{"title":"Universally Sacks-indestructible combinatorial families of reals","authors":"V. Fischer , L. Schembecker","doi":"10.1016/j.apal.2025.103566","DOIUrl":null,"url":null,"abstract":"<div><div>We introduce the notion of an arithmetical type of combinatorial family of reals, which serves to generalize different types of families such as mad families, maximal cofinitary groups, ultrafilter bases, splitting families and other similar types of families commonly studied in combinatorial set theory.</div><div>We then prove that every combinatorial family of reals of arithmetical type which is indestructible by the product of Sacks forcing <span><math><msup><mrow><mi>S</mi></mrow><mrow><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup></math></span> is in fact universally Sacks-indestructible, i.e. it is indestructible by any countably supported iteration or product of Sacks-forcing of any length. Further, under <span><math><mi>CH</mi></math></span> we present a unified construction of universally Sacks-indestructible families for various arithmetical types of families. In particular we prove the existence of a universally Sacks-indestructible maximal cofinitary group under <span><math><mi>CH</mi></math></span>.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 6","pages":"Article 103566"},"PeriodicalIF":0.6000,"publicationDate":"2025-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of Pure and Applied Logic","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0168007225000156","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"LOGIC","Score":null,"Total":0}
引用次数: 0
Abstract
We introduce the notion of an arithmetical type of combinatorial family of reals, which serves to generalize different types of families such as mad families, maximal cofinitary groups, ultrafilter bases, splitting families and other similar types of families commonly studied in combinatorial set theory.
We then prove that every combinatorial family of reals of arithmetical type which is indestructible by the product of Sacks forcing is in fact universally Sacks-indestructible, i.e. it is indestructible by any countably supported iteration or product of Sacks-forcing of any length. Further, under we present a unified construction of universally Sacks-indestructible families for various arithmetical types of families. In particular we prove the existence of a universally Sacks-indestructible maximal cofinitary group under .
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.