Abstract
We introduce a pure type system (PTS) λZ with four sorts and show that this PTS captures the proof-theoretic strength of Zermelo’s set theory. For that, we show that the embedding of the language of set theory into λZ via the ‘sets as pointed graphs’ translation makes λZ a conservative extension of IZ+AFA+TC (intuitionistic Zermelo’s set theory plus Aczel’s antifoundation axiom plus the axiom of transitive closure)—a theory which is equiconsistent to Zermelo’s. The proof of conservativity is achieved by defining a retraction from λZ to a (skolemised version of) Zermelo’s set theory and by showing that both transformations commute via the axioms AFA and TC.
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Miquel, A. (2006). λZ: Zermelo’s Set Theory as a PTS with 4 Sorts. In: Filliâtre, JC., Paulin-Mohring, C., Werner, B. (eds) Types for Proofs and Programs. TYPES 2004. Lecture Notes in Computer Science, vol 3839. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11617990_15
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DOI: https://doi.org/10.1007/11617990_15
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