The sets $V_\alpha$ in the cumulative von Neumann hierarchy, defined by transfinite induction:
$\begin{align} V_0 &= \varnothing \\ V_{\alpha+1} &= \mathcal{P}(V_\alpha) \\ V_\lambda &= \bigcup_{\beta < \lambda} V_\beta \quad \text{for limit ordinals } \lambda. \end{align}$
are known to be models of the various axioms of $\mathsf{ZFC}$ for various values of $\alpha$. (E.g. $\alpha > \omega$ limit $\implies V_\alpha \vDash \mathsf{ZFC}$ minus replacement, $\kappa$ strongly inaccessible $\implies V_\kappa \vDash \mathsf{ZFC}$).
What do the $V_\alpha$ say about the various alternative axioms for set theory / statements independent of $\mathsf{ZF}$ (e.g. $\mathsf{AC}, \mathsf{AD}, \mathsf{CH}, \mathsf{GCH}, \mathsf{V\!=\!L}$, Martin's axioms, large cardinal axioms, forcing axioms, etc), for various values of $\alpha$?