Partial Universes and the Axioms of $ZF$ Set Theory Without Choice In his Senior Thesis, Samuel Coskey answered the question of which axioms of $ZFC$ hold at each stage of the cumulative hierarchy.  Here is the list of his results:

Axioms that always hold:  Extensionality, Foundation, Union, Axiom Schema of Separation, Choice.
Axioms that hold in $V_{\alpha}$ iff $\alpha$$\gt$0:  Empty Set.
Axioms that hold in $V_{\alpha}$ iff $\alpha$$\gt$$\omega$:  Infinity.
Axioms that hold at limit ordinals:  Power Set, Pairing.
Axioms that hold at inaccessible cardinals: Axiom Schema of Replacement.

My question is simply this:

What changes (if anything) in these results if Choice is dropped?

 A: If you drop choice, then of course choice is no longer in the "always holds" category, but there's nothing to determine at what level of the cumulative hierarchy it first fails.  The first failure of choice could occur just a few levels past $\omega$, or it could occur far beyond the first inaccessible cardinal.  (The vagueness of "a few levels" comes from the details of how you formulate choice; formulations that are equivalent in ZF need not be equivalent when basic things like pairing are unavailable.)
As far as I can see, the only other thing that might change in the results you quoted is that definitions of inaccessibility that are equivalent in ZFC are no longer equivalent in ZF, so you need to be careful about which version of the definition you use.  For more details than you probably want about various definitions of inaccessibility, see my joint paper with Ioanna Dimitriou and Benedikt Löwe, "Inaccessible caridnals without the axiom of choice" [Fundamenta Mathematicae 194 (2007) 179-189, DOI: 10.4064/fm194-2-3].
