Mathematical induction for subsets of the natural numbers?

Question

Is there some systematic way to reformulate mathematical induction to work to prove propositions on general subsets of the natural numbers?

Own work: I'm thinking one could maybe use modulo arithmetic or divisibility to encode intervals or periodicity:

1. Prove something true for every number on an interval $i\in [a,b] \subset \mathbb N$, write $i = a+(b-a)k$ and then let $k = 0$ be part of the induction assumption.
2. Prove something true periodically by $i = a (\mod b)$ or tie the set of numbers to cyclic groups.

Answer 1

Let $S\subset\mathbb{Z}$ and let $S$ have a first element $n_0$.

For each $n\in S$ let $n^\prime$ denote the next element of $S$.

Then if the proposition $P(n_0)$ is true and if for $n\in S$, $P(n)$ implies $P(n^\prime)$ then $P(k)$ is true for all $k\in S$.

Answer 2

It's easy using the notion of "next smallest." The key to mathematical induction is that for every natural number the is a well-defined "next" number. For subsets of the natural numbers this is the case as well, although in general it isn't as simple as adding $1$. However, given a set $S\subseteq\mathbb{N}$, if you prove that $P(a)$ holds and you prove that $P(x)\Rightarrow P(n(x))$ holds, where $a$ is the smallest element of $S$ and $n:S\to S$ gives the next smallest element, then you've proven that $P$ holds on all of $S$.

This is related to proof by infinite descent.