# Can withdrawal of the Power set axiom enable absoluteness for set size statements?

If we look to models of $$\small \sf ZFC$$ we can have two transitive models $$N,M$$ such that for some formula $$\phi$$ in the language of $$\small \sf ZFC$$ we have:

$$\forall \mathcal M ([\mathcal M\models ZFC] \to \mathcal M\models \exists!x (\phi))$$

$$M\models \exists x (\phi \land |x| = \kappa)$$

$$N\models \exists x (\phi \land |x| \neq \kappa)$$,

for example left $$\phi$$ be $$x=P(\omega)$$ and let $$\kappa= \omega_1$$. I'll refer to the above as "size statements about some sets theorized to uniquely exist by $$\small \sf ZFC$$, are not absolute".

Now examine this theory, the question is:

Can we have the above situation raised with it also? In other words, can it theorize the unique existence of some sets the size statements about which are not absolute?

Yes. Surely we can, under reasonable assumptions, find two transitive models $$M, N$$ of this theory, one of which (say $$M$$) has a powerset of $$\omega$$ and the other does not. Let $$\varphi(x)$$ be the formula which asserts that $$x$$ is $$\mathcal P(\omega)$$ if it exists and $$\emptyset$$ else. The given theory proves that there is a unique $$x$$ which satisfies $$\varphi$$. But now the respective witnesses in $$M$$ and $$N$$ have vastly different sizes. A variation of this trick works (essentially) whenever the given theory is incomplete (given that the question even makes sense for that theory).