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Full intuitionistic Zermelo-Fraenkel set theory, IZF, is obtained from constructive Zermelo-Fraenkel set theory, CZF, by adding the full separation axiom scheme and the power set axiom. The strength of CZFplus full separation is the same as that of second order arithmetic, using a straightforward realizability interpretation in classical second order arithmetic and the fact that second order Heyting arithmetic is already embedded in CZFplus full separation. This paper is concerned with the strength of CZFaugmented by the power set axiom, \({\mathbf{CZF}}_{\mathcal{P}}\). It will be shown that it is of the same strength as Power Kripke–Platek set theory, \(\mathbf{KP}(\mathcal{P})\), as well as a certain system of type theory, \({\mathbf{MLV}}_{\mathbf{P}}\), which is a calculus of constructions with one universe. The reduction of \({\mathbf{CZF}}_{\mathcal{P}}\)to \(\mathbf{KP}(\mathcal{P})\)uses a realizability interpretation wherein a realizer for an existential statement provides a set of witnesses for the existential quantifier rather than a single witness. The reduction of \(\mathbf{KP}(\mathcal{P})\)to \({\mathbf{CZF}}_{\mathcal{P}}\)employs techniques from ordinal analysis which, when combined with a special double negation interpretation that respects extensionality, also show that \(\mathbf{KP}(\mathcal{P})\)can be reduced to CZFwith the negative power set axiom. As CZFaugmented by the latter axiom can be interpreted in \({\mathbf{MLV}}_{\mathbf{P}}\)and this type theory has a types-as-classes interpretation in \({\mathbf{CZF}}_{\mathcal{P}}\), the circle will be completed.

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... We will not provide a proof for the above proposition, but the readers may consult with [5] or [6] for its proof. We also note here that [46] showed that Subset Collection does not increase the proof-theoretic strength of CZF − while [52] showed that the Axiom of Power Set does. ...

... (Types have treelike structures.) Another construction on that line is functional realizability, which we define below for the sake of completeness: (3) Set realizability, which appears in [52] and [53]. This exploits the computational nature of sets to construct an interpretation. ...

We investigate large set axioms defined in terms of elementary embeddings over constructive set theories, focusing on $\mathsf{IKP}$ and $\mathsf{CZF}$. Most previously studied large set axioms, notably the constructive analogues of large cardinals below $0^\sharp$, have proof-theoretic strength weaker than full Second-order Arithmetic. On the other hand, the situation is dramatically different for those defined via elementary embeddings. We show that by adding to $\mathsf{IKP}$ the basic properties of an elementary embedding $j\colon V\to M$ for $\Delta_0$-formulas, which we will denote by $\mathsf{\Delta_0\text{-}BTEE}_M$, we obtain the consistency of $\mathsf{ZFC}$ and more. We will also see that the consistency strength of a Reinhardt set exceeds that of $\mathsf{ZF+WA}$. Furthermore, we will define super Reinhardt sets and $\mathsf{TR}$, which is a constructive analogue of $V$ being totally Reinhardt, and prove that their proof-theoretic strength exceeds that of $\mathsf{ZF}$ with choiceless large cardinals.

... Semi-intuitionistic set theories with Bounded Separation but containing the Power Set axiom were proposed by Pozsgay [19,20] and then studied more systematically by Tharp [40], Friedman [7] and Wolf [43]. Such theories are naturally related to systems derived from topos-theoretic notions and to type theories (e.g., see [36]). Mac Lane has singled out and championed a particular fragment of ZF, especially in his book Form and Function [14]. ...

