitive recursive inverses ( )0, ( )1, that satisfy (〈n,m〉N)0 = nand (〈n,m〉N)1 = m. KP should also have a symbol T for Kleene’s T -predicate and the resultextracting function U . Let e(n) ' k be a shorthand for ∃m[T (e, n, m) ∧U(m) = k]. Further, let〈n,m, k〉N := 〈〈n,m〉N, k〉N, 〈n,m, k, l〉N := 〈〈n,m, k〉N, l〉N, etc. We use e, d, f, n, m, l, k, s, t, j, ias metavariables for natural numbers. Occasionally, “ ⇒ ” is used for “then” and “ ⇔ ” for “iff”.Let

π(n,m) = 〈0, 〈n,m〉N〉Nσ(n,m) = 〈1, 〈n,m〉N〉Npl(n,m) = 〈2, 〈n,m〉N〉N

i(n,m, k) = 〈3, 〈n,m, k〉N〉NN = 〈4, 0〉N

Nk = 〈4, k + 1〉Nsup(n,m) = 〈5, 〈n, , 〉Nm〉N

8To increase readability and intelligibility, we will use the formulation a la Russel for type theories with universes(cf. [ML 84]).

17

Employing 4.2 it follows Uδ |= k set and

∀j ∈N[Uδ |= j ∈ k ⇒ Uδ |= e(j) set] and∀j, i∈N[Uδ |= i = j ∈ k ⇒ Uδ |= e(i) = e(j)].

Hence Uδ+1 |=“ e is a family of types over k” according to 4.1(ii). Thus U |=“e is a family of typesover k.” 2

In order to define its interpretation in KP, we need a detailed account of the syntax of ML1V.Here we will follow [Be 85], Ch. XI; however, for the readers convenience, we shall recall most ofthe definitions.

If B is any expression, and x1, . . . , xn are variables, we form the expression (x1, . . . , xn)B. Thesymbol

4≡ will be used for the relation on expressions satisfying

((x1, . . . , xn)B)(x1, . . . , xn)4≡ B

and A4≡ C for expressions A and C which differ only in the renaming of bound variables (cf.

[Be 85],XI6).

Definition 4.5 (cf. [Be 85],XI20.3) The constants of ML1V are: Π,Σ, I, +,N,0, sN,r,λ,ap, E,i,j,D, J,R,TV,U,V and for each natural number m, Nm and Rm. The terms are generated by:

1. Every constant and variable is a term;

2. If t and s are terms, then t(s) and (t, s) are terms;

3. If t is a term, then (x1, . . . , xn)t is a term, where the xi are variables.

Free and bound occurrences of variables in terms are defined as usual, letting abstraction, i.e. theformation of (x1, . . . , xn)t bind the variables x1, . . . , xn. We now would like to assign to everyterm t of ML1W a corresponding term t of the theory KP by replacing the abstract applicationof ML1W with recursive application. However, the technicality here is that recursive applicationcannot be expressed on the level of terms within KP. Therefore, we first assign to every term tof ML1W an application term t∗ of the theory EON of [Be 85],VI.2 or equivalently of the theoryAPP as described in [TD 88],Ch.9,Sect.3. It is then a straightforward matter to translate a formulaof the form t∗ ∈ X into a legitimate formula of KP.

Definition 4.6 The theory EON consists of T0 without classification variables and without theconstants cn, i, j. In particular, EON has no axioms ECA, (IG), (J).

Note that t he Abstraction Lemma 1.5 and the Recursion Theorem 1.6 are also theorems of EON.Both devices will be used in the next definition.

Definition 4.7 We now assign to each term t of ML1V an application term t∗ of the theory EON.Occurrences of λ in the definition of t∗ denote the λ-operator introduced in 1.5. We shall write(x, y) for pxy and, inductively, x1, . . . , xk+1 for p(x1, . . . , xk)xk+1. For constants c we define c∗

by:

20

0∗ is 0Π∗ is λx.λy.(0, x, y)Σ∗ is λx.λy.(1, x, y)+∗ is λx.λy.(2, x, y)I∗ is λx.λy.(3, x, y, z)

N∗ is (4, 0)N∗

k is (4, k + 1)U∗ is (6, 0)V∗ is (7, 0)s∗N is sNr∗ is 0λ∗ is λx.x (i.e. skk)

ap∗ is λx.λy.xysup∗ is λx.λy.(5, x, y)

E∗ is λx.λy.y(p0x,p1x)i∗ is λx.(0, x)j∗ is λx.(1, x)

D∗ is λx.λy.λz(0,p0x, y(p0x), z(p1x))J∗ is λx.λy.y

R∗k is λm.λx0 · · ·λxk−1.ek(m,x0, . . . , xk − 1), where ek is chosen so that EON proves

ek(m,x0, . . . , xk − 1) '

xm if m < kΩ otherwise,

where Ω := λx.xx(λx.xx) (signifying undefinedness). R∗ is an application term of EON in-troduced by the Recursion Theorem (cf. 1.6) to satisfy (provably in EON) R∗(a, b, 0) = a andR∗(a, b, sNx) ' b(x,R∗(a,b,x)). T∗

V is a term introduced by the recursion theorem of EON tosatisfy

T∗V(sup∗(a,b), λx.λy.λz.e(x,y, z)) ' e(a,b, (λx.x)(λv.T∗

V(ap∗(b,v)), λx.λy.λz.e(x,y, z))).

For complex terms of ML1V we define:

((x1, . . . , xn)t)∗ is λx1 · · ·λxn.t∗;(t(s))∗ is t∗s∗;(t, s)∗ is pt∗s∗.

Definition 4.8 Rather than translating application terms of EON into the set–theoretic languageof KP, we define the translation of expressions of the form t ' u, where t is an application term ofEON and u is a variable. First, we select indices k, s, p, p0, p1, sN, pN and d for partial recursivefunctions so that:

21

k(n)(m) ' ns(n)(m)(l) ' n(l)(m(l))

p(n)(m) ' 2n · 3m = 〈n,m〉Np0(n) ' (n)0p1(n) ' (n)1sN(n) ' n + 1pN(0) ' 0

pN(n + 1) ' n

d(n)(m)(x)(y) '

x if n = my if n 6= m

The definition of (t ' u)∧ proceeds along the way that t was built up.

(v ' u)∧ is v = u ∧ v ∈ ω if v is a variable;(0 ' u)∧ is 0 = u;(c ' u)∧ is c = u if c is one of the constants k, s,p,p0,p1, sN,pN,d.

A complex application term of EON has the form ts.Let

(ts ' u)∧ := ∃x, y ∈ ω[(t ' x)∧ ∧ (s ' y)∧ ∧ x(y) ' u].

