Finite Undecidability in Fields II: PAC, PRC and PpC Fields

TL;DR

This paper proves that all PAC and PRC fields are finitely undecidable, and shows bounded PpC fields cannot be finitely axiomatized, extending previous work on field decidability and model theory.

Contribution

It adapts existing arguments to establish finite undecidability for PAC and PRC fields and demonstrates limitations for PpC fields regarding axiomatizability.

Findings

PAC and PRC fields are finitely undecidable.

Bounded PpC fields are not finitely axiomatizable.

Extension of undecidability results to new classes of fields.

Abstract

A field $K$ in a ring language $\mathcal{L}$ is finitely undecidable if $\mbox{Cons}(\Sigma)$ is undecidable for every nonempty finite $\Sigma \subseteq \mbox{Th}(K; \mathcal{L})$. 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 P$p$C fields, and show no bounded P$p$C field is finitely axiomatisable. This work is drawn from the author's PhD thesis and is a sequel to arXiv:2210.12729.