The axiom of induction, which induction is based on can be stated as (for example in Peano axioms):
$$P(0)\wedge\forall x (P(x) \rightarrow P(S(x))) \rightarrow \forall x P(x)$$
Where P is some property of a number and S is the successor function.
This poses an interesting question, can the following theorems, which intuitively feel like they must obviously be valid be proven from the Peano axioms, (or in general).
Theorem 1. Consider the the induction ranging over only a subset of the natural numbers rather than over all of them. By having a similar induction rule for the subset, can we state that the property holds for the subset? Example: can we prove by induction that all even numbers have a property P by proving that 1 has that property and then that any even number k+2 has that property (assuming that even number k has that property).
Theorem 2. Can we say that if the induction proof fails only for some numbers, the result nonetheless holds for the rest. Example: $n+1 = n+(n-4)/(n-4)$ can be proven for all case, except for when $n=4$, so we should be able to say that the induction proof holds as long as n is not 4.
Theorem 3. Same as theorem 2 except now the induction would hold only after n, so that it wouldn't hold for the base case n=0.