EON is interpreted in KP by setting

(App(s, t, r))¦ := ∃u ∈ ω [(st ' u)∧ ∧ (r ' u)∧](s = t)¦ := ∃u ∈ ω [(s ' u)∧ ∧ (t ' u)∧]

(s ∈ N)¦ := ∃u ∈ ω [(s ' u)∧](φ2ψ)¦ := φ¦2ψ¦ (2 ∈ ∧,∨,→)

(∀xφ(x))¦ := ∀x ∈ ω φ(x)¦.

Definition 4.9 The set terms of ML1V are defined inductively by

1. N and Nk are set terms (for each integer k);

2. If A and B are set terms, so is (A+B);

3. IF B(x) and A are set terms, and x is not free in A or in B, then Π(A,B) and Σ(A,B) areset terms;

4. If A is a set term and t, s are any terms of ML1V, then I(A, s, t) is a set term;

5. U and V are set terms;

6. If A is a set term and B4≡ A, then B is a set term.

Definition 4.10 (Interpretation of ML1V in KP) By induction on the complexity of the set termA we shall assign to each judgement Φ of ML1V of the form u ∈ A or u = v ∈ A (u, v variables)a formula (Φ)∧ of KP with the same free variables. (ux ∈ A)∧ and (ux = uy ∈ A)∧ will be usedas shorthand for ∃z ∈ω[u(x) ' z ∧ (z ∈ A)∧] and ∃z ∈ω[u(x) ' z ∧ u(y) ' z ∧ (z ∈ A)∧],respectively. Likewise, (u ∈ A(vx))∧ abbreviates ∃z ∈ω[v(x) ' z ∧ (u ∈ A(z))∧], etc. The clausesin the definition are as follows:

22

(u ∈ Π(A,B))∧ is ∀x∈ω[(x ∈ A)∧ → (ux ∈ B(x))∧] ∧∀x, y ∈ω[(x = y ∈ A)∧ → (ux = uy ∈ B(x))∧]

(u = v ∈ Π(A,B))∧ is ∀x∈ω[(x ∈ A)∧ → (ux = vx ∈ B(x))∧] ∧(u ∈ Π(A,B))∧ ∧ (v ∈ Π(A, B))∧

(u ∈ Σ(A,B))∧ is (p0u ∈ A)∧ ∧ (p1u ∈ B(p0u))∧

(u = v ∈ Σ(A,B))∧ is (p0u = p0v ∈ A)∧ ∧ (p1u = p1v ∈ B(p0u))∧

(u ∈ (A+B))∧ is [p0u = 0 ∧ (p1u ∈ A)∧] ∨ [p0u = 1 ∧ (p1u ∈ B)∧](u = v ∈ (A+B))∧ is [p0u = 0 ∧ p0v = 0 ∧ (p1u = p1v ∈ A)∧] ∨

[p0u = 1 ∧ p0v = 1 ∧ (p1u = p1v ∈ B)∧](u ∈ I(A, b, c))∧ is u = 0 ∧ (b = c ∈ A)∧

(u = v ∈ I(A, b, c))∧ is u = 0 ∧ v = 0 ∧ (b = c ∈ A)∧

(u ∈ N)∧ is u ∈ ω

(u = v ∈ N)∧ is u = v ∧ u ∈ ω

(u ∈ Nk)∧ is u ∈ ω ∧ u ∈ k

(u = v ∈ Nk)∧ is u ∈ ω ∧ u = v ∧ u ∈ k

(u ∈ U)∧ is U |= u set

(u = v ∈ U)∧ is U |= u = v

(u ∈ V)∧ is V |= u set

(u = v ∈ V)∧ is V |= u = v

If s and t are arbitrary terms of ML1V and A is a set term of ML1V, we set:

(t ∈ A)∧ is ∃u∈ω[(t∗ ' u)∧ ∧ (u ∈ A)∧],(s = t ∈ A)∧ is ∃u, v ∈ω[(s∗ ' u)∧ ∧ (t∗ ' v)∧ ∧ (u = v ∈ A)∧].

For set terms A and B we define (A = B)∧ by

∀u∈ω[(u ∈ A)∧ ↔ (u ∈ B)∧] ∧ ∀u, v ∈ω[(u = v ∈ A)∧ ↔ (u = v ∈ B)∧].

Theorem 4.11 (Soundness of the Interpretation of ML1V in KP.) If Φ is a judgement of ML1Vnot of the form “A set,” then KP ` (Φ)∧.

Proof : First note that if an expression of the form A set, s ∈ A, s = t ∈ A, or A = B appearsin a derivation of ML1V, then A is a set term in the sense of Definition 4.9, as is readily seen byinduction on derivations in ML1V.9 This ensures that any judgement of ML1V gets translatedunder ∧. Secondly, it should be clear that the above interpretation replaces the abstract applicationof ML1V by recursive application in a faithful way, i.e. the equations which the rules of ML1Vprescribe for the constants of ML1V are satisfied by their translations. Since ML1V derivationsmay involve hypothetical judgements, 4.11 has to be stated in a more general form so as to beable to carry out the proof by induction on derivations in ML1V. For the details we refer to

9Incidentally, there are more set terms than those that appear in such derivations.

23

[Be 85],XI.20.3.1, where this is done for the interpretation of ML0 in EON. Proposition 4.4 yieldsthat the particular rules for U are sound with respect to the interpretation ∧. The soundness ofV–introduction is implied by 4.4(vii). Next we shall verify the soundness of V–elimination.

To facilitate the writing we shall write ψ(t) in place of ∃u[(t ' u)∧ ∧ ψ(u)], when t is a term ofEON and ψ(u) a set-theoretic formula. Suppose V |= c0 set and , for all a, f, g ∈ ω, if

U |= a set, (4)∀n∈ω[U |= n ∈ a → V |= f(n) set],∀n,m∈ω[U |= n = m ∈ a → V |= f(n) = f(m)],∀n∈ω[U |= n ∈ a → φ(g(n), f(n))],thenφ(d∗(a, f, g), sup∗(a, f)

),

where d(x1, x2, x3)10 is a term of ML1V and φ(u, v) an arbitrary set-theoretic formula. We wantto show φ

((TV(y, (x1, x2, x3)d(x1, x2, x3)))∗, y

)[c0/y], that is to say

φ(T∗

V(c0, λx1.λx2.λx3.d∗(x1,x2,x3))∗, c0

).

As Vα |= c0 set for some α, we may proceed by induction on α. Vα |= c0 set implies that thereare β < α and a0, f0 such that

c0 = sup(a0, f0) = sup∗(a0, f0), (5)Uβ |= a0 set,

∀n∈ω[U |= n ∈ a0 → Vβ |= f0(n) set],∀n,m∈ω[U |= n = m ∈ a0 → Vβ |= f0(n) = f0(m)].

The inductive assumption then yields

∀n∈ω[U |= n ∈ a0 → φ(T∗

V(f0(n), λx1.λx2.λx3.d∗(x1,x2,x3)), f0(n))];

hence,

∀n∈ω[U |= n ∈ a0 → φ((λ∗λv.T∗

V(v, λx1.λx2.λx3.d∗(x1,x2,x3)))(n), f0(n))].

