- A. Berarducci, A. Fornasiero, and J. D. Hamkins, “Is the twin prime conjecture independent of Peano Arithmetic?,” Mathematics arXiv, 2021.

[Bibtex]`@ARTICLE{BerarducciFornasieroHamkins:Is-the-twin-prime-conjecture-independent-of-PA, author = {Alessandro Berarducci and Antongiulio Fornasiero and Joel David Hamkins}, title = {Is the twin prime conjecture independent of Peano Arithmetic?}, journal = {Mathematics arXiv}, year = {2021}, volume = {}, number = {}, pages = {}, month = {}, note = {Under review}, abstract = {}, keywords = {under-review}, source = {}, doi = {}, eprint = {2110.08640}, archivePrefix = {arXiv}, primaryClass = {math.LO}, url = {http://jdh.hamkins.org/is-the-twin-prime-conjecture-independent-of-peano-arithmetic/}, }`

Download the article at arXiv:2110.08640

**Abstract.** We show that there is an arithmetical formula $\varphi$ such that ZF proves that $\varphi$ is independent of PA and yet, unlike other arithmetical independent statements, the truth value of $\varphi$ cannot at present be established in ZF or in any other trusted metatheory. In fact we can choose an example of such a formula $\varphi$ such that ZF proves that $\varphi$ 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.