How to generalize induction from this definition?

Question

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:

1. $\Bbb N$ is inductive.
2. 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?

Answer 1

We can always "cheat," and if the base case is for example $4$, we can let $P^\ast(n)$ be the proposition $P(n+3)$.