PAC 和 PRC 领域中的有限不可判定性

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic 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 $Σ \subseteq Th (K; L)$ . 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].

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来源期刊

Annals of Pure and Applied Logic 数学-数学

CiteScore

1.40

自引率

12.50%

发文量

审稿时长

200 days

期刊介绍： The journal Annals of Pure and Applied Logic publishes high quality papers in all areas of mathematical logic as well as applications of logic in mathematics, in theoretical computer science and in other related disciplines. All submissions to the journal should be mathematically correct, well written (preferably in English)and contain relevant new results that are of significant interest to a substantial number of logicians. The journal also considers submissions that are somewhat too long to be published by other journals while being too short to form a separate memoir provided that they are of particular outstanding quality and broad interest. In addition, Annals of Pure and Applied Logic occasionally publishes special issues of selected papers from well-chosen conferences in pure and applied logic.