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

Dec 2022

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 withCherlin-van den Dries-Macintyre/Ershov (for PAC fields), Haran (for PRCfields), and Efrat (for PpC fields) to prove all PAC, PRC, and (bounded) PpCfields are finitely undecidable. This work is drawn from the author's PhDthesis and is a sequel to arXiv:2210.12729.

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