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Working in $\sf ZF - Fnd$, add the following axiom:

AntiFoundation: $\forall x: x \neq \emptyset \to \exists! y: y \in y \land y \sim x$

where "$\sim$" stands for existence of a bijection.

In this theory we can easily define cardinality as:

$|X|= \kappa \iff \kappa \sim X \land ( \kappa=\emptyset \lor \kappa \in \kappa)$

The question is if this would imply any kind of Choice? If yes, what would be that form of Choice?

The backround of this question is related to this question and in particular to this answer to it, which asserts that some form of Choice would exist even if its for the empty structure (i.e.; Cardinality).

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    $\begingroup$ My guess would be that any model of ZF is a well-founded part of some model of ZF-Fnd satisfying your new axiom. So no, it probably doesn't imply any amount of choice. $\endgroup$
    – Wojowu
    Commented Jan 16, 2022 at 13:01
  • $\begingroup$ @Wojowu, that's what I thought also. I've added the context of this question, which is Asaf's answer to my question about structures, but that question was in $\sf ZF$, and this goes beyond it, so I'm not sure if this can manage to avoid choice? $\endgroup$
    – Zuhair Al-Johar
    Commented Jan 16, 2022 at 15:28

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