November 15
Rasmus Blanck, University of Gothenburg
Incompleteness results for arithmetically definable extensions of strong fragments of PA

In this talk, I will present generalisations of some incompleteness results along two axes: r.e. theories are replaced by $\Sigma_{n+1}$-definable ones, and the base theory is pushed down as far as it will go below PA. Such results are often easy to prove from suitably formulated generalisations of facts used in the original proofs. I will present a handful of such facts, including versions of the arithmetised completeness theorem and the Orey–Hájek characterisation, to show what additional assumptions our theories must satisfy for the results to generalise. Two salient classes of theories emerge in this context: (a) $\Sigma_n$-sound extensions of I$\Sigma_n$ + exp, and (b) $\Pi_n$-complete, consistent extensions of I$\Sigma_{n+1}$. Finally, I will discuss some results that fail to generalise to $\Sigma_{n+1}$-definable theories, as well as an open problem related to Woodin's theorem on the universal algorithm.

The presentation is based on the following paper: