March 12
Albert Visser,
Utrecht University
Restricted completions
This talk reports on research in collaboration with Ali Enayat and Mateusz Łełyk.
Steffen Lempp and Dino Rossegger asked: is there a consistent completion of ${\sf PA}^-$ that is axiomatised by sentences of bounded quantifier-alternation complexity? We show that there is no such restricted completion. We also show that, if one changes the measure of complexity to being $\Sigma_n$, there is a restricted completion. Specifically, we show that the true theory of the non-negative part of $\mathbb Z[X]$ can be axiomatised by a single sentence plus a set of $\Sigma^0_1$-sentences.
In our talk we will sketch these two answers. One of our aims is to make clear is that the negative answer for the case of quantifier-alternation complexity simply follows from Rosser's Theorem viewed from a sufficiently abstract standpoint.