Proof by induction confusion

Question

When we do proof by induction our inductive step is to assume that the statement $P(n)$ holds for $n=k$, we then show that $P(k) \implies P(k+1)$, which my textbook writes as "If its true for $n=k$, show its true for $n=k+1$", now I would like to ask this question, why are we allowed to write both $n=k$ and $n=k+1$ without it implying $k=k+1$? Sorry if this question is trivial/not appropriate for stackexchange. Also can we say that $P(n)$ is defined for all $n \geq 0$, but not necessarily true?

Answer 1

What you are doing is sort of this:

Statement $1:$ At night, the sky is black.

Statement $2:$ During daytime, the sky is blue.

Statement $1$ contains "the sky is black".

Statement $2$ contains "the sky is blue".

Both statements are true.

Everything so far has been correct. Now comes the logical fallacy:

Logical fallacy:

Statement $3$: Therefore, black is blue.

The "logical fallacy" here, is that Statement $1$ and Statement $2$ are referring to different times of day, and so it makes no sense to "put them together" into Statement $3$. Indeed, we know Statement $3$ is false...

Answer 2

I think you misunderstand what is meant here by "setting $n=k$". As you know, the goal is to show that some statement $P$ holds for all natural numbers $\mathbb{N}$, or more explicitly that $P(n)$ is true for all $n \in \mathbb{N}$. We establish in the base case that it holds for $1$, i.e. that $P(1)$ is true. The induction step is then to show that if $P(k)$ is true, then $P(k+1)$ is also true. Since we know that $P(1)$ is true, we would then get that $P(2), P(3), P(4),\dots$ are also true, showing that the statement $P(n)$ holds for all $n \in \mathbb{N}$. So, what they mean here by "$P(n)$ is true for $n = k$" is just an elaborate way of saying "$P(k)$ is true". They write it like that since it's more in the form of the "formal statement".