Intuitionistic Zermelo-Fraenkel Set Theory Research Articles

Choice and independence of premise rules in intuitionistic set theory

Choice and independence of premise principles play an important role in characterizing Kreisel's modified realizability and Gödel's Dialectica interpretation. In this paper we show that a great many intuitionistic set theories are closed under the corresponding rules for finite types over N. It is also shown that the existence property (or existential definability property) holds for statements of the form ∃yσφ(y), where the variable y ranges over objects of finite type σ. This applies in particular to CZF (Constructive Zermelo-Fraenkel set theory) and IZF (Intuitionistic Zermelo-Fraenkel set theory), two systems known not to have the general existence property. On the technical side, the paper uses a method that amalgamates generic realizability for set theory with truth, whereby the underlying partial combinatory algebra is required to contain all objects of finite type.

Ordinal analysis and the set existence property for intuitionistic set theories.

On account of being governed by constructive logic, intuitionistic theories [Formula: see text] often enjoy various existence properties. The most common is the numerical existence property (NEP). It entails that an existential theorem of [Formula: see text] of the form [Formula: see text] can be witnessed by a numeral [Formula: see text] such that [Formula: see text] proves [Formula: see text]. While NEP holds almost universally for natural intuitionistic set theories, the general existence property (EP), i.e. the property of a theory that for every existential theorem, a provably definable witness can be found, is known to fail for some prominent intuitionistic set theories such as Intuitionistic Zermelo-Fraenkel set theory (IZF) and constructive Zermelo-Fraenkel set theory (CZF). Both of these theories are formalized with collection rather than replacement as the latter is often difficult to apply in an intuitionistic context because of the uniqueness requirement. In light of this, one is clearly tempted to single out collection as the culprit that stymies the EP in such theories. Beeson stated the following open problem: 'Does any reasonable set theory with collection have the existence property? and added in proof: The problem is still open for IZF with only bounded separation.' (Beeson. 1985 Foundations of constructive mathematics, p. 203. Berlin, Germany: Springer.) In this article, it is shown that IZF with bounded separation, that is, separation for formulas in which only bounded quantifiers of the forms [Formula: see text] are allowed, indeed has the EP. Moreover, it is also shown that CZF with the exponentiation axiom in place of the subset collection axiom has the EP. Crucially, in both cases, the proof involves a detour through ordinal analyses of infinitary systems of intuitionistic set theory, i.e. advanced techniques from proof theory. This article is part of the theme issue 'Modern perspectives in Proof Theory'.

EXTENSIONAL REALIZABILITY AND CHOICE FOR DEPENDENT TYPES IN INTUITIONISTIC SET THEORY

AbstractIn [17], we introduced an extensional variant of generic realizability [22], where realizers act extensionally on realizers, and showed that this form of realizability provides inner models of $\mathsf {CZF}$ (constructive Zermelo–Fraenkel set theory) and $\mathsf {IZF}$ (intuitionistic Zermelo–Fraenkel set theory), that further validate $\mathsf {AC}_{\mathsf {FT}}$ (the axiom of choice in all finite types). In this paper, we show that extensional generic realizability validates several choice principles for dependent types, all exceeding $\mathsf {AC}_{\mathsf {FT}}$ . We then show that adding such choice principles does not change the arithmetic part of either $\mathsf {CZF}$ or $\mathsf {IZF}$ .

Extensional realizability for intuitionistic set theory

Abstract In generic realizability for set theories, realizers treat unbounded quantifiers generically. To this form of realizability, we add another layer of extensionality by requiring that realizers ought to act extensionally on realizers, giving rise to a realizability universe $\mathrm{V_{ex}}(A)$ in which the axiom of choice in all finite types, ${\textsf{AC}}_{{\textsf{FT}}}$, is realized, where $A$ stands for an arbitrary partial combinatory algebra. This construction furnishes ‘inner models’ of many set theories that additionally validate ${\textsf{AC}}_{{\textsf{FT}}}$, in particular it provides a self-validating semantics for ${\textsf{CZF}}$ (constructive Zermelo–Fraenkel set theory) and ${\textsf{IZF}}$ (intuitionistic Zermelo–Fraenkel set theory). One can also add large set axioms and many other principles.

