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 $T_i \nvdash \neg \varphi$.
Categories: math.LO