Employing this and (5), the general assumption (4) yields

φ(d∗(a0, f0, λ∗(λv.T∗

V(v, λx1.λx2.λx3.d∗(x1,x2,x3))), sup∗(a0, f0))),

where g corresponds to λ∗(λv.T∗V(v, λx1.λx2.λx3.d∗(x1,x2,x3))). In view of the translation of

TV under ∗, the latter is equivalent to

φ(T∗V(sup∗(a0, f0), λx1.λx2.λx3.d∗(x1,x2,x3)), sup∗(a0, f0)),

and thusφ(T∗

V(c0, λx1.λx2.λx3.d∗(x1,x2,x3)), c0)10To be precise, d∗(a, f, g) is short for (d(x1, x2, x3))

∗[a/x1, f/x2, g/x3].

24

since c0 = sup∗(a0, f0). 2

The preceding proof also yields the soundness of V–equality.Type theories with a universe U lend themselves to a natural strengthening by the principle of

U–induction. U–induction was employed by Aczel ([A 82],[A 86]) for the interpretation of certainpresentation axioms in type theory.

Definition 4.12 The rule of U–induction formalizes that the universe U is inductively defined.To put this rule on paper we resort to linear notation; the judgements to the right within bracketsare discharged assumptions.

Set C(z) [z ∈ U]a0 ∈ UdN0 ∈ C(N0)dN1 ∈ C(N1)dN ∈ C(N)dΠ(A,F, c, e) ∈ C(Π(A,F ))[A ∈ U, F ∈ A → U, c ∈ C(A), e ∈ Π(A,F )]dΣ(A,F, c, e) ∈ C(Σ(A,F ))[A ∈ U, F ∈ A → U, c ∈ C(A), e ∈ Σ(A,F )]d+(A,B, c, d) ∈ C(A + B)[A ∈ U, B ∈ U, c ∈ C(A), d ∈ C(B)]dI(A, c, a, b) ∈ C(I(A, a, b))[A ∈ U, c ∈ C(A), a ∈ A, b ∈ A]TU(a0, dN0 , dN1 , dN , (A, F, c, e)dΠ, (A, F, c, e)dΣ, (A,B, c, d)d+, (A,B, a, b)dI) ∈ C(a0)

If hU(a0) denotes the term to the left of C(a0), the equality rules are given by (where (λx)t :=λ((x)t))

hU(Ni) = dNi

hU(N) = dN

hU(Π(A,F )) = dΠ(A, F, hU(A), (λx)hU(F (x)))hU(Σ(A,F )) = dΣ(A,F, hU(A), (λx)hU(F (x)))

hU(A + B) = d+(A,B, hU(A), hU(B))hU(I(A, a, b)) = dI(A, hU(A), a, b).

The language of type theories with U–elimination needs a little adjustment in that we would ratherconsider N2,N3, . . . as defined by N2 := N1 + N1, N3 := N1 + N2, etc. The assignment ∗ of 4.7and the interpretation ∧ of 4.10 can be extended so as to accommodate the constant TU. Moreprecisely, there is a term T∗

U of EON such that, using the abbreviations λ~x := λx1.λx2.λx3 andH(a) := T∗

U(a,d0,d1,d2, λ~x.f , λ~x.g, λ~x.h, λ~x.l) we can prove in EON plus ∀x (x εN) that

H(a) '

d0 if a = (4, 1)d1 if a = (4, 2)d2 if a = (4, 0)f(a1, a2,H(a1), (λx.x)(λu.H(a2(u)))) if a = (0, (a1, a2))g(a1, a2,H(a1), (λx.x)(λu.H(a2(u)))) if a = (1, (a1, a2))h(a1, a2,H(a1),H(a2)) if a = (2, (a1, a2))l(a1,H(a1), a2, a3) if a = (3, (a1, a2, a3))Ω if else.

The definition of T∗U uses the Recursion Theorem 1.6 and definition by cases, which is justified

under the assumption ∀x(x εN) (cf. [Be 85],XI.2). Note that (∀x(x εN))¦ is derivable in KP andthus KP ` ψ¦ holds for any theorem ψ of EON + ∀x(x εN).

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Theorem 4.13 ML1V+U–induction is interpretable in KP via ∧.

Proof : The soundness of U–introduction under ∧ follows readily by induction on the stages of therelation U |= n set. 2

Theorem 4.14 The following theories have the same proof–theoretic strength:

KP, ID,ML1V,ML1Vi,ML1W + U–induction,ML1V + U–induction,CZF,CZF minus Subset Collection,CZF + DC + ΠΣI −AC + ΠΣI −PA.

Here ML1Vi denotes an intensional version of ML1V as described in [TD 88], Ch.11, Sect. 8. Thechoice principles DC (Dependent choices), ΠΣI −AC (ΠΣI− axiom of choice), and ΠΣI −PA(ΠΣI−presentation axiom) have been isolated in [A 82].

Proof : We use T → T ′ to mean that T is interpretable in T ′. T ≡ T ′ is used to signify proof–theoretic equivalence. Let IDi denote the intuitionistic theory of noninterated inductive definitions.It is an old result that IDi ≡ ID (cf. [BFPS 81]). Moreover, from [J 82] it follows that ID ≡ KP.Drawing on Lemma 3.1, which shows how to formalize inductive definitions in CZF, we can considerIDi as a subtheory of CZF, thus IDi → CZF. On closer scrutiny we observe that Subset Collectionwas not used in the proof of Lemma 3.1. Whence IDi is already interpretable in CZF minus SubsetCollection.

By [TD 88], Ch. 11, Sect. 8 we get CZF → ML1Vi. Furthermore, ML1Vi is a subtheory ofML1V, and [P 93] provides an interpretation of ML1V in ML1 + U–induction. From [A 82] weobtain an interpretation of CZF+DC+ΠΣI −AC+ΠΣI −PA in ML1V+U–induction. Finally,as ML1V+U–induction is interpretable in KP, by Theorem 4.13, the proof–theoretic equivalenceof all the theories follows. 2

5 Interpretation of ML1WV in KPi

Before we determine an upper bound for the strength of ML1W, we shall investigate a weaker theoryML1WV that allows W–formation only for small types. We take interest in this theory since ithas the same strength (as we will show) as some well–known classical theories like ∆1

2– CA + BIand permits us to determine the exact strength of Aczel’s constructive set theory CZF + REA.

Definition 5.1 ML1WV results from ML1W by omitting the rule of W–formation and addingthe rules pertaining to V, i.e. V–formation, V–introduction, V–elimination, and V–equality.

Remark 5.2 First, notice that ML1WV still allows one to derive restricted W–formation, i.e.

(res–W–formation)A ∈ U

[x ∈ A]F (x) ∈ U

W(A,F ) set,

as U–introduction provides

A ∈ U[x ∈ A]

F (x) ∈ UW(A,F ) ∈ U

26

and U–formation givesW(A,F ) ∈ UW(A,F ) set.

