Can every set in the Von-Neumann universe be obtained from finite ZFC steps

Question

For me, ZFC feels like saying either something is a set, or from a set, we know another thing is a set. On the other hand, something like doing the powerset operations for $\omega$ a number of times seems different.

I have been wondering for some time. Does this mean that all hereditary well-founded sets are exactly those that can be obtained from applying ZFC finite times?

I have read about the von Neumann Universe, which defines the class of hereditary of well founded sets. I know if a set is in ZFC, then after finite some "finite steps of applying ZFC", it is still in the von-neumann universe. However, I cannot think of a way to get the sets such as $V_{\lambda}$, where $\lambda$ is a limit ordinal from just applying ZFC finite times.

Furthermore, In the wikipedia article on the Von Neumann universe, it says that A crucial fact about this definition is that there is a single formula $\phi(\alpha,x)$ in the language of ZFC that states "the set x is in $V_{\alpha}$". I am confused what is the single formula described here.

Answer 1

I think I know what you are driving at, but the question is not well phrased. By "sets that can be obtained from applying ZFC finitely many times", I suspect you mean the set $W_\omega$ defined this way: $$\begin{align*}W_0&=\omega\\ W_{n+1}&=W_n\cup P(W_n)\\ W_\omega&=\bigcup_{n<\omega}W_n\end{align*}$$ Or something along those lines. However, these all belong to $V_{\omega 2}$, which is fairly low in the von Neumann hierarchy.

So the answer to your question is no, if I understand what you are asking.

Answer 2

The concept you are looking for is transfinite recursion, which is a way we can "do something infinitely many times". For the constructions you're describing, we need to do transfinite recursion along the ordinals, which makes use of the axiom schema of replacement.

If we grant for a moment that $V_n = P^n(\emptyset)$ makes sense for $n\in \omega$ (though we'll see this formally makes use of recursion, and is in some ways "the hard part"), then in order to iterate further, we want to "collect at the limit level $\omega$" to form $V_\omega=\bigcup_{n\in\omega}V_n.$ Formally, we have a class map $n\mapsto V_n,$ and we use an instance of replacement to construct the set $\{V_n: n\in \omega\}$ and then the union axiom to take the union of this set to obtain $V_\omega.$

Assuming we haven't done anything fishy to define the map $n\mapsto V_n$, we're not doing anything fishy to define $V_\omega$. We just did two more steps, both easily justifiable from ZFC.

So how do we formally define the map? First we prove this, by induction on $n\in \omega$:

Given $n\in \omega,$ there is a unique function $f$ whose domain is $n+1$, such that $f(0) = \emptyset$ and $f(k+1)= P(f(k))$ for all $k\in n.$

Write this statement that we've proven as $\forall n\in \omega \;\exists!f\;\varphi(n,f).$

Then we can write down a formula $\psi(n, z)$ which says

$n\in \omega$ and $z=f(n)$ for the unique $f$ such that $\varphi(n,z)$ holds.

and then easily prove $\forall n\in \omega \;\exists! z\; \psi(n,z).$ So $\psi$ is our class-function $n\mapsto V_n.$

As requested, we can write "$y\in V_\omega$" as: $\exists n\in \omega\; (\psi(n,z)\land y\in z).$ (However, note that we don't even need to know $V_\omega$ is a set in order to do this, only to have the map $n\mapsto V_n$.)

The generalization to ordinals bigger than $\omega$ is very similar and can be looked up in standard textbooks. It really only amounts to adding a clause about 'what to do at limit stages' (take the union in this case) to the definition of $\varphi(n,f).$ Justifying class function $\alpha\mapsto V_\alpha$ whose domain is all of the ordinals, rather than just some ordinal is a similar exercise.

Finally, replacement's role in "collecting the limit stages" is essential in a sense: for instance, without replacement, it is consistent that $V_\omega$ does not exist! On the other hand, without replacement we can define a relation on $\omega$ that is isomorphic to $(V_\omega, \in)$, and more generally we can often find sets isomorphic to seemingly innocuous things (like the ordinal $\omega+\omega$) that actually require replacement to define. This might seem odd, but it is the kind of thing you should expect to happen when you remove from set theory its means of enforcing that different representations of the same structure should be interchangeable.