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.