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arXiv:1808.05204math.LO

Comparing material and structural set theories

Michael Shulman

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Abstract

We study elementary theories of well-pointed toposes and pretoposes, regarded as category-theoretic or "structural" set theories in the spirit of Lawvere's "Elementary Theory of the Category of Sets". We consider weak intuitionistic and predicative theories of pretoposes, and we also propose category-theoretic versions of stronger axioms such as unbounded separation, replacement, and collection. Finally, we compare all of these theories formally to traditional membership-based or "material" set theories, using a version of the classical construction based on internal well-founded relations.

arXiv comments
41 pages. This is a revised version of the first half of arxiv:1004.3802. v2: Final version to appear in APAL. v3: Resubmission under CC-BY-NC-ND to comply with publisher policy; no changes from previous CC-BY version
Journal reference
DOI
doi:10.1016/j.apal.2018.11.002
Subjects
math.LOLogic math.CTCategory Theory

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  1. Are Real Numbers Real? (2017) 1 comment

    • This makes sense in ZFC. ZFC is a material set theory and is the most common set theory (and foundation). It's different in a Structural Set Theory. Michael Shulman: "Comparing material and structural set theories" is really nice. https://arxiv.org/abs/1808.05204 "In material…

      hackandthinkView on HNParent context

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