Secondly, observe that ML1WV can be viewed as a subtheory of ML1W by letting V := (WX ∈U).X.

Close inspection of the proofs in [A 86] reveals the following results.

Proposition 5.3

1. CZF+DC+REA+ΠΣI −AC is interpretable (via Aczel’s interpretation) in ML1WV.

2. If U–induction11 is added to ML1WV, then also the principle ΠΣI −PA is true under thisinterpretation. 2

Corollary 5.4 The proof-theoretic strength of ML1WV is at least that of ∆12– CA + BI.

Proof : This follows from Theorem 3.9 and Proposition 5.3 as T0 has the same strength as∆1

2– CA + BI, employing [JP 82] and [J 83]. 2

The remainder of this section is devoted to interpreting ML1WV in an extension of KP.

Definition 5.5 By KPi we denote the theory that entails KP and has an axiom INACCr signifyingthat for each set there is an admissible set that contains it.

Remark 5.6 According to [JP 82], KPi has the same proof–theoretic strength as ∆12– CA + BI.

In actuality, both theories prove the same statements of second order arithmetic.

Lemma 5.7 (KPi) (∆0 positive inductive definitions) Let φ(u, x, a1, . . . , an) be a ∆0 formula withfree variables among those shown such that each occurrence of x in φ is positive. Set Γφ,~a,b(x) =u ∈ b : φ(u, x, a1, . . . , an). Let F be the function defined by ∆ recursion along the ordinalssatisfying

F (α,~a, b) = Γφ,~a,b(⋃F (β,~a, b) : β < α).

Finally, put Iφ,~a,b =⋃F (α,~a, b) : α ∈ ON. Then Iφ,~a,b is a set. We say that Iφ,~a,b is the set

inductively defined by φ in the parameters ~a, b. Iφ,~a,b is closed under Iφ,~a,b, i.e. Γφ,~a,b(Iφ,~a,b) ⊆ Iφ,~a,b.Moreover, the assignment of Iφ,~a,b to each set ~a, b is ∆ definable.

Proof : Select an admissible set A such that ~a, b ∈ A . Let IA =⋃F (ξ,~a, b) : ξ ∈ A . Note that

IA is Σ–definable over A as A is admissible. We show F (α,~a, b) ⊆ IA by transfinite induction on α.Suppose

⋃F (β,~a, b) : β < α ⊆ IA . Now if u ∈ F (α,~a, b), then u ∈ b and φ(u,⋃F (β,~a, b) : β <

α,~a). Thus φ(u, IA ,~a) as positivity entails monotonicity. This yields A |= φ(u,∃β[· ∈ F (β)],~a),11Here U—induction has to be adapted to ML1WV in the obvious way: add

dW(A, F, c, e) ∈ C(W(A, F ))[A ∈ U, F ∈ A → U, c ∈ C(A), e ∈ W(A, F )]

to the hypotheses of the rule in Definition 4.12 and augment the function TU by the argument (A, F, c, e)dW. As toequality, we have to add hU(W(A, F )) = dW(A, hU(A), (λx)hU(F (x))).

27

where the latter formula arises from φ(u, x,~a) by replacing each occurrence of v ∈ x in φ by∃β[v ∈ F (β)]. Since the occurrences of x are all positive, φ(u,∃β[· ∈ F (β)],~a) becomes a Σformula. Hence, applying Σ reflection in A (cf. [Ba 75],I.4.3), there is a δ in A such that

A |= φ(u,∃β < δ[· ∈ F (β)],~a),

yielding u ∈ F (δ,~a, b) ⊆ IA . Thus we have shown IA = Iφ,~a,b, confirming that Iφ,~a,b is a set. Theabove proof also shows that Γφ,~a,b(IA ) ⊆ IA , so Iφ,~a,b is closed under Γφ,~a,b. Moreover we have

y = Iφ,~a,b iff ∃A [“A is admissible” ∧ ~a, b ∈ A ∧ y =⋃F (ξ,~a, b) : ξ ∈ A ],

showing that the class function (~a, b 7→ Iφ,~a,b) is Σ and therefore also ∆ definable. 2

Definition 5.8 (KPi) Next we want to extend the inductive definition of U and V of Definition4.1 so as to allow for an interpretation of res–W–formation. To distinguish the new relations,say RW,α

1 , . . . , RW,α7 , from those in 4.1, we shall write UW

α |= and VWα |= in place of Uα |= and

Vα |=, respectively. Let w(m,n) = 〈8, 〈n,m〉N〉N. We now define the relations UWα |= n = m ∈ k,

UWα |= n = m, UW

α |= “f is a family of types over k”, VWα |= k set, and VW

α |= n = m by the oldclauses of the corresponding relations in Definitions 4.1, together with the new clauses:

If UWα |= a set, and UW

α |= “ e is a family of types over a ”, then

1. UWα+1 |= w(a, e) set,

2. UWα+1 |= x ∈ w(a, e) iff xSx,

3. UWα+1 |= x = y ∈ w(a, e) iff xSy,

where S is inductively defined (on N) by the rule:

UWα |= x = y ∈ a ∧ ∀i, j[UW

α |= i = j ∈ e(x) ⇒ u(i)Sv(j)] ⇒ sup(x, u)S sup(y, v).(6)

Finally, define UW |= k set, VW |= k set, UW |= k = n, and VW |= k = n analogously toDefinition 4.3.

We are to justify that the preceding definition falls under the scope of Σ recursion (on the basisof KPi). However, this follows from Lemma 5.7 as S is inductively defined by a ∆0 formula in theparameters N, RW,α

1 , . . . , RW,α7 . 2

The analogues of Lemma 4.2 and Proposition 4.4 hold for the new relations. We shall verify thatthe new relation UW |= w(a, e) set provides for an interpretation of the W–type of ML1WV.

Lemma 5.9 (KPi) Suppose UW |= “e is a family of types over a”.

(i) If UW |= n ∈ a and UW |= b ∈ (e(n) → w(a, e)), 12 then UW |= sup(n, b) ∈ w(a, e).

(ii) (w–induction) If φ(u) is a set–theoretic formula such that, for all x, b, it holds that UW |= x ∈a, UW |= b ∈ (e(x) → w(a, e)

)and ∀m[UW |= m ∈ e(x) ⇒ φ(m)] imply φ(sup(x, b)),

then ∀n[UW |= n ∈ w(a, e) ⇒ φ(n)].12e(n) → w(a, e) denotes π

(e(n), λx.w(a, e)), where λx.w(a, e) stands for k(w(a, e)). Note that

k(w(a, e))(x) ' w(a, e).

