Power set axiom

Question

I want to write it in the following language $L=\{\in,=\}\cup \{u_{0},u_{1},\ldots\}$. Here is an attempt $\forall x~\exists y~\forall z~\in x~[z=p\wedge p\subset y]$. I also want to show whether it fails at $(V_{\omega},\in)$ and $(V_{\omega+1},\in)$. $(V_{\omega},\in)$ models it because $\omega$ is a limit ordinal. Any hint for the $(V_{\omega+1},\in)$? thanks

Answer 1

Um, your attempt contains $\subset$ which isn't in the language you want to stick to ... so it fails already there. (It also fails because it contains a free variable $p$ that comes out of nowhere, and even if those two are fixed it seems to become something like the union axiom, not the power set axiom).

The usual power set axiom (which you ought to be able to find in any text on axiomatic set theory, so I don't think I'm giving anything away here) is

$$ \forall x \exists y \forall z ( z\in y \iff z\subseteq x ) $$ where $\subseteq$ is an abbreviation, so the actual axiom is $$ \forall x \exists y \forall z ( z\in y \iff \forall w(w\in z \implies w\in x )) $$

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Hint for the second part: $V_{\omega+1}$ contains $V_{\omega}$ as an element ...