PAC 和 PRC 领域中的有限不可判定性

Pub Date : 2024-05-20 DOI:10.1016/j.apal.2024.103465
Brian Tyrrell
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引用次数: 0

摘要

如果对于每个非空有限 Σ⊆Th(K;L),环语言 L 中的场 K 是有限不可判定的,那么 Cons(Σ) 就是不可判定的。我们改编了起源于谢林-范登德里斯-麦金泰尔/埃尔绍夫(针对 PAC 场)和哈兰(针对 PRC 场)的论证,证明所有 PAC 场和 PRC 场都是有限不可判定的。我们描述了将这一证明应用于 PpC 场时遇到的困难,并证明了没有一个有界 PpC 场是有限公理可证的。这项工作来自作者的博士论文[44,第 4 章]。
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Finite undecidability in PAC and PRC fields

A field K in a ring language L is finitely undecidable if Cons(Σ) is undecidable for every nonempty finite ΣTh(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].

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