David Belanger, Chi Tat Chong, Rupert Hölzl, Frank Stephan
{"title":"Independence and Induction in Reverse Mathematics","authors":"David Belanger, Chi Tat Chong, Rupert Hölzl, Frank Stephan","doi":"arxiv-2408.09796","DOIUrl":null,"url":null,"abstract":"We continue the project of the study of reverse mathematics principles\ninspired by cardinal invariants. In this article in particular we focus on\nprinciples encapsulating the existence of large families of objects that are in\nsome sense mutually independent. More precisely, we study the principle MAD\nstating that a maximal family of pairwise almost disjoint sets exists; and the\nprinciple MED expressing the existence of a maximal family of functions that\nare pairwise eventually different. We investigate characterisations of and\nrelations between these principles and some of their variants. It will turn out\nthat induction strength is an essential parameter in this context.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Logic","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.09796","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We continue the project of the study of reverse mathematics principles
inspired by cardinal invariants. In this article in particular we focus on
principles encapsulating the existence of large families of objects that are in
some sense mutually independent. More precisely, we study the principle MAD
stating that a maximal family of pairwise almost disjoint sets exists; and the
principle MED expressing the existence of a maximal family of functions that
are pairwise eventually different. We investigate characterisations of and
relations between these principles and some of their variants. It will turn out
that induction strength is an essential parameter in this context.
我们将继续研究受心算不变式启发的逆向数学原理。在这篇文章中,我们特别关注那些概括了在某种意义上相互独立的对象大家族存在性的原理。更确切地说,我们研究了表示存在成对几乎不相交集合的最大族的 MAD 原则;以及表示存在成对最终不同的函数的最大族的 MED 原则。我们研究了这些原理及其一些变体的特征和它们之间的关系。事实将证明,归纳强度是这方面的一个重要参数。