28

Proof : (i): Choose α such that UWα |= “e is a family of types over a”, UW

α |= n ∈ a and UWα+1 |=

b ∈ (e(n) → w(a, e)). Then UWα+1 |= w(a, e) setand ∀i, j[UW

α |= i = j ∈ e(n) ⇒ UWα+1 |=

b(i) = b(j) ∈ w(a, e)]. Letting S be defined as in (6), the latter reads ∀i, j[UW

α |= i = j ∈e(n) ⇒ b(i)Sb(j)], thus, since also UW

α |= n = n ∈ a, the inductive definition of S entailssup(n, b)S sup(n, b), which means UW

α+1 |= sup(n, b) ∈ w(a, e), and therefore UW |= sup(n, b) ∈w(a, e).

(ii): Pick α such that UWα |= “ e is a family of types over a ” . Then UW

α+1 |= w(a, e) set. Sinceall elements of w(a, e) are added at the same time as w(a, e) is declared a set, it suffices to show

∀n[UWα+1 |= n ∈ w(a, e) ⇒ φ(n)]. (7)

Again, let S be the relation from 5.8. (7) then reads:

∀n[nSn → φ(n)]. (8)

We show (8) by induction on the way S was built up, which comes down to transfinite inductionon the ordinals. Assume nSn. Then n = sup(x, u) for some x, u ∈ N, UW

α |= x = x ∈ a, and

∀i, j[i = j ∈ e(x) ⇒ u(i)Su(j)]. (9)

Thus, inductively we may assume

∀i[UWα |= i ∈ e(x) ⇒ φ(u(i))]. (10)

Moreover, by (9), we also have

UW |= u ∈ (e(x) → w(a, e)). (11)

Employing (10) and (11), and the assumption on φ, we get φ(sup(x, u)), thus φ(n). 2

Definition 5.10 (Interpretation of ML1WV in KPi) By now it should be obvious how to extend∧ to an interpretation of ML1WV in KPi. First, note that ML1WV has the new constants Wand TW that need to be translated into terms of EON. Set W∗ := λx.λy.(8, (x, y)) and let T∗

W

be determined by the recursion theorem of EON as to satisfy

T∗W(sup∗(a,b), λ∗(λx1.λx2.λx3.d(x1,x2,x3))) '

d(a, b, λ∗(λv.T∗W(ap∗(b,v),λ∗(λx1.λx2.λx3.d(x1,x2,x3))))).

To the definition of set terms in 4.9, the following clause has to be added:

If B(x) and A are set terms, and x is not free in A or in B, then W(A,B), is a setterm.

The interpretation ∧ of 4.10 has to be complemented/changed as follows:

(x ∈ W(A,B))∧ is ∃u[(W(A, B)∗ ' u)∧ ∧ UW |= x ∈ u](x = y ∈ W(A,B))∧ is ∃u[(W(A, B)∗ ' u)∧ ∧ UW |= x = y ∈ u]

(u ∈ U)∧ is UW |= u set(u = v ∈ U)∧ is UW |= u = v

(u ∈ V)∧ is VW |= u set(u = v ∈ V)∧ is VW |= u = v

29

Theorem 5.11 If φ is a judgement of ML1WV not of the form “A set,” then KPi ` (Φ)∧.

Proof : The soundness of the rules pertaining to W follows from Lemma 5.9 and the choice ofT∗

W, more precisely, U–introduction and U–elimination are taken care of by 5.9(i) and 5.9(ii),respectively. 2

In view of the inductive definition of UW |= n set, it is clear that the interpretation ∧ also validatesU–induction in the form adapted to ML1WV.

Theorem 5.12 If Φ is a judgement of ML1WV+U–induction not of the form “A set”, thenKPi ` (Φ)∧.

Theorem 5.13 The following theories ML1WV, ML1WV+U–induction, T0, ∆12– CA + BI, KPi

CZF + REA CZF + REA minus Subset Collection and CZF + REA + DC + ΠΣI −AC +ΠΣI −PA are of the same proof–theoretic strength.

Proof : This follows from 4.3, 5.4, 5.6, 5.11, 3.3 and 5.12. 2

6 The system ML1W

The system ML1WV has an inductive type that does not belong to U, i.e. the type V. Here thequestion arises whether V really contributes to the proof–theoretic strength of ML1WV.

Definition 6.1 We denote by ML1W the theory resulting from ML1WV by depriving it of thetype V.

Since ML1V is considerably stronger than ML1 by [A 77] and Theorem 4.14, one might gatherthat this also holds true of ML1WV and ML1W. However, this is not the case.

In this paper we have so far avoided advanced techniques from ordinal–theoretic proof theoryand rather combined “soft” proof–theoretic methods (like interpreting a theory in another theory)with results from the literature, however, some of them heavily rely on ordinal analyses of theories.Notwithstanding that it would be possible to carry out well–ordering proofs for ordinal notationsystems in ML1W that would show ML1W to be of the strength of ∆1

2– CA + BI, we shallcontinue in using interpretations for determining the strength of ML1W. Unfortunately, the typeV is of central importance in Aczel’s interpretation of CZF + REA in ML1WV. Our approachwill therefore be to design an intuitionistic theory of second order arithmetic that is sufficient forcarrying out the well–ordering proofs in [J 83] and can be interpreted in ML1W.

Definition 6.2 (The theory IARI) IARI is a theory in the language of second order arithmeticwith set variables. The constants and function symbols are 0 (zero), S (successor), + (plus), ·(times), and 〈 , 〉, ( )0, ( )1 for an injective pairing function and its inverses. The only predicatesymbol is = for equality on the natural numbers. The logical rules of IARI are those of intuitionisticsecond order arithmetic. The arithmetic axioms are the usual defining axioms for 0, S,+, · and∀n∀m[(〈n,m〉)0 = n ∧ (〈n,m〉)1 = m]. Equality for sets will be considered a defined notion, that isto say X = Y := ∀n[n ∈ X ↔ n ∈ Y ].

30

In addition to the usual axioms for intuitionistic second order logic, axioms are (the universalclosures of):

1. Induction:φ(0) ∧ ∀n[φ(n) → φ(n + 1)] → ∀nφ(n)

for all formulae φ.

2. Arithmetic Comprehension Schema:

∃X∀n[n ∈ X ↔ ψ(x)]

for ψ arithmetical (parameters allowed).

3. Replacement:∀X[∀n ∈ X∃ !Y φ(n, Y ) → ∃Z∀n ∈ X φ(n, (Z)n)]

for all formulas φ. Here φ(n, (Z)n) arises from φ(n,Z) by replacing each occurrence t ∈ Zin the formula by 〈n, t〉 ∈ Z.

4. Inductive Generation:

∀U∀X∃Y [WPU (X,Y ) ∧ (∀n[∀k(k <X n → φ(k)) → φ(n)] → ∀m ∈ Y φ(m))

],

for all formulas φ, where k <X n abbreviates 〈k, n〉 ∈ X and WPU (X, Y ) stands for

ProgU (X, Y ) ∧ ∀Z[ProgU (X, Z) → Y ⊆ Z]

with ProgU (X,Y ) being ∀n ∈ U [∀k(k <X n → k ∈ Y ) → n ∈ Y ].

