Logic Seminar

I will sketch a proof of the statement in the title and outline how it is related to Ehrenfeucht–Fraïssé games on C*-algebras. I will provide the relevant background on C*-algebras (and descriptive set theory) and explain how to construct a standard Borel category X that can play a role of their `moduli'. The theorem from the title is an application of the compactness theorem, for a suitable first-order theory whose models correspond to functors from X. If time permits, I will mention some related problems and connections with conceptual completeness for infinitary logic. This talk is based on several discussions with Ehud Hrushovski, Jennifer Pi, Mira Tartarotti, and Stuart White after a reading group on the paper "Games on AF-algebras" by Ben De Bondt, Andrea Vaccaro, Boban Velickovic and Alessandro Vignati.

Feferman proved in 1962 that any arithmetical theorem is a consequence of a suitable transfinite iteration of uniform reflections. This result is commonly known as Feferman's completeness theorem. The talk aims to give one or two new proofs of Feferman's completeness theorem that, we hope, shed new light on this mysterious and often overlooked result.

Moreover, one of the proofs furnishes sharp bounds on the order types of well-orders necessary to attain completeness.

(This is joint work with Fedor Pakhomov and Dino Rossegger.)