Satisfiability is False Intuitionistically: A Question from Dana Scott

Satisfiability or Sat $$^{1}$$ is the metatheoretic statement Every formally intuitionistically consistent set of first-order sentences has a model. The models in question are the Tarskian relational structures familiar from standard first-order model theory (Chang and Keisler in Model theory, 3rd edn, Dover Publications Inc., Mineola, 2012), but here treated within intuitionistic metamathematics. We prove that both IZF, intuitionistic Zermelo–Fraenkel set theory, and HAS, second-order Heyting arithmetic, prove Sat $$^{1}$$ to be false outright. Following the lead of Carter (Notre Dame J Form Log 49(1):75–95, 2008), we then generalize this result to some provably intermediate first-order logics, including the Rose logic (Trans Am Math Soc 61:1–19, 1953). These metatheorems distinguish the intuitionistic foundational significance of Sat $$^{1}$$ sharply from that of Sat $$^{0}$$ , the satisfiability claim for intuitionistic propositional logic. At the same time, they establish intuitionistic connections with and between Test, COMP $$^{0}$$ , and $$\mathbf{COMP }^{1}$$ . Here, Test is the scheme of Testability, and COMP $$^{0}$$ and $$\mathbf{COMP }^{1}$$ are completeness for intuitionistic propositional logic and predicate logic, respectively.

Certain Partial Conservativeness Properties of Intuitionistic Set Theory with the Principle of Double Complement of Sets

The Zermelo–Fraenkel set theory with the underlying intuitionistic logic (for brevity, we refer to it as the intuitionistic Zermelo–Fraenkel set theory) in a two-sorted language (where the sort 0 is assigned to numbers and the sort 1, to sets) with the collection scheme used as the replacement scheme of axioms (the ZFI2C theory) is considered. Some partial conservativeness properties of the intuitionistic Zermelo–Fraenkel set theory with the principle of double complement of sets (DCS) with respect to a certain class of arithmetic formulas (the class all so-called AEN formulas) are proved. Namely, let T be one of the theories ZFI2C and ZFI2C + DCS. Then (1) the theory T+ECT is conservative over T with respect to the class of AEN formulas; (2) the theory T+ECT+M is conservative over T+M{su−} with respect to the class of AEN formulas. Here ECT stands for the extended Church’s thesis, Mis the strong Markov principle, and M{su−} is the weak Markov principle. The following partial conservativeness properties are also proved: (3) T+ECT+M is conservative over T with respect to the class of negative arithmetic formulas; (4) the classical theory ZF2 is conservative over ZFI2C with respect to the class of negative arithmetic formulas.

Semantic Completeness of First-Order Theories in Constructive Reverse Mathematics

We introduce a general notion of semantic structure for first-order theories, covering a variety of constructions such as Tarski and Kripke seman- tics, and prove that, over Zermelo Fraenkel set theory (ZF), the completeness of such semantics is equivalent to the Boolean Prime Ideal theorem (BPI). In partic- ular, we deduce that the completeness of that type of semantics for non-classical theories is unprovable in intuitionistic Zermelo Fraenkel set theory IZF. Using results of Joyal and McCarty, we conclude also that the completeness of Kripke semantics is equivalent, over IZF, to the Law of Excluded Middle plus BPI. By results from Banaschewski and Bhutani, none of these two principles imply each other, and so this gives the exact strength of Kripke completeness theorem in the sense of constructive reverse mathematics.

Models of intuitionistic set theory in subtoposes of nested realizability toposes

In [8] Joyal and Moerdijk have shown that realizability toposes over partial combinatory algebras (pca) host classes of small maps giving rise to initial ZF-algebras providing models of intuitionistic Zermelo–Fraenkel set theory IZF. Here we show that this can be extended to a much wider class of realizability toposes as considered in [4] and [3].For this purpose we first show this result for nested realizability toposesRT(A,A#) induced by a pca A together with a sub-pca A# as considered implicitly in [4] and then show that it is preserved by restriction to subtoposes. This suffices since all toposes considered in [4] and [3] arise as subtoposes of some nested realizability toposes.

From the weak to the strong existence property

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, IZF, does not have the existence property, where IZF is formulated with Collection. By contrast, the version of intuitionistic Zermelo–Fraenkel set theory formulated with Replacement, IZFR, has the existence property. Moreover, IZF 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 CZF−, CZFE, and CZFP, i.e., respectively, constructive Zermelo–Fraenkel set theory (CZF) without subset Collection, CZF formulated with Exponentiation and also CZF augmented by the Power Set axiom (basically IZF 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.To bring about these results we introduce a form of realizability based on general set recursive functions where a realizer for an existential statement provides a set of witnesses for the existential quantifier rather than a single witness. Moreover, this notion of realizability needs to be combined with truth to yield the desired results.This form of realizability is also utilized, albeit shorn of its truth component, in showing partial conservativity results for CZF−, CZFE, and CZFP over their intuitionistic counterparts IKP, IKP(E), and IKP(P), respectively.As it turns out, the combination of the weak existence property and partial conservativity of CZF− over IKP plus a further ingredient can be used to show that CZF− actually has the existence property. The additional ingredient is an advanced technique from proof theory (cut elimination and ordinal analysis of IKP). Roughly the same techniques can be deployed in showing that CZFE and CZFP have the stronger existence property, too. However, this requires a new form of ordinal analysis for theories with Power Set and Exponentiation (cf. Rathjen (2011) [39]) and is beyond the scope of the current paper.

