Grothendieck Universes

Colin McLarty colin.mclarty at
Thu May 19 21:12:15 EDT 2022

On Thu, May 19, 2022 at 7:59 PM jodmos.horon <jodmos.horon at>

> This seems to be a claim that ZFC may be interpreted by structural set
> theories such as ECTS or SEARL augmented by axiom schemas. I just do not
> know of a reference where such a claim is written down and proven black on
> white.

Numerous articles sketching the proof at varying levels of detail give
perfectly detailed references.  Three of the most widely cited complete
proofs are:

Mitchell, William, "Boolean topoi and the theory of sets", *Journal of Pure
and Applied Algebra*, 1972, vol.2, pp. 261--74.

Osius, Gerhard, "Logical and set-theoretical tools in elementary topoi", in
the book *Model Theory and Topoi*, Springer Lecture Notes in Mathematics
445, pages 297--346.

Mac~Lane, Saunders and Moerdijk, Ieke in chapter V of the book *Sheaves in
Geometry and Logic*, Springer-Verlag, 1992.

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