While power Kripke–Platek set theory, ${\textbf{KP}}({\mathcal{P}})$, shares many properties with ordinary Kripke–Platek set theory, ${\textbf{KP}}$, in several ways it behaves quite differently from ${\textbf{KP}}$. This is perhaps most strikingly demonstrated by a result, due to Mathias, to the effect that adding the axiom of constructibility to ${\textbf{KP}}({\mathcal{P}})$ gives rise to a much stronger theory, whereas in the case of ${\textbf{KP}}$, the constructible hierarchy provides an inner model, so that ${\textbf{KP}}$ and ${\textbf{KP}}+V=L$ have the same strength.
This paper will be concerned with the relationship between ${\textbf{KP}}({\mathcal{P}})$ and ${\textbf{KP}}({\mathcal{P}})$ plus the axiom of choice or even the global axiom of choice, $\textbf{AC}_{\tiny {global}}$. Since $L$ is the standard vehicle to furnish a model in which this axiom holds, the usual argument for demonstrating that the addition of ${\textbf{AC}}$ or $\textbf{AC}_{\tiny {global}}$ to ${\textbf{KP}}({\mathcal{P}})$ does not increase proof-theoretic strength does not apply in any obvious way. Among other tools, the paper uses techniques from ordinal analysis to show that ${\textbf{KP}}({\mathcal{P}})+\textbf{AC}_{\tiny {global}}$ has the same strength as ${\textbf{KP}}({\mathcal{P}})$, thereby answering a question of Mathias. Moreover, it is shown that ${\textbf{KP}}({\mathcal{P}})+\textbf{AC}_{\tiny {global}}$ is conservative over ${\textbf{KP}}({\mathcal{P}})$ for $\varPi ^1_4$ statements of analysis.
The method of ordinal analysis for theories with power set was developed in an earlier paper. The technique allows one to compute witnessing information from infinitary proofs, providing bounds for the transfinite iterations of the power set operation that are provable in a theory.
As the theory ${\textbf{KP}}({\mathcal{P}})+\textbf{AC}_{\tiny {global}}$ provides a very useful tool for defining models and realizability models of other theories that are hard to construct without access to a uniform selection mechanism, it is desirable to determine its exact proof-theoretic strength. This knowledge can for instance be used to determine the strength of Feferman’s operational set theory with power set operation as well as constructive Zermelo–Fraenkel set theory with the axiom of choice.

This new book on mathematical logic by Jeremy Avigad gives a thorough introduction to the fundamental results and methods of the subject from the syntactic point of view, emphasizing logic as the study of formal languages and systems and their proper use. Topics include proof theory, model theory, the theory of computability, and axiomatic foundations, with special emphasis given to aspects of mathematical logic that are fundamental to computer science, including deductive systems, constructive logic, the simply typed lambda calculus, and type-theoretic foundations. Clear and engaging, with plentiful examples and exercises, it is an excellent introduction to the subject for graduate students and advanced undergraduates who are interested in logic in mathematics, computer science, and philosophy, and an invaluable reference for any practicing logician's bookshelf.

The main goal of this paper is to formulate a constructive analogue of Ackermann's observation about finite set theory and arithmetic. We will see that Heyting arithmetic is bi-interpretable with CZFfin, the finitary version of CZF. We also examine bi-interpretability between subtheories of finitary CZF and Heyting arithmetic based on the modification of Fleischmann's hierarchy of formulas, and the set of hereditarily finite sets over CZF, which turns out to be a model of CZFfin but not a model of finitary IZF.

The main goal of this paper is to formulate a constructive analogue of Ackermann's observation about finite set theory and arithmetic. We will see that Heyting arithmetic is bi-interpretable with $\mathsf{CZF^{fin}}$, the finitary version of $\mathsf{CZF}$. We also examine bi-interpretability between subtheories of finitary $\mathsf{CZF}$ and Heyting arithmetic based on the modification of Fleischmann's hierarchy of formulas, and the set of hereditarily finite sets over $\mathsf{CZF}$, which turns out to be a model of $\mathsf{CZF^{fin}}$ but not a model of finitary $\mathsf{IZF}$.