Remark 6.3 (IARI) Note that WPU (X, Y ) and WPU (X,Y ′) imply Y = Y ′, i.e. ∀n(n ∈ Y ↔n ∈ Y ′). Therefore, if WPU (X,Y ), then

∀n ∈ U [∀k <X nφ(k) → φ(n)] → ∀m ∈ Y φ(m)

holds for all formulae φ.

The latter principle will be referred to as “ induction over the well–founded part of <X” . In therest of this section we shall write WF(U,X) for the (extensionally) uniquely determined Y whichsatisfies WPU (X, Y ).

The main tool for performing the wellordering proof of [J 82] in IARI is the following principleof transfinite recursion.

Proposition 6.4 (IARI) If WPU (X,Y ) and ∀n ∈ Y ∀W∃!V ψ(n,W, V ), then there exists Z suchthat

∀n∈Y ψ(n,⋃(Z)k : k <X n, (Z)n).

Proof : The proof proceeds by induction over Y , i.e. by induction over the well–founded part of<X with respect to U , and is similar to the proof of definition by recursion in CZF (cf. Proposi-tion 2.1); here Replacement is used instead of Strong Collection, and Arithmetic Comprehensionreplaces Bounded Separation. 2

Next we go about interpreting IARI in ML1W.

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Definition 6.5 (Interpretation of IARI in ML1W)

Terms. Each function symbol for an n-place primitive recursive function is interpreted as a functionf¦ ∈ N →n N in type theory, where N →0 X := N and N →n+1 X := N → (N →n X). Wedefine

s¦ := (λx)sN(x),(Pn

k )¦ := (λx1) · · · (λxn)xk,

(Cnk )¦ := (λx1) · · · (λxn)sk

N(0), ( where s0N := 0; sk+1

N := ap(sN, skN))

((Subnm))¦[g, h1, . . . , hn] := (λx1) · · · (λxn)ap(g¦,ap(h¦1, x1, . . . , xn), . . . ,ap(h¦m, x1, . . . , xn))

((Recn[g, h]))¦ := (λx1) · · · (λxn+1)R(xn+1,ap(g¦, x1, . . . , xn), (x, y)ap(h¦, x1, . . . , xn, x, y)).

For terms we let 0¦ := 0, x¦ := x if x is a numerical variable, and if t has the form f(t1, . . . , tn) let

t¦ := f¦(t¦1, . . . , t¦n) := ap(. . .ap(f¦, t¦1), . . . , t¦n).

It is obvious that we should interpret the numerical variables of IARI in type theory as rangingover N. It might be less obvious how to model the set variables of IARI. This is done by definingthe weak power set of N , P(N), as the set of of predicates on N with truth conditions in theuniverse U, i.e. P(N) := N → U. For n ∈ N and X ∈ P(N), membership of n in X is defined by

n ∈X := ap(X, n).

With this notion of power set, comprehension with respect to a property φ(n) ∈ U (n ∈ N) isimmediate, by letting

n ∈ N : φ(n) := (λx)φ(x) ∈ P(N). (12)

The subset relation ( ⊂ ) and extensional equality ( = ) on P(N) are defined in the obvious way asfollows.

X ⊂Y := ∀n ∈ N(n ∈X → n ∈Y)

andX =Y := X ⊂Y ∧ Y ⊂X.

If Xi ∈ P(N) (i ∈ I) is a family of sets with I ∈ U, let

⋃Xi : i ∈ I := n ∈ N : ∃i ∈ I(n ∈Xi).

We shall write s =N t for I(N, s, t). Variables n, k, m are supposed to range over N, thus, e.g., ∀kabbreviates ∀k ∈ N. ∀n ∈X(. . . ) is shorthand for ∀n ∈ N(n ∈X → . . . ). Formulas of second orderarithmetic are now translated as type expressions as follows:

[s = t]¦ := s¦ =N t¦,[s ∈ X]¦ := s¦ ∈X,[φ ψ]¦ := φ¦ ψ¦ for ∈ ∧,∨,→,

⊥¦ := I(N, sN(0), 0),[Qy φ]¦ := (Qy ∈ N)φ¦ for Q ∈ ∀, ∃,

[QX φ]¦ := (QX ∈ P(N))φ¦ for Q ∈ ∀, ∃.

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Definition 6.6 If F is a family of propositions (i.e. types) over the type A we call F a speciesover A.

In the following deductions in type theory are presented in an informal style. The following lemmadeals with the interpretation of Replacement.

Lemma 6.7 Suppose A ∈ P(N). Let F be a species over N×P(N) such that

∀n ∈A ∃X ∈ P(N)F(n,X) true, (13)

∀n ∈A ∀X, Y ∈ P(N) [X =Y ∧ F(n,X) → F(n,Y)] true, (14)

∀n ∈A∀X,Y ∈ P(N) [F(n,X) ∧ F(n,Y) → X =Y] true. (15)

Then there exists C ∈ P(N) such that, with g(n) := m ∈ N : ∃k ∈C(〈n,m〉¦ =N k),∀n ∈AF (n, g(n)) true.

Proof : From (13) it follows ∀n[∃v ∈ A(n) → ∃X ∈ P(N)F(n,X)] true, thus ∀n∀v ∈ A(n)∃X ∈P(N)F(n,X) true, and hence13

∀z ∈ (Σv ∈ NA(v))∃X ∈ P(N)F(p0(z),X) true.

As a result, the axiom of choice in type theory provides us with a function h ∈ (Σv ∈ NA(v)) →P(N) so that

∀z ∈ (Σv ∈ NA(v))F (p0(z), h(z)) true.

Now define g ∈ N → P(N) by

g(n) :=⋃h(z) : z ∈ (Σv ∈ NA(v)) ∧ p0(z) =N n.

As a consequence of (15) we have

∀z, z′ ∈ (Σv ∈ NA(v)) [p0(z) =N p0(z′) → h(z) =h(z′)] true. (16)

Now if n ∈A true we may pick z ∈ (Σv ∈ NA(v)) such that p0(z) =N n true. Then g(n) =h(z) trueby (16) and, as F (n, h(z)) true, we obtain F (n, g(n)) true using (14). Finally, set

C := z ∈ N : ∃v ∈ N∃u ∈ g(v)(〈v, u〉¦ =N z) 2

The next lemma addresses Inductive Generation.

Lemma 6.8 Let U,X ∈ P(N) and define n <X m := 〈n,m〉¦ ∈X. Then there exists Y ∈ P(N)such that

∀n ∈U [∀k(k <X n → k ∈Y ) → n ∈Y ] true (17)

and for each species G on N,

∀n ∈U [∀k(k <X n → G(k)) → G(n)] → ∀n ∈Y G(n) true. (18)13p0,p1 are defined by p0(c) := E(c, (x, y).x) and p1(c) := E(c, (x, y).y), where E is the eliminatory constant

related to Σ.

