Full Souslin trees at small cardinals

IF 1.2 2区数学 Q1 MATHEMATICS Journal of the London Mathematical Society-Second Series Pub Date : 2024-06-28 DOI:10.1112/jlms.12957

Assaf Rinot, Shira Yadai, Zhixing You

{"title":"Full Souslin trees at small cardinals","authors":"Assaf Rinot, Shira Yadai, Zhixing You","doi":"10.1112/jlms.12957","DOIUrl":null,"url":null,"abstract":"A <math>\n <semantics>\n <mi>κ</mi>\n <annotation>$\\kappa$</annotation>\n </semantics></math>-tree is full if each of its limit levels omits no more than one potential branch. Kunen asked whether a full <math>\n <semantics>\n <mi>κ</mi>\n <annotation>$\\kappa$</annotation>\n </semantics></math>-Souslin tree may consistently exist. Shelah gave an affirmative answer of height a strong limit Mahlo cardinal <math>\n <semantics>\n <mi>κ</mi>\n <annotation>$\\kappa $</annotation>\n </semantics></math>. Here, it is shown that these trees may consistently exist at small cardinals. Indeed, there can be <math>\n <semantics>\n <msub>\n <mi>ℵ</mi>\n <mn>3</mn>\n </msub>\n <annotation>$\\aleph _3$</annotation>\n </semantics></math> many full <math>\n <semantics>\n <msub>\n <mi>ℵ</mi>\n <mn>2</mn>\n </msub>\n <annotation>$\\aleph _2$</annotation>\n </semantics></math>-trees such that the product of any countably many of them is an <math>\n <semantics>\n <msub>\n <mi>ℵ</mi>\n <mn>2</mn>\n </msub>\n <annotation>$\\aleph _2$</annotation>\n </semantics></math>-Souslin tree.","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"110 1","pages":""},"PeriodicalIF":1.2000,"publicationDate":"2024-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.12957","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the London Mathematical Society-Second Series","FirstCategoryId":"100","ListUrlMain":"https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/jlms.12957","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

Abstract

A $κ$ -tree is full if each of its limit levels omits no more than one potential branch. Kunen asked whether a full $κ$ -Souslin tree may consistently exist. Shelah gave an affirmative answer of height a strong limit Mahlo cardinal $κ$ . Here, it is shown that these trees may consistently exist at small cardinals. Indeed, there can be $ℵ_{3}$ many full $ℵ_{2}$ -trees such that the product of any countably many of them is an $ℵ_{2}$ -Souslin tree.

查看原文

微信好友朋友圈 QQ好友复制链接

本刊更多论文

小红雀的全苏树林

如果κ $kappa$ -树的每个极限层都没有遗漏一个以上的潜在分支，那么这棵树就是完整的。库能（Kunen）问，一棵完整的κ $kappa$ -Souslin 树是否可能一直存在。谢拉给出了一个肯定的答案，即高度强极限马赫洛红心κ $\kappa $ 。这里，我们证明了这些树在小红心时可能一直存在。事实上，可以有ℵ 3 $\aleph _3$很多棵完整的ℵ 2 $\aleph _2$-树，使得其中任意可数棵树的乘积都是一棵ℵ 2 $\aleph _2$-苏林树。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文去求助

来源期刊

Journal of the London Mathematical Society-Second Series 数学-数学

CiteScore

1.90

自引率

0.00%

发文量

186

审稿时长

6-12 weeks

期刊介绍： The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.