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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]. ...
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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
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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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