Definition 1:
$(0\in S, n\in S \implies n+1\in S) \implies n\in S \forall n≥0$.
Definition 2:
$(P(0), P(n)\implies P(n+1)) \implies P(n) \forall n≥0$.
To prove the equivalence of these definitions, the only thing needed is the "relabelling" $n \mapsto P(n), S\mapsto T$, where $T$ is the set of truths?
To prove definition 2 from definition 1 let $S = \{n \ | \ P(n)\}$ and to prove definition 1 from definition 2 let $P(n)$ be the statement "$n \in S$".