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

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 $\phi$ of arithmetic such that $\phi ⊢\mathsf{R}$ and for all $i\in \omega$, $T_i⊬\phi$ and $T_i⊬\mathrm{\neg }\phi$.