Is there a formula for general induction?

Question

When I read about mathematical induction, there is no general formula, just a notion that is described:

Show true for $n = 1$

Assume true for $n = k$

Show true for $n = k + 1$

Conclusion: Statement is true for all $n \geq 1$

Can we formalize the above notion into a "pluggable" formula where we just plug in the values of what we want to prove?

Answer 1

Any formula would depend on the specific case.

In logic notation,

Mathematical induction as an inference rule can be formalized as a second-order axiom. The axiom of induction is, in logical symbols,

$$\forall P.\,[[ P(0)\land \forall (k\in \mathbb {N}).\,[P(k)\Rightarrow P(k+1)]]\Rightarrow \forall (n\in \mathbb {N}).\,P(n)]$$

where $P$ is any predicate and $k$ and $n$ are both natural numbers.