TITLE: How can induction work on non-standard natural numbers?
QUESTION [7 upvotes]: When we consider the Peano axioms minus the induction scheme, we can have strange, but still quite understandable models in which there are "parallel strands" of numbers, as I imagine in the picture below:
$\quad\quad\quad$
This mental image makes it at least plausible that induction might not work in all models of this set of axioms: "knocking over the domino at zero, there is no reason that any domino in the parallel strand will ever fall over".
But if we add the induction scheme, we still have non-standard models besides $\Bbb N$.
I cannot wrap my head around how any of these might work.
Yes, I know, you append $\Bbb Q$-many copies of $\Bbb Z$ and so on.
But the real problem for me is, how can induction work (that is, prove statements about all non-standard numbers) if "no domino in a parallel strand is ever tipped over by the chain of dominos starting from $0$"?
In such a model, is it just "coincidence" that all statements satisfied by $0$ and its successors also hold for the additional numbers?

REPLY [2 votes]: I think the big problem here is posing the question faithfully.
In one sense of course (and I think the other answers are addressing this) it's no coincidence that nonstandard models of $\mathsf{PA}$ satisfy the first-order induction scheme: that's literally part of the definition of "model of $\mathsf{PA}$." But despite this not all models feel the same. Per the domino picture, the (second-order) principle of well-foundedness provides a "satisfying justification" for the first-order induction scheme in $\mathbb{N}$. Nonstandard models of $\mathsf{PA}$ - which of course includes the first-order induction scheme - are obviously not well-founded. Put another way:

No nonstandard model of $\mathsf{PA}$ can satisfy the first-order induction scheme for "the right reason."

And this is exactly the cost of trying to "first-orderize" a categorical description of an infinite structure: you'll wind up (per compactness) opening the door to structures which satisfy your approximate version for wrong reasons.

So as soon as we see that well-foundedness characterizes $\mathbb{N}$ up to isomorphism, we know that any of its "first-orderizations" will have "unintended models." Should we use the term "coincidence" at this point? I'm not sure: unintended phenomena are not always coincidences!
I think one reasonable question which arises at this point is whether some first-order approximation to true well-foundedness could itself be a higher-order principle of interest on its own:

Are there any "reasonably simple" second-order sentences which (over the discrete ordered semiring axioms, say) imply the first-order induction scheme but which are strictly weaker than well-foundedness?

(The "reasonably simple"-bit is important: "Every first-order $\mathsf{PA}$ axiom is true" can be expressed by a single second-order sentence, but is pretty silly.) Tentatively I'll say no, that the only "natural" justification for the first-order induction scheme is the idea of genuine well-foundedness. At the same time I can't quite bring myself to use the word "coincidence" here: the passage from second-order to first-order induction is definitely not arbitrary, and so being a nonstandard model of $\mathsf{PA}$ still feels very meaningful to me (moreso than, say, being a model of $I\Sigma_{17}$ - why $17$ and not $18$?). But here we move into the realm of mathematical aesthetics, so I'll stop there.