"Infinitely deep" sets In the Von Neumann Universe, a generation such as $V_{\omega+\omega}$ seems to occur "after" applying the power set operation infinitely many times, so intuitively it feels like there must be sets which are "infinitely deep" in the sense of never being able to reach $\varnothing$ with a finite number of "unwrappings".
On the other hand, the above is both vague and false, since the Axiom of Regularity implies that there's no infinite descending chain of sets like that. Where is my intuition about $V_{\omega+\omega}$ going wrong?
 A: Let's pick something concrete, start with $a_0=\omega$, and $a_{n+1}=\{a_n\}$. Then each $a_n\in V_{\omega+\omega}$.
We start "peeling" off braces, say from $a_{42}$, then after $42$ steps, we have reached $\omega$, and now the next step will require us to go down all the way to some finite set.
The point here, really, is that sets with a limit von Neumann rank collect sets from unboundedly many stages below that limit stage. So you can peel down some finitely many braces, but once you've hit a step of limit rank, the next step will have to be choosing an element from a significantly smaller rank, peel some finitely many steps, and keep going down.
Again, if you think of this in terms of rank, start from $\omega+42$, after finitely many steps down, we are staring at $\omega$ right in the face. But the next step is finite. Similarly, start with $\omega_1+54$, after finitely many steps, we are at $\omega_1$, then we pick some countable $\alpha$, maybe it was some $\delta+10^{40}$ for some limit ordinal $\delta$, then we keep going down, hitting $\delta$, and then we have to decrease even further, etc. After finitely many steps, we'll have reached $\omega$, and then the next step is just a finite integer.
