Von Neumann hierarchy has a critical role in set theory. It is well-known that $V_\alpha$ is a model of $\mathsf{ZC}$ if $\alpha$ is a limit ordinal. Furthermore, $V_\alpha$ satisfies the cumulative hierarchy axiom ($\mathsf{CHA}$), an axiom claiming the existence of sequence $\langle V_\eta\mid \eta<\xi\rangle$ for each ordinal $\xi$ with that every set is contained in some $V_\eta$, which is apparently not a theorem of $\mathsf{ZC}$. (Caution.$\mathsf{CHA}$ is not equivalent to the claim that $V_\xi$ exists for all ordinal $\xi$. See Hamkins' answer.)
However, $V_\alpha$ does not satisfy Replacement even for $\Delta_1$-formulas.
Theorem.$V_{\omega+\omega}$ does not satisfy Replacement for $\Delta_1$-formulas.
Proof. Consider
\begin{align} \phi(x,y) :\equiv \exists n<\omega\exists f [x<n+1\land f\colon (n+1)\to V \land f(0)=\omega \\ \land \forall m<n [f(m+1) = f(m)+1] ]\land f(x)=y. \end{align}
We can see that the above formula is equivalent to\begin{align} \forall f \forall n<\omega [x<n+1\land f\colon (n+1)\to V \land f(0)=\omega \\ \land \forall m<n [f(m+1) = f(m)+1] ]\to f(x)=y. \end{align}
Furthermore, $\forall x\in\omega\exists! y\phi(x,y)$, but Replacement for $\phi$ results in $\{\omega+n\mid n<\omega\}$ that is not a member of $V_{\omega+\omega}$.
(I previously claimed that $\Delta_0$-Replacement is invalid over $V_{\omega+\omega}$, but I found there is a gap in my proof.)
Then we can ask the following questions:
Is there any non-trivial consequences of Replacement that are valid over any $V_{\alpha}$ for limit $\alpha>\omega$? (According to Mathias's The Strength of Mac Lane Set Theory, $\mathsf{ZC}$ proves Replacement for stratified formulas.)
Can we characterize the theory of $V_\alpha$ for limit $\alpha>\omega$?
The second question needs some clarification. Assume that $\mathsf{ZFC}$ is consistent. My question is we can characterize the theory $T$ defined by
A sentence $\sigma$ is a member of $T$ if and only if, for any models $M\models \mathsf{ZFC}$ and $\alpha\in M$ such that $M\models \alpha>\omega\text{ is a limit ordinal}$, $M\models (V_\alpha\models \sigma)$.
Obviously $T\supseteq \mathsf{ZC+CHA}$. Furthermore, if we assume $\mathsf{ZFC}$+ there is a worldly cardinal is consistent, then
- $T\nvdash\lnot\sigma$ for axioms $\sigma$ of $\mathsf{ZFC}$: If $M$ is a model of $\mathsf{ZFC}$ and $M\models \kappa \text{ is worldly}$, then $V^M_\kappa\models \mathsf{ZFC}$.
- If $\mathsf{ZFC}$ proves we can force $\sigma$, then $T\nvdash\lnot\sigma$. Again, let $M$ be a countable model of $\mathsf{ZFC}$ and $M\models \kappa \text{ is worldly}$. By the assumption, we can find a forcing poset $\mathbb{P}\in V^M_\kappa$ such that $V^M_\kappa$ thinks $\mathbb{P}$ forces $\sigma$. Now consider the $\mathbb{P}$-generic extension $M[G]$ of $M$. Then $V^{M[G]}_\kappa = V^M_\kappa[G]$ and $V^M_\kappa[G]\models \mathsf{ZFC}+\sigma$.
These results provide some upper bound for $T$. Could we find a better upper bound for $T$?
