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What definitions or equivalencies between definitions for standard set theory objects (such as large cardinals) do not hold or do not carry through in the expected manner to the world without choice? Also, equivalences that DO carry over, but that you might expect wouldn't, would be interesting as well.

For instance I recently heard that the equivalence between the diagonal intersection and the regressive function definitions of normal ultrafilter does not hold without AC, though under closer inspection I don't see why it doesn't. Clarification on this would be helpful as well.

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2 Answers 2

I don't think you need to go to anything as exotic as large cardinals.

AC states that every surjection has a right inverse. So if there's a surjection $B \to A$ then there's an injection $A \to B$. The converse holds (as long as $A$ is nonempty) whether or not you assume AC. Hence, if you assume AC then for nonempty sets $A$ and $B$,

there is an injection A --> B iff there is a surjection B --> A

But I guess these two equivalent definitions of "$A \leq B$" fail to be equivalent if you don't assume AC.

It's certainly true that AC is equivalent (in the presence of the other axioms) to cardinal comparability --- the statement that for all sets $A$ and $B$, either there exists an injection $A \to B$ or there exists an injection $B \to A$. At the moment I can't see how to turn that into a proof that the above equivalence fails without AC. Maybe someone else can.

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Well, without AC the standard definition of cardinality fails, in the sense that not every set has one.

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