This article is concerned with classifying the provably total set-functions of Kripke-Platek set theory, KP, and Power Kripke-Platek set theory, KP(P), as well as proving several (partial) conservativity results. The main technical tool used in this paper is a relativisation technique where ordinal analysis is carried out relative to an arbitrary but fixed set x. A classic result from ordinal analysis is the characterisation of the provably recursive functions of Peano Arithmetic, PA, by means of the fast growing hierarchy [10]. Whilst it is possible to formulate the natural numbers within KP, the theory speaks primarily about sets. For this reason it is desirable to obtain a characterisation of its provably total set functions. We will show that KP proves the totality of a set function precisely when it falls within a hierarchy of set functions based upon a relativised constructible hierarchy stretching up in length to any ordinal below the Bachmann-Howard ordinal. As a consequence of this result we obtain that IKP + forall x,y (x in y or x not in y) is conservative over KP for forall-exists-formulae, where IKP stands for intuitionistic Kripke-Platek set theory. In a similar vein, utilising [56], it is shown that KP(P) proves the totality of a set function precisely when it falls within a hierarchy of set functions based upon a relativised von Neumann hierarchy of the same length. The relativisation technique applied to KP(P) with the global axiom of choice, AC-global, also yields a parameterised extension of a result in [58], showing that KP(P) + AC-global is conservative over KP(P) + AC and CZF + AC for Pi_2 in powerset statements. Here AC stands for the ordinary axiom of choice and CZF refers to constructive Zermelo-Fraenkel set theory.

Until the 1970s, proof theoretic investigations were mainly concerned with theories of inductive definitions, subsystems of analysis and finite type systems. With the pioneering work of Gerhard Jaeger in the late 1970s and early 1980s, the focus switched to set theories, furnishing ordinal-theoretic proof theory with a uniform and elegant framework. More recently it was shown that these tools can even sometimes be adapted to the context of strong axioms such as the powerset axiom, where one does not attain complete cut elimination but can nevertheless extract witnessing information and characterize the strength of the theory in terms of provable heights of the cumulative hierarchy. Here this technology is applied to intuitionistic Kripke-Platek set theories IKP(P) and IKP(E), where the operation of powerset and exponentiation, respectively, is allowed as a primitive in the separation and collection schemata. In particular, IKP(P) proves the powerset axiom whereas IKP(E) proves the exponentiation axiom. The latter expresses that given any sets A and B, the collection of all functions from A to B is a set, too. While IKP(P) can be dealt with in a similar vein as its classical cousin, the treatment of IKP(E) posed considerable obstacles. One of them was that in the infinitary system the levels of terms become a moving target as they cannot be assigned a fixed level in the formal cumulative hierarchy solely based on their syntactic structure. As adumbrated in an earlier paper, the results of this paper are an important tool in showing that several intuitionistic set theories with the collection axiom possess the existence property, i.e., if they prove an existential theorem then a witness can be provably described in the theory, one example being intuitionistic Zermelo-Fraenkel set theory with bounded separation.

The paper relativizes the method of ordinal analysis developed for Kripke–Platek set theory to theories which have the power set axiom. We show that it is possible to use this technique to extract information about Power Kripke–Platek set theory, KP(P)KP(P).
As an application it is shown that whenever KP(P)+ACKP(P)+AC proves a Π2P statement then it holds true in the segment VτVτ of the von Neumann hierarchy, where τ stands for the Bachmann–Howard ordinal.

A hallmark of many an intuitionistic theory is the existence property, EP, i.e., if the theory proves an existential statement then there is a provably definable witness for it. However, there are well known exceptions, for example, the full intuitionistic Zermelo–Fraenkel set theory, , does not have the existence property, where is formulated with Collection. By contrast, the version of intuitionistic Zermelo–Fraenkel set theory formulated with Replacement, , has the existence property. Moreover, does not even enjoy a weaker form of the existence property, wEP, defined by the slackened requirement of finding a provably definable set of witnesses for every existential theorem. In view of these results, one might be tempted to put the blame for the failure of the existence properties squarely on Collection. However, in this paper it is shown that several well known intuitionistic set theories with Collection have the weak existence property. Among these theories are , , and , i.e., respectively, constructive Zermelo–Fraenkel set theory ( ) without subset Collection, formulated with Exponentiation and also augmented by the Power Set axiom (basically with only bounded separation). As a result, the culprit preventing the weak existence property from obtaining must consist of a combination of Collection and unbounded Separation.

Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. Admissible set theory is a major source of interaction between model theory, recursion theory and set theory, and plays an important role in definability theory. In this volume, the seventh publication in the Perspectives in Logic series, Jon Barwise presents the basic facts about admissible sets and admissible ordinals in a way that makes them accessible to logic students and specialists alike. It fills the artificial gap between model theory and recursion theory and covers everything the logician should know about admissible sets. © 1975 Springer-Verlag Berlin Heidelberg and © 2016 Association for Symbolic Logic under license to Cambridge University Press.

The reader of this chapter is presumed to have some familiarity with the practice of constructive mathematics, acquired for example by studying the first half of either Bishop [1967] or Bridges [1979], together with the preceding chapter of this book. He or she will then be in a position to reflect fruitfully on the foundations of the subject. Every approach to the foundations of mathematics begins by analyzing the fundamental notions that occur again and again in the actual mathematics, and in terms of which the mathematics can be understood.

In this paper we will define a computation-theory called cursion, which will be a theory of partial set-recursive functions, defined on sets and with sets as values. We will use natural numbers as indices.E-re- The original purpose was to develop a theory on the companion of a normal functional of type k+2 such that semirecursion over type k in F and the theory are the same. The motivation for this was that this set-recursion theory might accept priority-arguments, arguments giving results about degrees of functionals. Some results from that program are given in Normann [I21 and [13]. The recursion theory we developed for that purpose, happened to be of a more general nature, and not quite unnatural even if one does not have the applications on degrees of functionals in mind. Moschovakis [ l o ] has constructed essentially the same theory, using inductive schemes and fixpoint operators. A computation theory on a structure must satisfy certain fundamental properties, composition of recursive functions gives a recursive function, you may diagonalize or compute on indices ([e)(eq,x>r (e,,](x>. which are so deeply connected with the structure that they obviously must be computable. In addition there may be some finiteness-properties, search-operators, stage-comparison etc. giving the theory its particular flavour.

Saunders Mac Lane has drawn attention many times, particularly in his book Mathematics: Form and Function, to the system ZBQC of set theory of which the axioms are Extensionality, Null Set, Pairing, Union, Infinity, Power Set, Restricted Separation, Foundation, and Choice, to which system, afforced by the principle, TCo, of Transitive Containment, we shall refer as MAC. His system is naturally related to systems derived from topos-theoretic notions concerning the category of sets, and is, as Mac Lane emphasises, one that is adequate for much of mathematics. In this paper we show that the consistency strength of Mac Lane's system is not increased by adding the axioms of Kripke–Platek set theory and even the Axiom of Constructibility to Mac Lane's axioms; our method requires a close study of Axiom H, which was proposed by Mitchell; we digress to apply these methods to subsystems of Zermelo set theory Z, and obtain an apparently new proof that Z is not finitely axiomatisable; we study Friedman's strengthening of , and the Forster–Kaye subsystem KF of MAC, and use forcing over ill-founded models and forcing to establish independence results concerning MAC and ; we show, again using ill-founded models, that proves the consistency of ; turning to systems that are type-theoretic in spirit or in fact, we show by arguments of Coret and Boffa that KF proves a weak form of Stratified Collection, and that is a conservative extension of MAC for stratified sentences, from which we deduce that proves a strong stratified version of ; we analyse the known equiconsistency of MAC with the simple theory of types and give Lake's proof that an instance of Mathematical Induction is unprovable in Mac Lane's system; we study a simple set theoretic assertion—namely that there exists an infinite set of infinite sets, no two of which have the same cardinal—and use it to establish the failure of the full schema of Stratified Collection in Z; and we determine the point of failure of various other schemata in MAC. The paper closes with some philosophical remarks.

