Is the twin prime conjecture independent of Peano Arithmetic?

[bibtex key=”BerarducciFornasieroHamkins:Is-the-twin-prime-conjecture-independent-of-PA”]

Download the article at arXiv:2110.08640

Abstract. We show that there is an arithmetical formula φ such that ZF proves that φ is independent of PA and yet, unlike other arithmetical independent statements, the truth value of φ cannot at present be established in ZF or in any other trusted metatheory. In fact we can choose an example of such a formula φ such that ZF proves that φ is equivalent to the twin prime conjecture. We conclude with a discussion of notion of trustworthy theory and a sharper version of the result.

This work grows in part out of an answer I posted on MathOverflow in 2012.