{"title":"A criterion for uniform finiteness in the imaginary sorts","authors":"Will Johnson","doi":"10.1007/s00153-021-00803-5","DOIUrl":null,"url":null,"abstract":"<div><p>Let <i>T</i> be a theory. If <i>T</i> eliminates <span>\\(\\exists ^\\infty \\)</span>, it need not follow that <span>\\(T^{\\mathrm {eq}}\\)</span> eliminates <span>\\(\\exists ^\\infty \\)</span>, as shown by the example of the <i>p</i>-adics. We give a criterion to determine whether <span>\\(T^{\\mathrm {eq}}\\)</span> eliminates <span>\\(\\exists ^\\infty \\)</span>. Specifically, we show that <span>\\(T^{\\mathrm {eq}}\\)</span> eliminates <span>\\(\\exists ^\\infty \\)</span> if and only if <span>\\(\\exists ^\\infty \\)</span> is eliminated on all interpretable sets of “unary imaginaries.” This criterion can be applied in cases where a full description of <span>\\(T^{\\mathrm {eq}}\\)</span> is unknown. As an application, we show that <span>\\(T^{\\mathrm {eq}}\\)</span> eliminates <span>\\(\\exists ^\\infty \\)</span> when <i>T</i> is a C-minimal expansion of ACVF.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3000,"publicationDate":"2021-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Archive for Mathematical Logic","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00153-021-00803-5","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"Arts and Humanities","Score":null,"Total":0}
引用次数: 1
Abstract
Let T be a theory. If T eliminates \(\exists ^\infty \), it need not follow that \(T^{\mathrm {eq}}\) eliminates \(\exists ^\infty \), as shown by the example of the p-adics. We give a criterion to determine whether \(T^{\mathrm {eq}}\) eliminates \(\exists ^\infty \). Specifically, we show that \(T^{\mathrm {eq}}\) eliminates \(\exists ^\infty \) if and only if \(\exists ^\infty \) is eliminated on all interpretable sets of “unary imaginaries.” This criterion can be applied in cases where a full description of \(T^{\mathrm {eq}}\) is unknown. As an application, we show that \(T^{\mathrm {eq}}\) eliminates \(\exists ^\infty \) when T is a C-minimal expansion of ACVF.
期刊介绍:
The journal publishes research papers and occasionally surveys or expositions on mathematical logic. Contributions are also welcomed from other related areas, such as theoretical computer science or philosophy, as long as the methods of mathematical logic play a significant role. The journal therefore addresses logicians and mathematicians, computer scientists, and philosophers who are interested in the applications of mathematical logic in their own field, as well as its interactions with other areas of research.