Is the twin prime conjecture independent of Peano Arithmetic?

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. We are looking for an arithmetical formula φ whose truth value (in the standard model N) is at present unknown and such that ZF (Zermelo-Fraenkel set theory) proves that φ is independent of PA (first order Peano Arithmetic). A solution of the riddle has been obtained independently by the first two authors in a recent arXiv preprint and by the third author in an older post on Mathoverflow, which actually provides a sharper version given in Theorem 1 We take the opportunity to reflect on the nature of the incompleteness results and make some philosophical considerations. Let us first observe that the formula Con(ZF) expressing the consistency of ZF would not work. Although Con(ZF) is independent of PA, the proof of this fact cannot be carried out in ZF. To see why, recall that an independence result involves both the unprovability of the formula and the unprovability of its negation. In our case one direction poses no problems: ZF does prove that Con(ZF) is unprovable in PA. The problem lies in the other direction: ZF, if consistent, does not prove that ¬Con(ZF) is unprovable in PA. This depends on the fact that ¬Con(ZF) has complexity Σ1, so if true it must be provable in PA, and if unprovable it must be false, provably in ZF (and also in PA itself). Thus if ZF proves that ¬Con(ZF) is unprovable in PA, then ZF proves Con(ZF), contradicting Gödel’s second incompleteness theorem. More generally, we cannot take for φ any formula of complexity Π1 (such as Goldbach’s conjecture). The argument is the same: if ZF proves that a Π1 formula φ is independent of PA, then ZF proves φ, so we would have established the truth value of φ on the basis of a trusted metatheory, which is against our desiderata. By simmetry any formula of complexity Σ1 does not work. Before presenting the solution to our riddle, the reader is invited to make an attempt himself. In the meantime we try to clarify the notion of trustworthy metatheory. Let us agree that if we trust a theory T we must at least believe that T does not prove false arithmetical sentences, or, said differently, we must believe that the standard model N of PA is also a model of the arithmetical consequences of T . Now, if we Date: 1 Nov. 2021. 2020 Mathematics Subject Classification. Primary 03B10; Secondary 03F40; 03F30.