On first-order arithmetic truth

TL;DR

This paper challenges the standard interpretation of first-order number theory (PA), arguing that the class of PA theorems is not well-defined due to logical inconsistencies revealed by a modified Godel's proof.

Contribution

It demonstrates that the common semantic understanding of PA theorems leads to a contradiction, questioning the foundational interpretation of first-order arithmetic.

Findings

The class of PA theorems is not well-defined under the standard interpretation.

A modified Godel's proof shows the inconsistency in defining PA's theorems.

The standard interpretation of PA cannot be semantically well-founded.

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.