The induction principle is used to prove a sequence of propositions $P_0, P_1, P_2, P_3, ...$.
And it proceeds as follows
(i) Verify $P_0$. (ii) Assuming truth of $P_k$ for some $k$, verify $P_{k+1}$.
In this way, denoting by $S$ the set of those $k$ for which $P_k$ is true, then $S$ verifies both (i) and (ii) in the induction principle. Hence S coincides with N, so all propositions $P_k$ are true.
Can one describe this using mathematical symbolisms rather than prose ? What does one do exactly ?