I have read this answer about the well ordering principle and the induction principle. It especially says that "any proper axiomatization of $\mathbb N$ in modern logic does not involve set-induction or set-well-ordering". Then, it provides the right axiom schemas:
Induction:
$Q(0) ∧ ∀k{∈}ℕ\ ( \ Q(k)⇒Q(k+1) \ ) ⇒ ∀k{∈}ℕ\ ( \ Q(k) \ )$, for every property $Q$ on $ℕ$.Well-ordering:
$∃k{∈}ℕ\ ( \ Q(k) \ ) ⇒ ∃m{∈}ℕ\ ( \ Q(m) ∧ ∀k{∈}ℕ\ ( \ Q(k)⇒m≤k \ ) \ )$, for every property $Q$ on $ℕ$.
This resonates with a question I had in mind: I have noted that the induction principle (especially in the "induction step" step) is not equivalent when it is formulated with the set-induction principle and when it is formulated with the property way.
Indeed, let $P$ be a property on $\mathbb N$. And let $S=\{n\ge 1 \mid P(n) \text{ is true} \}$.
Then, my thought was that, supposing $n\in S$ and proving from there, that $n+1 \in S$ is not the same as writing $P(n)$ and, by a succesion of implication, arrive at $P(n+1)$.
Indeed, the former supposes that $P(n)$ is true and the latter does not. So the former requires an extra assumption that we don't need. Even worse: in the former case again, we could be supposing that $P(n)$ is true for some $n$, when in fact the property on the whole $\mathbb N$ might be false. Supposing true something that is false is non-sense according to me (but I might be wrong).
Accordingly I have two questions:
- the first: "is there any link between the stackexchange answer I quoted and the thing I explain after ?"
- More practical question: "when redacting, should we avoid writing the Induction Step in the set-induction form and instead write it using only the property formulation ?"
Thanks in advance.