33

Proof : Let A = (Σn ∈ N)U(n). Let B be the species on A defined by B(v) := (Σk ∈N)(k <X p0(v)). Then B ∈ A → P(N). Set W := W(A,B). Define Γ : P(N) → P(N) by

Γ(Z) := n ∈ N : n ∈U ∧ ∀k(k <X n → k ∈Z).Define F : W → P(N) by transfinite recursion on W so that

F (sup(a, f)) := Γ(⋃F (f(s)) : s ∈ B(a))

for a ∈ A, f ∈ B(a) → W . Finally set

Y :=⋃F (s) : s ∈ W.

Then Y ∈ P(N). To verify (17), assume n ∈U true and ∀k(k <X n → n ∈Y ) true. Pick x0 ∈ U(n)and let v0 := (n, x0). Then v0 ∈ A and ∀z ∈ B(v0)∃s ∈ W [p0(z) ∈F (s)] true. Employing theaxiom of choice in type theory, there is an f ∈ B(v0) → W such that

∀z ∈ B(v0)[p0(z) ∈F (f(z))] true.

Therefore we get

∀k(k <X n → k ∈⋃F (f(z)) : z ∈ B(v0)) true.

Thus n ∈F (sup(v0, f)) true. As F (sup(v0, f)) ⊂Y true, this implies n ∈Y true.To verify (18) let G be a species over N such that

∀n ∈U [∀k(k <X n → G(k)) → G(n)] true. (19)

We use transfinite induction over W to verify ∀s ∈ W [∀k ∈F (s) G(k))] true which comprehends(18). So assume a ∈ A, f ∈ B(a) → W and ∀x ∈ B(a)∀k ∈F (f(x))G(k) true. We wish to show∀n ∈F (sup(f, a))G(n) true. So assume n ∈F (sup(f, a)) true. This means

n ∈ Γ(⋃F (f(s)) : s ∈ B(a)) true,

thus n ∈U true and ∀k(k <X n → ∃x ∈ B(a) [k ∈F (f(x))]) true. As ∀x ∈ B(a)∀k ∈F (f(x))G(k)true, we obtain ∀k[k <X n → G(k)] true. Therefore G(n) true using (19) and n ∈U true. 2

In actuality, the intensional version14 MLi1W of ML1W suffices for the interpretation of IARI.

Theorem 6.9 Any theorem of IARI is true on interpretation in MLi1W, that is to say, for each

theorem φ of IARI, φ¦ is a proposition of MLi1W and MLi

1W ` t ∈ φ¦, for a suitable closed termt.

Proof : By induction on the length of deductions in IARI. A natural deduction formulationof IARI is most convenient. That ¦ validates intuitionistic arithmetic is, for example, provedin [Be 85],XI.17 and [TD 88],Ch. 11, Sect. 4. It is also routine to check that ¦ validates the(intuitionistic) laws for the second order quantifiers. As to equality, note that

∀u, v ∈ N ∀X, Y ∈ P(N) [u =N v → (u ∈X ↔ v ∈Y)]14For a definition of the intensional versions of type theories see [TD 88],Ch.11,Sect.5.

34

holds true in MLi1W by the very definitions of =N and ∈ . Therefore, if IA is obtained from

IARI by retaining only the part without Arithmetic Comprehension, Replacement, and InductiveGeneration, the theorem holds with IA in place of IARI. We thus only have to check thoseaxioms. As to Arithmetic Comprehension, note that if φ(x, ~y, ~Z) is an arithmetic formula then, forn, ~m ∈ N and ~X ∈ P(N), φ(n, ~m, ~X)¦ is a small type, that is to say φ(n, ~m, ~X)¦ ∈ U; and thereforethe validity of Arithmetic Comprehension is confirmed by (12). Replacement is taken care of byLemma 6.7 and Inductive Generation follows from Lemma 6.8. 2

To determine a lower bound for the strength of IARI we show that the well–ordering proof of[J 83] can be carried out in IARI, thereby confirming that IARI has the same proof–theoreticstrength as ∆1

2– CA + BI.In what follows we assume familiarity with [J 83]. We shall only dwell on those parts of the

latter paper that require an alternative development in IARI. Just by glancing through [J 83] itbecomes clear that everything up to Lemma 2.14 is provable in IARI, where of course classificationsare conceived as sets of natural numbers and the role of T0’s Inductive Generation is now takenover by the Inductive Generation of IARI, that is to say i(X,R) must be replaced with WF(X, R).

The first place that needs to undergo a major change is [J 83] Lemma 2.15 since it draws onthe operation w which is defined there by invoking the recursion theorem of T0. Instead, we shalluse transfinite recursion over distinguished sets in the form of Proposition 6.4.

Definition 6.10 (IARI) In keeping with [J 83], let (OT, <OT) denote the notation system. Weidentify <OT with the set 〈n,m〉 : n <OT m. Let Q be a distinguished set. Then WF(Q,<OT

) = Q. So we can define a set ZQ by transfinite recursion over Q by

(ZQ)a =

(ZQ)Sa if a /∈ CT0

WF(u ∈ OT : Sa = 0, <OT) if a = 0WF(Ma, <) if a ∈ CT ,

where for u ∈ CT ,

Mu = x ∈ OT : Sx ≤OT u; ∀v ∈ Wu [Kv ⊂ Wu],Wu =

(ZQ)u∗ if u ∈ L0⋃(ZQ)u[x] : x ∈ Q; x <OT t(u) if u /∈ L0.

In order for the latter to be a legitimate transfinite recursion over Q we must guarantee that, foru ∈ Q, if x ∈ Q and x <OT t(u), then u[x] ∈ Q and u[x] <OT u. But this follows from [J 83]Lemma 2.6.

Remark 6.11 (IARI) For distinguished sets Q,P and x ∈ Q ∩ P , it follows from [J 83], Lemma2.13 that (Q)x = (P )x. Thus we may define an operation C on the whole of

W :=⋃Q : Q is distinguished

by letting

C(x) = (ZQ)x for some Q with x ∈ Q.

Lemma 6.12 (IARI) For all a ∈ W, C(a) is a distinguished set.

35

Proof : Let Q be a distinguished set such that a ∈ Q. Use transfinite induction on Q to verify thatC(a) is distinguished. Note that, for all a ∈ Q, (ZQ)a = C(a). The proof of the lemma is then thesame as for the statement “a ∈ Q → ∃XC(X, a)” in [J 83] Theorem 2.4. 2

If one now replaces the assertion “∃XC(X, a)” with “C(a) is distinguished” and “W” with “W” inthe remainder of [J 83] (from Theorem 2.4 till the end), the same proofs will work. The upshot isthat IARI has at least the proof–theoretic strength of ∆1

2– CA + BI. Hence, in view of Theorem5.13, we have ascertained the strength of ML1W.

Theorem 6.13 ML1W has the same proof–theoretic strength as the theories listed in Theorem5.13, in particular the same strength as ∆1

2– CA + BI. The same holds for MLi1W.

