Serial properties, selector proofs and the provability of consistency

Abstract

The consistency of a theory means that each of its formal derivations $D_{0}, D_{1}, D_{2}, \ldots $ is free of contradictions. For Peano Arithmetic PA, after the standard coding of derivations by numerals, PA-consistency is directly represented by the consistency scheme $\textsf{Con}^{S}_{\textsf{PA}}$, which is a series of arithmetical statements ‘$n$ is not a code of a derivation of $\ (0=1)$’ for numerals $n=0,1,2,\ldots $. We note that the consistency formula $\textsf{Con}_{\textsf{PA}}$, $\forall x$ ‘$x$ is not a code of a derivation of $(0=1)$,’ is strictly stronger in PA than PA-consistency and corresponds to some other property, which we call uniform consistency. When studying the provability of consistency in PA we ought to work not with the consistency formula $\textsf{Con}_{\textsf{PA}}$ but rather with the consistency scheme $\textsf{Con}^{S}_{\textsf{PA}}$, which adequately represents PA-consistency. This paper introduces the Hilbert-inspired notion of proof of an infinite series of formulas in a theory and proves PA-consistency in the form $\textsf{Con}^{S}_{\textsf{PA}}$ in PA. These findings show that PA proves its consistency whereas, by Gödel’s second incompleteness theorem, PA cannot prove its uniform consistency.