• 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.