Mathematical induction of weak form and different versions

Question

The statement $P(n)$ is true for all $n\in\mathbb{N}$ if the following conditions are satisfied,

• $P(1)$ is true
• For all $k\in\mathbb{N}$, if $P(k)$ is true, then $P(k+1)$ is true. (*)

While in some literatures, they use another second point

• If $P(k)$ is true for some $k\in\mathbb{N}$, then $P(k+1)$ is true. (**)

What are the differences? Whenever I try to understand the both versions by looking at the domino example, they seem to equivalent. I suppose that (**) has been rewritten from (*), even though the wordings are different. The first version can be proven by the axioms of Peano. Though I prefer using the second version to prove some claims by induction.

Answer 1

Let's consider a variant of

If $P(k)$ is true for some $k \in \mathbb{N}$, then $P(k+1)$ is true, $~~~~~~(**)$

namely the one we obtain by omitting the word "some:"

If $P(k)$ is true for $k \in \mathbb{N}$, then $P(k+1)$ is true.

Do we believe that it captures the same condition as $(**)$? If so we may then ask whether any natural number is excluded. None, of course, because the only condition imposed on $k$ is membership in $\mathbb{N}$. That means that the condition holds for all natural numbers $k$. We can make it explicit by writing

For all $k \in \mathbb{N}$, if $P(k)$ is true, then $P(k+1)$ is true.

Rewriting this condition as a sentence of first-order logic finally gives:

$\forall k \in \mathbb{N} \,.\, P(k) \rightarrow P(k+1)$.

Suppose, however, that we believe that the omission of "some" changes the meaning of $(**)$. Perhaps the work "some" evokes the idea of existential quantification.

As you write in a comment, "Assume $P(k)$ is true for some $k$," which may be read, "Suppose there exists $k$ such that $P(k)$ is true."

The problem with this reading of $(**)$ is that existence is not what the second condition of induction is concerned with. The existence proof is the base case. When you prove $P(0)$, you prove that one $k$ for which $P(k)$ is true exists. With the inductive step you show that everywhere in the infinite line of dominoes, a falling tile knocks out the next.

Yet another way to look at $(**)$ is to rewrite it as follows:

If for some $k \in \mathbb{N}$, $P(k)$ is true, then $P(k+1)$ is true.

We have just rearranged things a bit, but now we are closer to translating this informal statement into a formal one. The new arrangement highlights that whatever quantifier binds $k$ applies to both occurrences of $k$ (the one in $P(k)$ as well as the one in $P(k+1)$). The statement is about the relation between $P(k)$ and $P(k+1)$, not about the existence of some $k$ such that $P(k)$ is true.