All Peano axioms except the induction axiom are statements in first-order logic. The induction axiom is written as $\forall X(0 \in X \land \forall n(n \in \mathbb{N} \rightarrow (n \in X \land n' \in X)) \rightarrow \mathbb{N} \subseteq X$, where $n'$ is the successor of $n$.

Now I want to look at an extension of first-order logic. I also allow $\exists^{\infty}$ [exists infinity many]. My notes states that the induction axiom can be rewritten only using this extension of first-order logic.

My question is: How?

Since I look at first-order logic I am not allowed to have a set $X$. Here I am allready stuck, because I think I can not just say $\exists{^\infty} x$ to have something like a set ... I think the solution requires two steps: First get rid of the set and then use $\exists^{\infty}$ to make it well-defined in my case.

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    $\begingroup$ It's kinda weird talking about Peano and first-order while quantifying over subsets (second-order) and referring to $\Bbb N$ in the formula. $\endgroup$ – Asaf Karagila Mar 18 '19 at 11:15
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    $\begingroup$ @AsafKaragila Well he did say all the axioms "expect" the induction axiom - if you assume "expect" was a typo for "except" it makes sense.... $\endgroup$ – David C. Ullrich Mar 18 '19 at 14:45

Note that induction is equivalent to well-ordering (more generally to well-foundedness). Namely, removing the induction axiom, a model of $\sf PA$ is well-ordered if and only if it satisfies the (second-order) induction axiom.

But well-ordering is equivalent to "there is no infinite decreasing chain". Finally, since in $\sf PA$ every non-zero element has a predecessor, this means that well-ordering is equivalent to stating that no element has infinitely many elements smaller than itself.

And this should be fairly straightforward to state using $\exists^\infty$.

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