## Two neat things

Today I stumbled across two lovely bits of math(s) I’d like to share.

The first is a fun topological proof that there are infinitely many prime numbers.

The second is a proof that the set of prime numbers is bounded…

[Wait for it…]

[Keep waiting…]

…in certain models of weak arithmetic. (I paused for Hotovy to say ‘What?!’ and shake his head in that goofy way. You know the one.) Have a look for yourself: MO21367. (Also, $\pi$ is rational in the model given in that answer.)

Yeah! I like it a lot. And it’s not so surprising that weird things happen (like $\pi$ being rational) in the absence of full induction.