1 results
CERTIFIED
$ \Sigma _1$-SENTENCES
- Part of
-
- Journal:
- The Journal of Symbolic Logic , First View
- Published online by Cambridge University Press:
- 12 December 2024, pp. 1-29
-
- Article
-
- You have access
- Open access
- HTML
- Export citation
-
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.
