I disagree with your example:
The rest of this proof is really complicated, and uses things you might not think are valid, so you might not buy it. But second order arithmetic demands that if you believe the rest of the proof, you should also agree that the twin primes conjecture holds.
But set theory says the same thing. And set theory, unlike second-order arithmetic, is probably strong enough to formalize the large and complicated proof in the first place. Even if there are elements in the proof that go beyond ZFC (large cardinals etc.), mathematicians are likely to view them as additional assumptions on top of what they see as set theory.
Consider a non-logician mathematician to whom the induction principle is not primarily a formal statement to be analyzed, but just, well, induction, a basic working tool. Given a large proof as you describe, ending in an application of induction. What would be the benefit, to the mathematician, of viewing that application as happening in second-order logic, as opposed to first-order set theory? Why would they want to use second-order anything?
To phrase this differently, second order arithmetic gives us a universal way of picking the natural numbers out in any situation. Different models of set theory may end up disagreeing about what's true in the natural numbers, but second order arithmetic is a consistent way of identifying the notion in that model of set theory which lines up with our intuitions for the what the natural numbers are.
I don't see how that works, either.
Let G be the arithmetical statement expressing the consistency of ZFC. There are models of set theory in which G is true, and models in which G is false. Are you saying that second-order arithmetic gives us a better way, a less ambiguous way, to study the truth of G? How would that work in practice?
The way I see it, different models of set theory agree on what natural numbers are, but disagree on what subsets of natural numbers exist. This ambiguity is not resolved by second-order arithmetic; rather, it's swept under the carpet. The "unique" model "pinpointed" by it is utterly at the mercy of the same ambiguity of what the set of subsets of N is, and the ambiguity reasserts itself the moment you start studying the semantics of second-order arithmetic which you will do through model theory, expressed within set theory. So what is it that you have gained?
So apparently whatever I thought was the natural numbers was some other bizzarre object; what the second order induction axiom does for a Platonist is pin down which Platonic object we're actually talking about.
To a Platonist, what you used was not the second order induction axiom; it was just the familiar principle of induction.
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Took the survey. I just missed last year's, so I was glad to get to participate this year.