{"title":"λ完备μ +$ \\mu ^+$ -c的强迫公理。","authors":"Saharon Shelah","doi":"10.1002/malq.201900020","DOIUrl":null,"url":null,"abstract":"<p>We consider forcing axioms for suitable families of μ-complete <math>\n <semantics>\n <msup>\n <mi>μ</mi>\n <mo>+</mo>\n </msup>\n <annotation>$\\mu ^+$</annotation>\n </semantics></math>-c.c. forcing notions. We show that some form of the condition “<math>\n <semantics>\n <mrow>\n <msub>\n <mi>p</mi>\n <mn>1</mn>\n </msub>\n <mo>,</mo>\n <msub>\n <mi>p</mi>\n <mn>2</mn>\n </msub>\n </mrow>\n <annotation>$p_1,p_2$</annotation>\n </semantics></math> have a <math>\n <semantics>\n <mrow>\n <msub>\n <mo>≤</mo>\n <mi>Q</mi>\n </msub>\n <mi>-</mi>\n <mi>lub</mi>\n </mrow>\n <annotation>$\\le _{{\\mathbb {Q}}}\\text{-}{\\rm lub}$</annotation>\n </semantics></math> in <math>\n <semantics>\n <mi>Q</mi>\n <annotation>${\\mathbb {Q}}$</annotation>\n </semantics></math>” is necessary. We also show some versions are really stronger than others.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Forcing axioms for λ-complete \\n \\n \\n μ\\n +\\n \\n $\\\\mu ^+$\\n -c.c.\",\"authors\":\"Saharon Shelah\",\"doi\":\"10.1002/malq.201900020\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We consider forcing axioms for suitable families of μ-complete <math>\\n <semantics>\\n <msup>\\n <mi>μ</mi>\\n <mo>+</mo>\\n </msup>\\n <annotation>$\\\\mu ^+$</annotation>\\n </semantics></math>-c.c. forcing notions. We show that some form of the condition “<math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>p</mi>\\n <mn>1</mn>\\n </msub>\\n <mo>,</mo>\\n <msub>\\n <mi>p</mi>\\n <mn>2</mn>\\n </msub>\\n </mrow>\\n <annotation>$p_1,p_2$</annotation>\\n </semantics></math> have a <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mo>≤</mo>\\n <mi>Q</mi>\\n </msub>\\n <mi>-</mi>\\n <mi>lub</mi>\\n </mrow>\\n <annotation>$\\\\le _{{\\\\mathbb {Q}}}\\\\text{-}{\\\\rm lub}$</annotation>\\n </semantics></math> in <math>\\n <semantics>\\n <mi>Q</mi>\\n <annotation>${\\\\mathbb {Q}}$</annotation>\\n </semantics></math>” is necessary. We also show some versions are really stronger than others.</p>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2022-02-03\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/malq.201900020\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/malq.201900020","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
We consider forcing axioms for suitable families of μ-complete -c.c. forcing notions. We show that some form of the condition “ have a in ” is necessary. We also show some versions are really stronger than others.