CERTIFIED Σ1-SENTENCES

Abstract

In this paper, we study the employment of Σ1-sentences with certificates, i.e., Σ1-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 R0. We also develop the new idea of a theory being R0p-sourced. Using this notion, we can transfer a number of salient results from R0 to a variety of other theories.