The Undecidability of k-Provability

Abstract Buss, S.R., The undecidability of k-provability, Annals of Pure and Applied Logic 53 (1991) 75-102. The k-provability problem is, given a first-order formula o and an integer k, to determine if o has a proof consisting of k or fewer lines (i.e., formulas or sequents). This paper shows that the k-provability problem for the sequent calculus is undecidable. Indeed, for every r.e. set X there is a formula o(x) and an integer k such that for all n,o(Sn0) has a proof of ⩽k sequents if and only if n ϵ X.