A variant of realizability for Heyting arithmetic which validates Church’s thesis with uniqueness condition, but not the general form of Church’s thesis, was introduced by Lifschitz (Proc Am Math Soc 73:101–106, 1979). A Lifschitz counterpart to Kleene’s realizability for functions (in Baire space) was developed by van Oosten (J Symb Log 55:805–821, 1990). In that paper he also extended Lifschitz’ realizability to second order arithmetic. The objective here is to extend it to full intuitionistic Zermelo–Fraenkel set theory, IZF. The machinery would also work for extensions of IZF with large set axioms. In addition to separating Church’s thesis with uniqueness condition from its general form in intuitionistic set theory, we also obtain several interesting corollaries. The interpretation repudiates a weak form of countable choice, AC
ω,ω
, asserting that a countable family of inhabited sets of natural numbers has a choice function. AC
ω,ω
is validated by ordinary Kleene realizability and is of course provable in ZF. On the other hand, a pivotal consequence of AC
ω,ω
, namely that the sets of Cauchy reals and Dedekind reals are isomorphic, remains valid in this interpretation. Another interesting aspect of this realizability is that it validates the lesser limited principle of omniscience.

The main objective of this paper is to show that a certain formulae-as- classes interpretation based on generalized set recursive functions provides a self- validating semantics for Constructive Zermelo-Fraenkel Set theory, CZF. It is argued that this interpretation plays a similar role for CZF as the constructible hierarchy for classical set theory, in that it can be employed to show that CZF is expandable by several forms of the axiom of choice without adding more consistency strength.

Constructive set theory is a possible framework for the formalization of constructive mathematics. The chapter describes the formal system CZF+DC and its type theoretic interpretation. An inductive definition usually involves the characterization of a collection of objects as the smallest collection satisfying certain closure conditions. Such a characterization can be made explicit in one of at least two ways. The first way is to define the collection as the intersection of all collections that satisfies the closure conditions. Such an explicit definition is thoroughly impredicative in that the collection is defined using quantification over all collections. The second way is to build up the collection from below as the union of a hierarchy of stages. These stages of the inductive definition are indexed using some suitable notion of “ordinal.” The paradigm for a direct understanding of an inductive definition is that for the collection of natural numbers, which is characterized as the smallest collection containing zero and closed under the successor function.

Dedicated to the memory of my friend and colleague, Karel de Leeuw.

Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. This volume, the second publication in the Perspectives in Logic series, is an almost self-contained introduction to higher recursion theory, in which the reader is only assumed to know the basics of classical recursion theory. The book is divided into four parts: hyperarithmetic sets, metarecursion, α-recursion, and E-recursion. This text is essential reading for all researchers in the field. © 1990 Springer-Verlag Berlin Heidelberg and 2016 Association for Symbolic Logic under license to Cambridge University Press.

Constructive Zermelo-Fraenkel Set Theory, CZF, has emerged as a standard ref-erence theory that relates to constructive predicative mathematics as ZFC relates to classical Cantorian mathematics. A hallmark of this theory is that it possesses a type-theoretic model. Aczel showed that it has a formulae-as-types interpretation in Martin-Löf's intuitionist theory of types [14, 15]. This paper, though, is concerned with a rather different interpretation. It is shown that Kleene realizability provides a self-validating semantics for CZF, viz. this notion of realizability can be formalized in CZF and demonstrably in CZF it can be verified that every theorem of CZF is realized. This semantics, then, is put to use in establishing several equiconsistency re-sults. Specifically, augmenting CZF by well-known principles germane to Russian constructivism and Brouwer's intuitionism turns out to engender theories of equal proof-theoretic strength with the same stock of provably recursive functions.

The objective of this paper is to assay several forms of the axiom of choice that have been deemed constructive. In addition to their deductive relation-ships, the paper will be concerned with metamathematical properties effected by these choice principles and also with some of their classical models.

