Comment by LudwigNagasena 4 days ago
How is it different from using ZF as a meta-theory to study ZF(C)? Is there anything special about category theory vis-à-vis ZF as a meta-theory? You're not arguing about ontology or the nature of truth, because you've picked category theory as your ontology just like you could pick ZF or ZFC.
But then you would think there is a right answer to “what is true” about categories, and you would face AC again.
For sure there would have to be a meta theory of some sort. But, I think that many classical mathematicians would be happy with that meta theory to be weaker, rather than stronger. I don't think there is a need for that theory to have AC. After all, thanks to its independence of from the other axioms, AC does not add logical strength to a classical theory.
(For die-hard constructivists, such as myself, the story is of course different. But that story is for a another time. I am presenting the classical view here.)
Category theory gives a structural framework for discussing these things. The various categories live side by side and can be related with functors. This allows a broader view and makes it easier perhaps, to understand that there isn’t a right answer to “what is true” about sets in the absolute.