## LESSWRONGLW

I think this critique undervalues the insight second order arithmetic gives for understanding the natural numbers.

If what you wanted from logic was a complete, self-contained account of mathematical reasoning then second order arithmetic is a punt: it purports to describe a set of rules for describing the natural numbers, and then in the middle says "Just fill in whatever you want here, and then go on". Landsburg worries that, as a consequence, we haven't gotten anywhere (or, worse, have started behind where we started, since no we need an accou... (read more)

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)

1Cyan7yI object to the notion that the hypernaturals [http://en.wikipedia.org/wiki/Hyperinteger] are bizarre. Down with ω-consistency! /geek