Certified ${\Sigma }_1$-sentences

Abstract

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 ${\mathsf{R}}_{0\mathsf{p}}$-sourced. Using this notion, we can transfer a number of salient results from ${\mathsf{R}}_0$ to a variety of other theories.

34 pages

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