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Using only axiom of specification, can we create all possible subsets of any set?

For finite sets, it’s true I guess. But for infinite sets, what can we say? If it were true of infinite sets, we’d have not needed the axiom of powers.

But we do need that. Does that imply that statements of first-order logic are not enough to create (via specification) all the subsets of infinite sets?

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  • The Power ser axiom (as per your previous post) is needed in order to assert that the set of all subsets of a set $A$ exists, i.e. that we can collect all subsets of $A$ into a single set $\mathcal P(A)$. – Mauro ALLEGRANZA Jun 28 '19 at 13:34
  • When you say "create" do you mean define it ? In order to do this, we can use Specification axiom, using a formula that "specifies" the corresponding subset of $A$. – Mauro ALLEGRANZA Jun 28 '19 at 13:36
  • Your argument about a finite set is also wrong. If you have non-standard integers, then they are internally finite, but externally they are infinite sets. And the induction that you have in mind is external, so it does not apply to non-standard finite sets anyway. But it's worse than that. Specification allow parameters. – Asaf Karagila Jun 28 '19 at 13:38
  • Also, to add on what @Mauro has been writing, https://math.stackexchange.com/questions/1446509/zf-difference-between-pow-and-sep might be relevant for your recent questions. – Asaf Karagila Jun 28 '19 at 13:40

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