We interpret intuitionistic theories of (iterated) strictly positive inductive definitions (s.p.-ID
i
s) into Martin-Lf's type theory. The main purpose being to obtain lower bounds of the proof-theoretic strength of type theories furnished with means for transfinite induction (W-type, Aczel's set of iterative sets or recursion on (type) universes). Thes.p.-ID
i
s are essentially the wellknownID
i
-theories, studied in ordinal analysis of fragments of second order arithmetic, but the set variable in the operator form is restricted to occur only strictly positively. The modelling is done by constructivizing continuity notions for set operators at higher number classes and proving that strictly positive set operators are continuous in this sense. The existence of least fixed points, or more accurately, least sets closed under the operator, then easily follows.

We consider the question on the number of decidable models of a decidable theory. By the theorem on omitting recursive types,
a decidable theory having an undecidable prime model has a countable set of decidable nonisomorphic models; moreover, the
family of all recursive types is not countable (cf. Corollary 3.3.3).

It is shown how the strong ordinal notation systems that figure in proof theory and have been previously defined by employing large cardinals, can be developed directly on the basis of their recursively large counterparts. Thereby we provide a completely new approach to well-ordering proofs as will be exemplified by determining the proof-theoretic ordinal of the systemKPM of [R91].

By adding to Martin-LSf's intuitionistic theory of types a ‘type of sets’ we give a constructive interpretation of constructive set theory. constructive version of the classical conception of the cumulative hierarchy of sets. This interpretation is a

Let TC be intuitionistic higher-order arithmetic or intuitionistic ZF (with Replacement), both with Relativized Dependent Choice, or just Countable Choice. We show that TC[boxvr]∄x. A(x) (closed) gives TC[boxvr]A(t) for some comprehension term t. This solves a problem left open by Myhill in [4].

Constructive Zermelo–Fraenkel set theory, CZF, can be interpreted in Martin-Löf type theory via the so-called propositions-as-types interpretation. However, this interpretation validates more than what is provable in CZF. We now ask ourselves: is there a reasonably simple axiomatization (by a few axiom schemata say) of the set-theoretic formulae validated in Martin-Löf type theory? The answer is yes for a large collection of statements called the mathematical formulae. The validated mathematical formulae can be axiomatized by suitable forms of the axiom of choice.The paper builds on a self-interpretation of CZF (developed in [M. Rathjen, The formulae-as-classes interpretation of constructive set theory, in: Proof Technology and Computation (Proceedings of the International Summer School Marktoberdorf 2003) IOS Press, Amsterdam, 2004 (in press)]) that provides an “inner” model of CZF which also validates the so-called -axiom of choice, . The crucial technical step taken in the present paper is to investigate the absoluteness properties of this model under the hypothesis .It is also shown that CZF plus the -axiom of choice possesses the disjunction property, the numerical existence property and the existence property for an important group of formulae.

While it is known that intuitionistic ZF set theory formulated with Replacement, IZFR, does not prove Collection, it is a longstanding open problem whether IZFR and intuitionistic set theory ZF formulated with Collection, IZF, have the same proof-theoretic strength. It has been conjectured that IZF proves the consistency of IZFR. This paper addresses similar questions but in respect of constructive Zermelo–Fraenkel set theory, CZF. It is shown that in the latter context the proof-theoretic strength of Replacement is the same as that of Strong Collection and also that the functional version of the Regular Extension Axiom is as strong as its relational version.Moreover, it is proved that, contrary to IZF, the strength of CZF increases if one adds an axiom asserting that the trichotomous ordinals form a set.Unlike IZF, constructive Zermelo–Fraenkel set theory is amenable to ordinal analysis and the proofs in this paper make pivotal use thereof in the guise of well-ordering proofs for ordinal representation systems.

CZF + Separation is shown to be equiconsistent with second-order arithmetic,
using realizability.

V. Lifschitz defined in 1979 a variant of realizability which validates Church's thesis with uniqueness condition, but not the general form of Church's thesis. In this paper we describe an extension of intuitionistic arithmetic in which the soundness of Lifschitz' realizability can be proved, and we give an axiomatic characterization of the Lifschitz-realizable formulas relative to this extension. By a "q-variant" we obtain a new derived rule. We also show how to extend Lifschitz' realizability to second-order arithmetic. Finally we describe an analogous development for elementary analysis, with partial continuous application replacing partial recursive application.

