Certified $\Sigma_1$-sentences

In this paper, we study the employment of $\Sigma_1$-sentences with certificates, i.e., $\Sigma_1$-sentences where a number of principles is added to ensure that the witness is sufficiently number-like. We develop certificates in some detail and illustrate their use by reproving some classical results and proving some new ones. An example of such a classical result is Vaught's theorem of the strong effective inseparability of $\mathsf{R}_0$. We also develop the new idea of a theory being ${\sf R}_{0{\sf p}}$-sourced. Using this notion, we can transfer a number of salient results from $\mathsf{R}_0$ to a variety of other theories.