# Can we define the tower of iterated power sets in ZFC

set-theory

Let $$S$$ be a set and $$n$$ be a positive integer.
One can safely say that
$$\mathcal{P}^n(S)\overset{\mathrm{def}}{=}\overset{\mathrm{n\;times}\;\;\;\;\;}{\mathcal{P}\mathcal{P}\cdots\mathcal{P}(S)}$$ exists, thus for every $$n\in\mathbb{N}$$, we can define the set ($$\mathcal{P}^0(S)\overset{\mathrm{def}}{=}S$$)
$$\bigcup_{i=0}^n\mathcal{P}^i(S).$$
But things become different when we replace the $$n$$ there by $$+\infty$$.
Does the following set exists in ZFC?
$$\bigcup_{n\in\mathbb{N}}\mathcal{P}^n(S).$$
What I am trying to define seems quite recursive but I don't know how to do it in ZFC.
If we go straight to the recursion theorem, then a function $$f:\bigcup_{n\in\mathbb{N}}\mathcal{P}^n(S)\rightarrow\bigcup_{n\in\mathbb{N}}\mathcal{P}^n(S)$$ is needed, which is a circular argument.

I was pretty sure this was a duplicate, but right now I can't find this exact question having been asked earlier, so here goes:

Yes, you can do this - the key is the axiom scheme of replacement (or collection depending on how you've seen $$\mathsf{ZF}$$ presented).

Consider the following (English shorthand for a) formula $$\varphi(x,y,z)$$:

$$y$$ is a natural number and there is a finite sequence $$b$$ of length $$y$$ such that the initial term of $$b$$ is $$z$$, the last term of $$b$$ is $$x$$, and whenever $$i+1 we have $$b(i+1)=\mathcal{P}(b(i))$$.

Then "$$\varphi(x,y,z)$$" should be interpreted as "$$x=\mathcal{P}^y(z)$$." A quick argument shows that we can apply replacement (and infinity) to prove in $$\mathsf{ZF}$$ the following:

For every $$z$$ the class $$C_z:=\{x:\exists y\in\omega(\varphi(x,y,z))\}$$ is a set.

Applying the union axiom to $$C_z$$ then gives the desired set.

There are two key things worth noting here:

• The use of the "coding sequence" $$b$$ exactly parallels a similar technique in the context of arithmetic for talking about computations.

• By taking unions at limits there's no difficulty in extending this to arbitrary ordinals instead of just natural numbers, where $$y$$ is concerned - in $$\mathsf{ZF}$$ we can in fact make sense of $$\mathcal{P}^\alpha(S)$$ and $$\bigcup_{\beta<\alpha}\mathcal{P}^\beta(S)$$ for any set $$S$$ and ordinal $$\alpha$$.