Grothendieck Universes

Colin McLarty colin.mclarty at
Fri May 20 21:06:21 EDT 2022

On Fri, May 20, 2022 at 8:33 PM jodmos.horon <jodmos.horon at>

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

> I just checked that first document. I do not see the model Z1 presented
> there being as strong as ZFC.

Quite right. And that is a good thing because Z1 is not as strong as ZFC.

But you see that ZFC is an extension of Z1, in the same language, right?
And you see that ETCS is interinterpretable with Z1, right?  That means
every extension of either ETCS or Z1 is interinterpretable with a
corresponding extension of the other (all without changing the languages).
The paper makes exactly this comment on page 268 right after theorem 4.

The debate over whether ETCS and its extensions "can do'' what ZF (or its
fragment Z1) and its extensions can do, could have closed 50 years ago with
this proof.  The answer is proven yes.

Then discussion could advance to genuine questions of how well
either axiom system serves what purposes.  But a lot of people have
preferred not to know about this proof.


> But interesting read. Thank you.
> Jodmos Horon.
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