How do you symbolically represent the general principle of induction?

Question

Normally a specific function is given, and then it would be asked to prove the validity of that specific function with induction.

But how do you logically represent the general principle of induction for any given function, regardless of a specific case?

Answer 1

You are looking for the logical axiom of induction.

From Wikipedia's article on induction:

where P is any predicate and k and n are both natural numbers. In words, the basis P(0) and the inductive step (namely, that the inductive hypothesis P(k) implies P(k + 1)) together imply that P(n) for any natural number n. The axiom of induction asserts that the validity of inferring that P(n) holds for any natural number n from the basis and the inductive step. Note that the first quantifier in the axiom ranges over predicates rather than over individual numbers. This is a second-order quantifier, which means that this axiom is stated in second-order logic. Axiomatizing arithmetic induction in first-order logic requires an axiom schema containing a separate axiom for each possible predicate. The article Peano axioms contains further discussion of this issue.