Finite undecidability in PAC and PRC fields

Abstract

A field K in a ring language $L$ is finitely undecidable if $\text{Cons}\left(\Sigma \right)$ is undecidable for every nonempty finite $\Sigma \subseteq \text{Th}(K;L)$. We adapt arguments originating with Cherlin-van den Dries-Macintyre/Ershov (for PAC fields) and Haran (for PRC fields) to prove all PAC and PRC fields are finitely undecidable. We describe the difficulties that arise in adapting the proof to PpC fields, and show no bounded PpC field is finitely axiomatisable. This work is drawn from the author's PhD thesis [44, Chapter 4].