## LESSWRONGLW

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... (read more)

I think you're reading too much into what I'm saying. I'm not suggesting that second order arithmetic is useful as a mathematical framework to talk about reasoning, in the way that first-order logic can. I'm saying that second order arithmetic is a useful way to talk about what makes the natural numbers special.

I'm also not suggesting that second order arithmetic has anything deep to add relative to a naïve (but sufficiently abstract) understanding of induction, but given that many people don't have a sufficiently abstract understanding of induction, I t... (read more)