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:
- 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.
- Prove something true periodically by $i = a (\mod b)$ or tie the set of numbers to cyclic groups.