Chapter

# Constructive Zermelo-Fraenkel Set Theory, Power Set, and the Calculus of Constructions

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## Abstract

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. ...
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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]. ...
Article
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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.
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Chapter
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.
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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.
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Article
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
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