Remark 6.14 There is also a subsystem of ML1W that has exactly the strength of ∆12–CA. This

theory arises from ML1W by further demanding that W–elimination has to be restricted to familiesin U.

7 The Strength of ML1W

For an interpretation of ML1W in set theory one also has to deal with the W–types “at large,”i.e. those which are not ranging over U.

Definition 7.1 Let KP1W be defined as follows. The language of KP1W is the language of settheory augmented by infinitely many constants A 0,A 1,A 2, . . . . The axioms of KP1W consist ofthe axioms of KP and the formulae:

1. “A i is a transitive non-empty set” (i ∈ N)

2. A i ∈ A j if i < j

3. φA 0 if φ is an axiom of KPi

4. ψA i+1 if ψ is an axiom of KP (i ∈ N)

Definition 7.2 (Interpretation of ML1W in KP1W) This interpretation (we call it ∧ again) isdefined in the same way as in 5.10 except for the W-type whose interpretation we take from 5.8.

(u ∈ W(A,B))∧ := uSu,(u = v ∈ W(A,B))∧ := uSv,

where S is inductively defined (on N) by the rule:

(*) If (x = y ∈ A)∧ and ∀x′, y′ ∈N[(x′ = y′ ∈ B(x))∧ → u(x′)Sv(y′)], then

sup(x, u)S sup(y, v).

Note that in order for S to be definable in KP1W, 〈x, y〉 ∈ N×N : (x = y ∈ A)∧ and 〈x, y′, x〉 ∈N3 : (x′ = y′ ∈ B(x))∧ have to be sets. Thus in order for this interpretation to make sense, wehave to verify that each set term A of ML1W gets interpreted by a set, i.e. x : (x ∈ A)∧ is a setusing only means available in KP1W.

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Lemma 7.3 For C a set term we denote by |C| the number of steps it takes to generate C (bythe clauses of 4.9). If C is a set term of ML1W with free variables among ~z (= z1, . . . , zn), thenKP1W proves

〈u, ~z〉 ∈ Nn+1 : (u ∈ C)∧ ∈ A |C|

and〈u, v, ~z〉 ∈ Nn+2 : (u = v ∈ C)∧ ∈ A |C|.

Proof : We proceed by (meta) induction on the generation of C. The assertions are obvious whenC is N or Nk.

1. If C is U, then (u ∈ C)∧ is UW |= u set. Now, u : UW |= u set is Σ definable in KPiand therefore this class has a definition wherein all quantifiers are restricted to A 0. Thusu : UW |= u set is an element of A 1 by ∆0 separation in A 1. Similarly, 〈u, v〉 ∈ N2 :UW |= u = v ∈ A 1.

2. If C is Π(A,B), then 〈u, ~z〉 ∈ Nn+1 : (u ∈ C)∧ is

〈u, ~z〉 ∈ Nn+1 : ∀x∈N[(x ∈ A)∧ → (ux ∈ B(x))∧]∧∀x, y ∈N[(x = y ∈ A)∧ → (ux = uy ∈ B(x))∧].

By induction hypothesis, we obtain

〈u, ~z, x〉 ∈ Nn+1 : (x ∈ A)∧ ∈ A n,〈v, x, ~z〉 ∈ Nn+2 : (v ∈ B(x))∧ ∈ A n,

〈x, y, ~z〉 ∈ Nn+2 : (x = y =∈ A)∧ ∈ A n,〈v, w, x, ~z〉 ∈ Nn+3 : (v = w ∈ B(x))∧ ∈ A n,

where n = max(|A|, |B(x)|).15 As 〈u, ~z〉 ∈ Nn+1 : (u ∈ C)∧ is ∆0 definable, using the abovesets as parameters, this is a set in A n and thus also in A n+1, which is A |C|.

The argument for 〈u, v, ~z〉 ∈ Nn+2 : (u = v ∈ C)∧ ∈ A |C| proceeds along the same lines.The same arguments apply when C is a set term via one of the clauses (2)–(6) of 4.9

3. Suppose C is W(A,B). Inductively we get that

〈x, y, ~z〉 ∈ Nn+2 : (x = y ∈ A)∧ and 〈x, y′, ~z〉 ∈ Nn+2 : (x′ = y′ ∈ B)∧

are elements of A n with n = max(|A|, |B(x)|). Now let S~z be the relation inductively definedin 7.2(*). The proof of Lemma 5.7 reveals that 〈x′, y′, ~z〉 : x′S~zy

′ is Σ1 definable over A n,whence 〈x′, y′, ~z〉 : x′S~zy

′ ∈ A n+1. Consequently,

〈u, ~z〉 ∈ Nn+1 : (u ∈ W(A,B))∧

and〈u, v, ~z〉 ∈ Nn+2 : (u = v ∈ W(A,B))∧

are also elements of A n+1. 2

15This case shows the need for the extra parameters ~z in the formulation of the lemma

37

Theorem 7.4 If Φ is a judgement of ML1W not of the form “A set”, then KP1W ` Φ∧.

Proof : By Lemma 7.3, ∧ really provides a syntactic translation of Φ into a formula of set theory.On the strength of its definition, it is clear that the rules for the W–types are sound under thisinterpretation. 2

By now we know that KP1W is an upper bound for the proof–theoretic strength of ML1W.Actually, this bound is sharp according to Setzer’s thesis [S 93]. As regards a proof, we wouldhave to employ rather advanced techniques from Gentzen–style proof theory, including an ordinalanalysis of KP1W. However, once these techniques are digested it is fairly easy to provide anordinal analysis of KP1W that yields the right bound. Indeed, the paper [R 91] permits one toalmost read off the ordinal of KP1W which is ψΩ1(I + ω), where I is a notation for the firstinaccessible ordinal. The next step would then consist in showing that, for all (meta) n, ML1Wproves the well–foundedness of the notation system up to αn, where αn = ψΩ1ΩI+n. This wouldshow that ML1W has the proof–theoretic ordinal supαn : n < ω = ψΩ1ΩI+ω.

8 Conclusions

Similar results can be established for the theories MLnW. As a rule of thumb, the tower of nuniverses in MLnW corresponds to n recursively inaccessible universes in classical Kripke–Platekset theory.

References

[A 77] P. Aczel: The Strength of Martin–Lof ’s Intuitionistic Type Theory with One Universe,in: S. Miettinen, S. Vaananen, eds. Proceedings of Symposia in Mathematical Logic,Oulu, 1974, and Helsinki, 1975. Report No. 2, University of Helsinki, Department ofPhilosophy, 1977, 1–32.

[A 78] P. Aczel: The Type Theoretic Interpretation of Constructive Set Theory, in: Mac-Intyre, A. Pacholski, L., and Paris, J. (eds.), Logic Colloquium ’77, North–Holland,Amsterdam 1978.

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the strength of some martin{l˜of type theoriesrathjen/typeohio.pdfthe strength of some martin{l˜of...

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