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

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 $T_i \nvdash \neg \varphi$.