Is the twin prime conjecture independent of Peano Arithmetic?

  • 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.