I wrote a brief review of universes in Mizar.
Mizar has an axiom that for any set, there exists a *Tarski* universe containing it.
However, this is not necessarily the same as a *Grothendieck* universe.
It turns out, when we take a transitive set \(X\) and look for the smallest Tarski universe containing it \(X\in\mathbf{T}(X)\) and the smallest Grothendieck universe containing it \(X\in\mathbf{G}(X)\), the two coincide \(\mathbf{T}(X)=\mathbf{G}(X)\).
In general, though, we have \(\mathbf{T}(X)\subseteq\mathbf{G}(X)\).
What's more interesting is that we can form the smallest Grothendieck universe containing the natural numbers \(\mathbf{G}(\omega)\) and this is a model for ZF(C). Roland Coghetto wrote a couple recent articles proving all our favorite sets live in it. If you were to formalize the category \(\mathbf{Set}\) in Mizar, I'd suggest using \(\mathbf{G}(\omega)\) as the collection of objects underlying it (making it, effectively, the category \(\mathbf{ZFSet}\)).
#Mizar #proofassistant #logic #universe #mathematics
https://thmprover.wordpress.com/2024/12/23/universes-in-mizar/