Relating First-Order Set Theories and Elementary Toposes

AbstractWe show how to interpret the language of first-order set theory in an elementary topos endowed with, as extra structure, a directed structural system of inclusions (dssi). As our main result, we obtain a complete axiomatization of the intuitionistic set theory validated by all such interpretations. Since every elementary topos is equivalent to one carrying a dssi, we thus obtain a first-order set theory whose associated categories of sets are exactly the elementary toposes. In addition, we show that the full axiom of Separation is validated whenever the dssi is superdirected. This gives a uniform explanation for the known facts that cocomplete and realizability toposes provide models for Intuitionistic Zermelo–Fraenkel set theory (IZF).

Computational adequacy for recursive types in models of intuitionistic set theory

This paper provides a unifying axiomatic account of the interpretation of recursive types that incorporates both domain-theoretic and realizability models as concrete instances. Our approach is to view such models as full subcategories of categorical models of intuitionistic set theory. It is shown that the existence of solutions to recursive domain equations depends upon the strength of the set theory. We observe that the internal set theory of an elementary topos is not strong enough to guarantee their existence. In contrast, as our first main result, we establish that solutions to recursive domain equations do exist when the category of sets is a model of full intuitionistic Zermelo–Fraenkel set theory. We then apply this result to obtain a denotational interpretation of FPC, a recursively typed lambda-calculus with call-by-value operational semantics. By exploiting the intuitionistic logic of the ambient model of intuitionistic set theory, we analyse the relationship between operational and denotational semantics. We first prove an “internal” computational adequacy theorem: the model always believes that the operational and denotational notions of termination agree. This allows us to identify, as our second main result, a necessary and sufficient condition for genuine “external” computational adequacy to hold, i.e. for the operational and denotational notions of termination to coincide in the real world. The condition is formulated as a simple property of the internal logic, related to the logical notion of 1-consistency. We provide useful sufficient conditions for establishing that the logical property holds in practice. Finally, we outline how the methods of the paper may be applied to concrete models of FPC. In doing so, we obtain computational adequacy results for an extensive range of realizability and domain-theoretic models.

Replacement and collection in intuitionistic set theory

Intuitionistic Zermelo-Fraenkel set theory, which we call ZFI, was introduced by Friedman and Myhill in [3] in 1970. The idea was to have a theory with the same axioms as ordinary classical ZF, but with Heyting's predicate calculus HPC as the underlying logic. Since some classically equivalent statements are intuitionistically inequivalent, however, it was not always obvious which form of a classical axiom to take. For example, the usual formulation of the axiom of foundation had to be replaced with a principle of transfinite induction on the membership relation in order to avoid having excluded middle provable and thus making the logic classical. In [3] the replacement axiom is taken in its most common classical form:In the presence of the separation axiom,this is equivalent to the axiomIt is this schema Rep that we shall call the replacement axiom.Friedman and Myhill were able to show in [3] that ZFI has a number of desirable “constructive” properties, including the existence property for both numbers and sets. They were not able to determine, however, whether ZFI is proof-theoretically as strong as ZF. This is still open.Three years later, in [2], Friedman introduced a theory ZF− which is like ZFI except that the replacement axiom is changed to the classically equivalent collection axiom:He showed that ZF− is proof-theoretically of the same strength as ZF, and he asked whether ZF− is equivalent to ZFI.

Church's thesis, continuity, and set theory

AbstractUnder the assumption that all “rules” are recursive (ECT) the statement Cont(NN, N) that all functions from NN to N are continuous becomes equivalent to a statement KLS in the language of arithmetic about “effective operations”. Our main result is that KLS is underivable in intuitionistic Zermelo-Fraenkel set theory + ECT. Similar results apply for functions from R to R and from 2N to N. Such results were known for weaker theories, e.g. HA and HAS. We extend not only the theorem but the method, fp-realizability, to intuitionistic ZF.