Por Albert Visser (Utrecht University).

This talk reports partly on research in collaboration with Ali Enayat and Mateusz Łełyk.

In my talk I give an introduction to two fundamental weak arithmetical theories, PA^- and Q. I will briefly review some of the salient facts about these theories.

We will zoom in on the question whether these theories have completions of restricted complexity. We consider both depth-of-quantifier-alternation complexity and the \Sigma_n-hierarchy.  The theory PA^- has no complete extension of restricted depth-of-quantifier alternation complexity. The question whether Q has a complete extension of restricted depth-of-quantifier-alternation complexity remains open. Both Q and PA^- do have a complete extension of a single sentence plus a set of \Sigma_1-sentences.

We will sketch a proof.

Transmissão via Zoom.