New answers tagged foundations
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An axiomatic approach to the multiverse of sets
This might not precisely answer your question, but why not use topos theory? The category of (Grothendieck) topoi has as its objects generalized universes of sets whose internal logic does no not ...
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Smallest ordinal modelling $\aleph_1$?
About well-definedness of $X_1$, "being a model of ZFC" is definable since ZFC is a recursive theory, so we could construct some $\Sigma_1^0$ predicate $\textrm{isZFCAxiom}(e)$ for $e$ a ...
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