Definition: An inductive set $A$ is a one that satisfies: $1\in A$ and $k \in A\implies k+1\in A$.
If we characterize the natural numbers as the set which has the following properties:
- $\Bbb N$ is inductive.
- If $H$ is inductive then $\Bbb N \subseteq H$.
Then if we want to prove some proposition $P(n)$ we only need to show that $T=\{n\in \Bbb N: P(n)\}$ is inductive ($\iff T=\Bbb N)$.
From this definition however, it seems mandatory to have $1$ as the usually called 'base case'.
Does any of these definitions need to be modified to allow moving the base case?
We can always "cheat," and if the base case is for example $4$, we can let $P^\ast(n)$ be the proposition $P(n+3)$.