Constructive set theory started with Myhill's seminal 1975 article [8]. This paper will be concerned with axiomatizations of the natural numbers in constructive set theory discerned in [3], clarifying the deductive relationships between these axiomatizations and the strength of various weak constructive set theories. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

This paper is the third in a series collectively entitled Formal systems of intuitionistic analysis . The first two are [4] and [5] in the bibliography; in them I attempted to codify Brouwer's mathematical practice. In the present paper, which is independent of [4] and [5], I shall do the same for Bishop's book [1]. There is a widespread current impression, due partly to Bishop himself (see [2]) and partly to Goodman and the author (see [3]) that the theory of Gödel functionals, with quantifiers and choice, is the appropriate formalism for [1]. That this is not so is seen as soon as one really tries to formalize the mathematics of [1] in detail. Even so simple a matter as the definition of the partial function 1/ x on the nonzero reals is quite a headache, unless one is prepared either to distinguish nonzero reals from reals (a nonzero real being a pair consisting of a real x and an integer n with ∣ x ∣ > 1/ n ) or, to take the Dialectica interpretation seriously, by adjoining to the Gödel system an axiom saying that every formula is equivalent to its Dialectica interpretation. (See [1, p. 19], [2, pp. 57–60] respectively for these two methods.) In more advanced mathematics the complexities become intolerable.

Let ZF be the usual Zermelo-Fraenkel set theory formulated without identity, and with the collection axiom scheme. Let ZF ⁻ -extensionality be obtained from ZF by using intuitionistic logic instead of classical logic, and dropping the axiom of extensionality. We give a syntactic transformation of ZF into ZF ⁻ -extensionality.
Let CPC, HPC respectively be classical, intuitionistic predicate calculus without identity, whose only homological symbol is ∈. We use the ~ ~-translation, a basic tool in the metatheory of intuitionistic systems, which is defined by
The two fundamental lemmas about this ~ ~ -translation we will use are
For proofs, see Kleene [3, Lemma 43a, Theorem 60d].
This - would provide the desired syntactic transformation at least for ZF into ZF ⁻ with extensionality, if A ⁻ were provable in ZF ⁻ for each axiom A of ZF. Unfortunately, the ~ ~-translations of extensionality and power set appear not to be provable in ZF ⁻ . We therefore form an auxiliary classical theory S which has no extensionality and has a weakened power set axiom, and show in §2 that the ~ ~-translation of each axiom of Sis provable in ZF ⁻ -extensionality. §1 is devoted to the translation of ZF in S .

It is natural, given the usual iterative description of the universe of sets, to investigate set theories which in some way take account of the unfinished character of the universe. We do not here consider any arguments aimed at justifying one system over another, or at clarifying the basic philosophy. Rather, we look at an obvious candidate which is similar to a system discussed by L. Pozsgay in [1]. Pozsgay sketched the development of the ordinary theorems in such a system and attempted to show it equiconsistent with ZF. In this paper we show that the consistency of the system we call IZF can be proved in the usual ZF set theory.

In this paper we calibrate the exact proof--theoretic strength of Kruskal's theorem, thereby giving, in some sense, the most elementary proof of Kruskal's theorem. Furthermore, these investigations give rise to ordinal analyses of restricted bar induction. Introduction S.G. Simpson in his article [10], "Nonprovability of certain combinatorial properties of finite trees", presents proof--theoretic results, due to H. Friedman, about embeddability properties of finite trees. It is shown there that Kruskal's theorem is not provable in ATR 0 . An exact description of the proof--theoretic strength of Kruskal's theorem is not given. On the assumption that there is a bad infinite sequence of trees, the usual proof of Kruskal's theorem utilizes the existence of a minimal bad sequence of trees, thereby employing some form of Pi 1 1 comprehension. So the question arises whether a more constructive proof can be given. The need for a more elementary proof of Kruskal's theorem is especially felt ...

