{"title":"PAC 和 PRC 领域中的有限不可判定性","authors":"Brian Tyrrell","doi":"10.1016/j.apal.2024.103465","DOIUrl":null,"url":null,"abstract":"<div><p>A field <em>K</em> in a ring language <span><math><mi>L</mi></math></span> is <em>finitely undecidable</em> if <span><math><mtext>Cons</mtext><mo>(</mo><mi>Σ</mi><mo>)</mo></math></span> is undecidable for every nonempty finite <span><math><mi>Σ</mi><mo>⊆</mo><mtext>Th</mtext><mo>(</mo><mi>K</mi><mo>;</mo><mi>L</mi><mo>)</mo></math></span>. 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<em>p</em>C fields, and show no bounded P<em>p</em>C field is finitely axiomatisable. This work is drawn from the author's PhD thesis <span>[44, Chapter 4]</span>.</p></div>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0168007224000630/pdfft?md5=4a33b42fff6d541d26261561103e7ddd&pid=1-s2.0-S0168007224000630-main.pdf","citationCount":"0","resultStr":"{\"title\":\"Finite undecidability in PAC and PRC fields\",\"authors\":\"Brian Tyrrell\",\"doi\":\"10.1016/j.apal.2024.103465\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>A field <em>K</em> in a ring language <span><math><mi>L</mi></math></span> is <em>finitely undecidable</em> if <span><math><mtext>Cons</mtext><mo>(</mo><mi>Σ</mi><mo>)</mo></math></span> is undecidable for every nonempty finite <span><math><mi>Σ</mi><mo>⊆</mo><mtext>Th</mtext><mo>(</mo><mi>K</mi><mo>;</mo><mi>L</mi><mo>)</mo></math></span>. 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<em>p</em>C fields, and show no bounded P<em>p</em>C field is finitely axiomatisable. This work is drawn from the author's PhD thesis <span>[44, Chapter 4]</span>.</p></div>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-05-20\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.sciencedirect.com/science/article/pii/S0168007224000630/pdfft?md5=4a33b42fff6d541d26261561103e7ddd&pid=1-s2.0-S0168007224000630-main.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0168007224000630\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0168007224000630","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
A field K in a ring language is finitely undecidable if is undecidable for every nonempty finite . 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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