Incompleteness and undecidability of theories consistent with $\mathsf{R}$

TL;DR

This paper extends classical incompleteness results to any c.e. family of consistent theories extending arithmetic R, showing the existence of sentences undecidable in all these theories, thus highlighting inherent limitations.

Contribution

It proves a strengthened version of the first incompleteness theorem applicable to any c.e. family of theories extending R, unifying and strengthening prior results.

Findings

Existence of a sentence provable from R but undecidable in all theories in the family.

Demonstrates limitations of c.e. theories extending R in capturing all arithmetic truths.

Unifies and strengthens classical incompleteness theorems.

Abstract

We prove the following version of the first incompleteness theorem that simultaneously strengthens Mostowski's theorem and Vaught's theorem: For any c.e. family $\{ T_i \}_{i \in \omega}$ of consistent extensions of Tarski, Mostowski and Robinson's arithmetic $\mathsf{R}$, there exists a sentence $\varphi$ of arithmetic such that $\varphi \vdash \mathsf{R}$ and for all $i \in \omega$, $T_i \nvdash \varphi$ and $T_i \nvdash \neg \varphi$.