The original motivation1 for the work described in this paper was to determine the proof theoretic strength of the type theories implemented in the proof development systems Lego and Coq, [12],[4]. These type theories combine the impredicative type of propositions2, from the calculus of constructions, [5], with the inductive types and hierarchy of type universes of Martin-Löf’s constructive type theory, [13]. Intuitively there is an easy way to determine an upper bound on the proof theoretic strength. This is to use the ‘obvious’ types-as-sets interpretation of these type theories in a strong enough classical axiomatic set theory. The elementary forms of type of Martin-Löf’s type theory have their familiar set theoretic interpretation, the impredicative type of propositions can be interpreted as a two element set and the hierarchy of type universes can be interpreted using a corresponding hierarchy of strongly inaccessible cardinal numbers. The assumption of the existence of these cardinal numbers goes beyond the proof theoretic strength of ZFC. But Martin-Löf’s type theory, even with its W types and its hierarchy of universes, is not fully impredicative and has proof theoretic strength way below that of second order arithmetic. So it is not clear that the strongly inaccessible cardinals used in our upper bound are really needed. Of course the impredicative type of propositions does give a fully impredicative type theory, which certainly pushes up the proof theoretic strength to a set theory3, Z−, whose strength is well above that of second order arithmetic. The hierarchy of type universes will clearly lead to some further strengthening. But is it necessary to go beyond ZFC to get an upper bound?

are not powerful enough to describe all existing algorithms. That is why we focus our interest on well-founded recursive functions. For these functions there are no reduction rules associated, therefore we provide the x-point equality theorem that represents the reduction rule and can be used without any knowledge of the proof's structure. With our work, we generate this equality automatically and it becomes simpler to reason about well-founded recursive functions in Coq. In future work, we would like to study how this method can be extended to handle functions containing recursion operators and mutual recursion. We also want to simplify equations describing the computational behavior of these functions. Abstracts: Stefan Berghofer 3 Aarne Ranta and Marcin Benke (Chalmers Tekniska Hgskola) New Developments in the Alfa Proof-editor Demonstration on Saturday 9 December, afternoon The GF plugin provides an extension of the proof editor Alfa w

Contents 1 Introduction 1-1 2 Some Axiom Systems 2-1 2.1 Classical Set Theory . . . . . . . . . . . . . . . . . . . . . . . 2-1 2.2 Intuitionistic Set Theory . . . . . . . . . . . . . . . . . . . . . 2-2 2.3 Constructive Set Theory . . . . . . . . . . . . . . . . . . . . . 2-2 2.3.1 CZF 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-2 2.3.2 Constructive Zermelo Fraenkel, CZF . . . . . . . . . . 2-3 2.3.3 Basic Constructive Set Theory, BCST . . . . . . . . . 2-3 3 Elementary Mathematics in Constructive Set Theory 3-1 3.1 Class Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1 3.2 Pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-2 3.2.1 Ordered Pairs . . . . . . . . . . . . . . . . . . . . . . . 3-2 3.2.2 Cartesian Products of Classes . . . . . . . . . . . . . . 3-2 3.2.3 Relations and Functions between Classes . . . . . . . . 3-3 3.3 The class of Natural

Some applications of Kleene’s method for intuitionistic systems. In Cambridge summer school in mathematical logic, Lectures notes in mathematics

- H Friedman

Recursion in the universe of sets

- Y N Moschovakis

Types and sets: A study on the jump to full impredicativity, Laurea dissertation

- N Gambino

Formally intuitionistic set theories with bounded predicates decidable

- R S Wolf

Constructivism in mathematics, volumes I

- A S Troelstra
- D Van Dalen
- AS Troelstra

Power set recursion. Annals of Pure and Applied Logic71

- L Moss

Set recursion In Generalized recursion theory II, 303–320

- D Normann

Constructive set theory, book draft

- P Aczel
- M Rathjen

The weak existence property for intuitionistic set theories

- M Rathjen