Is there a set in ZFC that can not be obtained from Ordinals

Question

Is there a set in ZFC that can not be obtained from ordinals (defined according the Von Neumann definition) via set operations (union, intersection, set difference) and power set?

Answer 1

That depends on what you mean.

If you just iterate the power set operation starting from $\varnothing$, and taking unions at limit ordinals, you will generate the whole universe. But you will not "obtain" every set, because every set will be an element of some point in the hierarchy, rather than a step in that hierarchy.

But even here, when you reach a limit step, you'd need something that will let you collect the previous steps.

With just what you describe, the answer is very much negative. You will not have $V_\omega$, the set of hereditarily finite sets. To see that, note that $V_\omega$ is not an ordinal, and it is not a power set of an ordinal, or a second-power set of an ordinal, etc., because it is in fact the $\omega$th power set of any finite ordinal (i.e. $\bigcup\{\mathcal P^n(\varnothing)\mid n<\omega\}$). And so it is in fact not a subset of any finite power set either.

You can conclude from the above that it is not a union or an intersection of any such sets, because power sets, in general, are closed under unions and intersections.