The right choice? An intuitionistic exploration of Zermelo’s Axiom
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λZ: Zermelo’s Set Theory as a PTS with 4 Sorts
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 axi...
متن کاملLambda-Z: Zermelo’s Set Theory as a PTS with 4 Sorts
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 axi...
متن کاملlamda-Z: Zermelo's Set Theory as a PTS with 4 Sorts
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 axi...
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Ernst Zermelo is familiar to mathematicians as the creator of the controversial Axiom of Choice in 1904 and the theorem, based on the Axiom of Choice, that every set can be well ordered. Many will be aware that in 1908 he axiomatized set theory—in a form later modified by Abraham Fraenkel (1922) and then by Zermelo himself (1930). Some will know of Zermelo’s conflict with Ludwig Boltzmann over ...
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In this thesis, we give a substantial formalisation of classical set theory in the proof system Coq. We assume an axiomatisation of ZF and present a development of the theory containing relations, functions and ordinals. The implementation follows the structure of standard text books. In the context of this theory, we prove Zermelo’s Well-Ordering Theorem and the Axiom of Choice equivalent. In ...
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