{"title":"带有 CH 否定和弱强制公理的 C(X) 上的不连续同构","authors":"Yushiro Aoki","doi":"10.1112/jlms.12956","DOIUrl":null,"url":null,"abstract":"<p>In this paper, I introduce the properties <span></span><math>\n <semantics>\n <msub>\n <mi>EPC</mi>\n <msub>\n <mi>ℵ</mi>\n <mn>1</mn>\n </msub>\n </msub>\n <annotation>$\\mathrm{EPC}_{\\aleph _1}$</annotation>\n </semantics></math> and <span></span><math>\n <semantics>\n <mrow>\n <mi>ProjCes</mi>\n <mo>(</mo>\n <mi>E</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$\\mathrm{ProjCes}(E)$</annotation>\n </semantics></math> for forcing notions and show that it is consistent that the forcing axiom for <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>EPC</mi>\n <msub>\n <mi>ℵ</mi>\n <mn>1</mn>\n </msub>\n </msub>\n <mo>+</mo>\n <mi>ProjCes</mi>\n <mrow>\n <mo>(</mo>\n <mi>E</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$\\mathrm{EPC}_{\\aleph _1}+ \\mathrm{ProjCes}(E)$</annotation>\n </semantics></math> forcing notions holds, the continuum hypothesis fails, and an ultrapower of the field of reals has the property <span></span><math>\n <semantics>\n <msub>\n <mi>β</mi>\n <mn>1</mn>\n </msub>\n <annotation>$\\beta _1$</annotation>\n </semantics></math>. This provides a partial solution to H. Woodin's question concerning the existence of discontinuous homomorphisms on the Banach algebra of all complex-valued continuous functions on a compact space. Furthermore, we prove that the uniformization of a coloring of a ladder system on a stationary–costationary set <span></span><math>\n <semantics>\n <mi>E</mi>\n <annotation>$E$</annotation>\n </semantics></math> is an example of an <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>EPC</mi>\n <msub>\n <mi>ℵ</mi>\n <mn>1</mn>\n </msub>\n </msub>\n <mo>+</mo>\n <mi>ProjCes</mi>\n <mrow>\n <mo>(</mo>\n <msub>\n <mi>ω</mi>\n <mn>1</mn>\n </msub>\n <mo>∖</mo>\n <mi>E</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$\\mathrm{EPC}_{\\aleph _1}+ \\mathrm{ProjCes}(\\omega _1 \\setminus E)$</annotation>\n </semantics></math> forcing notion. As a corollary, it is consistent that a nonfree Whitehead group exists, the continuum hypothesis fails, and an ultrapower of the field of reals has the property <span></span><math>\n <semantics>\n <msub>\n <mi>β</mi>\n <mn>1</mn>\n </msub>\n <annotation>$\\beta _1$</annotation>\n </semantics></math>.</p>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Discontinuous homomorphisms on C(X) with the negation of CH and a weak forcing axiom\",\"authors\":\"Yushiro Aoki\",\"doi\":\"10.1112/jlms.12956\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>In this paper, I introduce the properties <span></span><math>\\n <semantics>\\n <msub>\\n <mi>EPC</mi>\\n <msub>\\n <mi>ℵ</mi>\\n <mn>1</mn>\\n </msub>\\n </msub>\\n <annotation>$\\\\mathrm{EPC}_{\\\\aleph _1}$</annotation>\\n </semantics></math> and <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>ProjCes</mi>\\n <mo>(</mo>\\n <mi>E</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$\\\\mathrm{ProjCes}(E)$</annotation>\\n </semantics></math> for forcing notions and show that it is consistent that the forcing axiom for <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>EPC</mi>\\n <msub>\\n <mi>ℵ</mi>\\n <mn>1</mn>\\n </msub>\\n </msub>\\n <mo>+</mo>\\n <mi>ProjCes</mi>\\n <mrow>\\n <mo>(</mo>\\n <mi>E</mi>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation>$\\\\mathrm{EPC}_{\\\\aleph _1}+ \\\\mathrm{ProjCes}(E)$</annotation>\\n </semantics></math> forcing notions holds, the continuum hypothesis fails, and an ultrapower of the field of reals has the property <span></span><math>\\n <semantics>\\n <msub>\\n <mi>β</mi>\\n <mn>1</mn>\\n </msub>\\n <annotation>$\\\\beta _1$</annotation>\\n </semantics></math>. This provides a partial solution to H. Woodin's question concerning the existence of discontinuous homomorphisms on the Banach algebra of all complex-valued continuous functions on a compact space. Furthermore, we prove that the uniformization of a coloring of a ladder system on a stationary–costationary set <span></span><math>\\n <semantics>\\n <mi>E</mi>\\n <annotation>$E$</annotation>\\n </semantics></math> is an example of an <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>EPC</mi>\\n <msub>\\n <mi>ℵ</mi>\\n <mn>1</mn>\\n </msub>\\n </msub>\\n <mo>+</mo>\\n <mi>ProjCes</mi>\\n <mrow>\\n <mo>(</mo>\\n <msub>\\n <mi>ω</mi>\\n <mn>1</mn>\\n </msub>\\n <mo>∖</mo>\\n <mi>E</mi>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation>$\\\\mathrm{EPC}_{\\\\aleph _1}+ \\\\mathrm{ProjCes}(\\\\omega _1 \\\\setminus E)$</annotation>\\n </semantics></math> forcing notion. As a corollary, it is consistent that a nonfree Whitehead group exists, the continuum hypothesis fails, and an ultrapower of the field of reals has the property <span></span><math>\\n <semantics>\\n <msub>\\n <mi>β</mi>\\n <mn>1</mn>\\n </msub>\\n <annotation>$\\\\beta _1$</annotation>\\n </semantics></math>.</p>\",\"PeriodicalId\":1,\"journal\":{\"name\":\"Accounts of Chemical Research\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":16.4000,\"publicationDate\":\"2024-06-30\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Accounts of Chemical Research\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12956\",\"RegionNum\":1,\"RegionCategory\":\"化学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"CHEMISTRY, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12956","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
Discontinuous homomorphisms on C(X) with the negation of CH and a weak forcing axiom
In this paper, I introduce the properties and for forcing notions and show that it is consistent that the forcing axiom for forcing notions holds, the continuum hypothesis fails, and an ultrapower of the field of reals has the property . This provides a partial solution to H. Woodin's question concerning the existence of discontinuous homomorphisms on the Banach algebra of all complex-valued continuous functions on a compact space. Furthermore, we prove that the uniformization of a coloring of a ladder system on a stationary–costationary set is an example of an forcing notion. As a corollary, it is consistent that a nonfree Whitehead group exists, the continuum hypothesis fails, and an ultrapower of the field of reals has the property .
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