Is the twin prime conjecture independent of Peano Arithmetic?

Abstract

We show that there is an arithmetical formula F such that ZF proves that F is independent of PA and yet, unlike other arithmetical independent statements, the truth value of F cannot at present be established in ZF or in any other trusted metatheory. In fact we can choose an example of such a formula F such that ZF proves that F is equivalent to the twin prime conjecture.

4 pages. Added Hamkins as coauthor and a sharper version of the main result (Theorem 1)