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Abstract: We extend the usual internal logic of a pretopos to a more generalinterpretation, called the stack semantics, which allows for -unbounded-quantifiers ranging over the class of objects of the topos. Using well-foundedrelations inside the stack semantics, we can then recover a membership-basedor -material- set theory from an arbitrary topos, including evenset-theoretic axiom schemas such as collection and separation which involveunbounded quantifiers. This construction reproduces the models ofFourman-Hayashi and of algebraic set theory, when the latter apply. It turnsout that the axioms of collection and replacement are always valid in the stacksemantics of any topos, while the axiom of separation expressed in the stacksemantics gives a new topos-theoretic axiom schema with the full strength ofZF. We call a topos satisfying this schema -autological.-



Author: Michael A. Shulman

Source: https://arxiv.org/







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