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Recall Muller's criteria in Sets, Classes and Categories: page 14 for a theory that founds Mathematics.

Arn't those captured simply by $$\sf Zermelo + \text {worldly cardinals exist} $$

define sets as classes with ranks lower than the first worldly cardinal and everything would go through!

A worldly cardinal is some cardinal $\kappa$ such that $V_\kappa\models \sf ZFC$

Zuhair
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  • You might be interested in https://math.stackexchange.com/questions/375085/on-the-large-cardinals-foundations-of-categories – Asaf Karagila Jun 24 '22 at 15:15
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    I see there is some difference between what the answer to your linked question is saying and what Muller is stating in his work. Unless I missed something, it is enough to just have POWERSET operator iterated upon a universe of sets to establish founding Cateorgies, powersets are not universes, and according to Muller we only need those iterated powers to be at FINITE distance from that universe. Now I don't know how to reconcile this with what the answer requires namely a countable sequence of UNIVERSES embedding seamelessly in an ascending order. – Zuhair Jun 24 '22 at 15:51

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