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I’m not entirely happy with the introduction (“Statement”) to the page axiom of choice. On the one hand, it implies that the axiom of choice is something to be considered relative to a given category $C$ (which is reasonable), but it then proceeds to give the external formulation of AC for such a $C$, which I think is usually not the best meaning of “AC relative to $C$”. I would prefer to give the Statement as “every surjection in the category of sets splits” and then discuss later that analogous statements for other categories (including both internal and external ones) can also be called “axioms of choice” — but with emphasis on the internal ones, since they are what correspond to the original axiom of choice (for sets) in the internal logic.
(I would also prefer to change “epimorphism” for “surjection” or “regular/effective epimorphism”, especially when generalizing away from sets.)
I agree.
Since no one objected, I went ahead and made this change.
Seems like a somewhat roundabout way of putting it: can’t we just say that for infinite $X$, that $X$ and $F(X)$ have the same cardinality? Am I missing something?
Okay, I guess never mind my question. The direction that the indicated statement plus ZF implies AC doesn’t look easy.
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