{"title":"On the Absoluteness of ℵ1-Freeness","authors":"D. Herden, A. V. Pasi","doi":"10.1007/s10469-023-09704-3","DOIUrl":null,"url":null,"abstract":"<div><div><p>ℵ<sub>1</sub>-free groups, Abelian groups for which every countable subgroup is free, exhibit a number of interesting algebraic and set-theoretic properties. We will give a complete proof that the property of being ℵ<sub>1</sub>-free is absolute; that is, if an Abelian group <i>G</i> is ℵ<sub>1</sub>-free in some transitive model <b>M</b> of ZFC, then it is ℵ<sub>1</sub>-free in any transitive model of ZFC containing <i>G</i>. The absoluteness of ℵ<sub>1</sub>-freeness has the following remarkable consequence: an Abelian group <i>G</i> is ℵ<sub>1</sub>-free in some transitive model of ZFC if and only if it is (countable and) free in some model extension. This set-theoretic characterization will be a starting point for further exploring the relationship between the set-theoretic and algebraic properties of ℵ<sub>1</sub>-free groups. In particular, we will demonstrate how proofs may be dramatically simplified using model extensions for ℵ<sub>1</sub>-free groups.</p></div></div>","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"61 5","pages":"351 - 362"},"PeriodicalIF":0.4000,"publicationDate":"2023-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Algebra and Logic","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10469-023-09704-3","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"LOGIC","Score":null,"Total":0}
引用次数: 1
Abstract
ℵ1-free groups, Abelian groups for which every countable subgroup is free, exhibit a number of interesting algebraic and set-theoretic properties. We will give a complete proof that the property of being ℵ1-free is absolute; that is, if an Abelian group G is ℵ1-free in some transitive model M of ZFC, then it is ℵ1-free in any transitive model of ZFC containing G. The absoluteness of ℵ1-freeness has the following remarkable consequence: an Abelian group G is ℵ1-free in some transitive model of ZFC if and only if it is (countable and) free in some model extension. This set-theoretic characterization will be a starting point for further exploring the relationship between the set-theoretic and algebraic properties of ℵ1-free groups. In particular, we will demonstrate how proofs may be dramatically simplified using model extensions for ℵ1-free groups.
期刊介绍：
This bimonthly journal publishes results of the latest research in the areas of modern general algebra and of logic considered primarily from an algebraic viewpoint. The algebraic papers, constituting the major part of the contents, are concerned with studies in such fields as ordered, almost torsion-free, nilpotent, and metabelian groups; isomorphism rings; Lie algebras; Frattini subgroups; and clusters of algebras. In the area of logic, the periodical covers such topics as hierarchical sets, logical automata, and recursive functions.
Algebra and Logic is a translation of ALGEBRA I LOGIKA, a publication of the Siberian Fund for Algebra and Logic and the Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences.
All articles are peer-reviewed.
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