As we all know, induction is used to prove statements concerning natural numbers. When we are making a proof by induction, we use the principle of mathematical induction or some variant of it. To my understanding, one of the principles we rely on goes as follows.
Let $G \subseteq \mathbb{N}$. Suppose that
(1) $1 \in G$;
(2) if $n \in G$, then $n+1 \in G$.
Then $G = \mathbb{N}$.
Note that defining $G$ can be cumbersome, so that isn't always done.
However, I have seen, and I believe you have as well, many occurences when this principle or others similar to it have been used as arguments to derive results for finite sets, not just naturals numbers. Say we want to prove that $P$ is true for all finite sets $X \in \mathcal{X}$. In the derivations I am talking about, $P$ is first proven for all $X \in \mathcal{X}$ such that $|X| = 1$. Next, it is shown that $P$ being true for all $X \in \mathcal{X}$ such that $|X| = n$ implies that $P$ is true for all $X \in \mathcal{X}$ such that $|X| = n + 1$. Then, it is concluded that $P$ is true for all $X \in \mathcal{X}$.
As I said, the induction principle is intended for natural numbers, yet it seems like we are using it to deduce something for finite sets, a different type of object. This begs the question: is there some other principle of induction concerning something else, or are we relying on some complicated theorem? What is going on here?
I have one possible explanation for this, but I don't think it's a good one. The induction principle is often motivated by the transitive the property of implications, so I can see how we could use it to "substitute" a finite amount of reasoning, as it feels like is done in the given example. This is why I can see how induction could be used as an argument that is informal but often acceptable.
