How would one go about proving an infinite set is a set?

Question

Say you wanted to prove the set Z such that Z = $\{\phi, \{\phi\}, \{\{\phi\}\}, ... \}$ is a set. We could construct each element of z using the pair axiom or the power set axiom (I think) but how would we go about proving it infinitely?

Answer 1

Step I: Prove that $\omega$ is a set, where $\omega$ is the smallest inductive set, or the closure of $\{\varnothing\}$ under the successor operator ($x\mapsto x\cup\{x\}$).

Step II: Prove the recursion theorem. Namely, you can define a function on $\omega$ by specifying what it is doing for $0$, and how to define the successor steps. Replacement and Union are your friends for this proof.

Step III: Use the recursion theorem. Clearly $F(0)=\varnothing$, how would you define $F(n+1)$?

Step IV: The range of a function is a set.

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Alternatively, prove Step I and use the specific instance of the recursion theorem and find a formula such that $\varphi(n,x)$ holds if and only if $x$ is the $n$th iteration of taking a singleton with $\varphi(\varnothing,\varnothing)$. Then apply Replacement.