On the standard interpretation of first-order number theory
Stephen Boyce
TL;DR
The paper addresses whether facts arising from the standard interpretation of first-order number theory can meaningfully determine theoremhood. It uses the arithmetized provability predicate $Pf(x,y)$, the Gödel sentence $\mathcal{G}$, and the diagonal lemma to construct a semantic definition of provability and derive a paradox under metatheoretic assumptions. The result shows that the class of $S$-theorems is not well-defined, challenging the foundations of metamathematics and the use of semantic notions to certify theorems. The work suggests a broader failure of metamathematical notions of proof and truth for formal theories, with implications beyond traditional truth theories, and notes that a purely syntactic argument can reach the same conclusion.
Abstract
The standard interpretation of first-order number theory (PA), according to the generally accepted view, associates well-defined set-theoretic entities with each and every well-formed formula of this system. But this implies that the class of PA theorems is semantically defined by a class sign of PA itself, (E x_2) Pf(x_2, x_1), in the following sense: with b' the PA numeral for the number b, (E x_2) Pf(x_2, b') is true under the standard interpretation if and only if b is the Godel number of a PA theorem. From this however it is easily established, by a modification of Godel's proof, that the class of PA theorems, and hence the standard interpretation of PA itself, is not well defined after all.