{"title":"P-理想二分法、马丁公理和纠缠集","authors":"Osvaldo Guzmán, Stevo Todorcevic","doi":"10.1007/s11856-024-2651-8","DOIUrl":null,"url":null,"abstract":"<p>We build a model of the <i>P</i>-ideal ichotomy (PID) and Martin’s axiom for <i>ω</i><sub>1</sub> (<span>\\({\\rm MA}_{{\\omega}_1}\\)</span>) in which there is a 2-entangled set of reals. In particular, it follows that the Open Graph Axiom or Baumgartner’s axiom for <i>ω</i><sub>1</sub>-dense sets are not consequences of <span>\\({\\rm PID} + {\\rm MA}_{{\\omega}_1}\\)</span>. We review Neeman’s iteration method using two type side conditions and provide an alternative proof for the preservation of properness.</p>","PeriodicalId":14661,"journal":{"name":"Israel Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2024-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The P-ideal dichotomy, Martin’s axiom and entangled sets\",\"authors\":\"Osvaldo Guzmán, Stevo Todorcevic\",\"doi\":\"10.1007/s11856-024-2651-8\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We build a model of the <i>P</i>-ideal ichotomy (PID) and Martin’s axiom for <i>ω</i><sub>1</sub> (<span>\\\\({\\\\rm MA}_{{\\\\omega}_1}\\\\)</span>) in which there is a 2-entangled set of reals. In particular, it follows that the Open Graph Axiom or Baumgartner’s axiom for <i>ω</i><sub>1</sub>-dense sets are not consequences of <span>\\\\({\\\\rm PID} + {\\\\rm MA}_{{\\\\omega}_1}\\\\)</span>. We review Neeman’s iteration method using two type side conditions and provide an alternative proof for the preservation of properness.</p>\",\"PeriodicalId\":14661,\"journal\":{\"name\":\"Israel Journal of Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2024-08-04\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Israel Journal of Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s11856-024-2651-8\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Israel Journal of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s11856-024-2651-8","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
The P-ideal dichotomy, Martin’s axiom and entangled sets
We build a model of the P-ideal ichotomy (PID) and Martin’s axiom for ω1 (\({\rm MA}_{{\omega}_1}\)) in which there is a 2-entangled set of reals. In particular, it follows that the Open Graph Axiom or Baumgartner’s axiom for ω1-dense sets are not consequences of \({\rm PID} + {\rm MA}_{{\omega}_1}\). We review Neeman’s iteration method using two type side conditions and provide an alternative proof for the preservation of properness.
期刊介绍：
The Israel Journal of Mathematics is an international journal publishing high-quality original research papers in a wide spectrum of pure and applied mathematics. The prestigious interdisciplinary editorial board reflects the diversity of subjects covered in this journal, including set theory, model theory, algebra, group theory, number theory, analysis, functional analysis, ergodic theory, algebraic topology, geometry, combinatorics, theoretical computer science, mathematical physics, and applied mathematics.
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