{"title":"在集合和函数的强链上","authors":"Tanmay C. Inamdar","doi":"10.1112/mtk.12183","DOIUrl":null,"url":null,"abstract":"<p>Shelah has shown that there are no chains of length ω<sub>3</sub> increasing modulo finite in <math>\n <semantics>\n <mrow>\n <msup>\n <mrow></mrow>\n <msub>\n <mi>ω</mi>\n <mn>2</mn>\n </msub>\n </msup>\n <msub>\n <mi>ω</mi>\n <mn>2</mn>\n </msub>\n </mrow>\n <annotation>${}^{\\omega _2}\\omega _2$</annotation>\n </semantics></math>. We improve this result to sets. That is, we show that there are no chains of length ω<sub>3</sub> in <math>\n <semantics>\n <msup>\n <mrow>\n <mo>[</mo>\n <msub>\n <mi>ω</mi>\n <mn>2</mn>\n </msub>\n <mo>]</mo>\n </mrow>\n <msub>\n <mi>ℵ</mi>\n <mn>2</mn>\n </msub>\n </msup>\n <annotation>$[\\omega _2]^{\\aleph _2}$</annotation>\n </semantics></math> increasing modulo finite. This contrasts with results of Koszmider who has shown that there are, consistently, chains of length ω<sub>2</sub> increasing modulo finite in <math>\n <semantics>\n <msup>\n <mrow>\n <mo>[</mo>\n <msub>\n <mi>ω</mi>\n <mn>1</mn>\n </msub>\n <mo>]</mo>\n </mrow>\n <msub>\n <mi>ℵ</mi>\n <mn>1</mn>\n </msub>\n </msup>\n <annotation>$[\\omega _1]^{\\aleph _1}$</annotation>\n </semantics></math> as well as in <math>\n <semantics>\n <mrow>\n <msup>\n <mrow></mrow>\n <msub>\n <mi>ω</mi>\n <mn>1</mn>\n </msub>\n </msup>\n <msub>\n <mi>ω</mi>\n <mn>1</mn>\n </msub>\n </mrow>\n <annotation>${}^{\\omega _1}\\omega _1$</annotation>\n </semantics></math>. More generally, we study the depth of function spaces <math>\n <semantics>\n <mrow>\n <msup>\n <mrow></mrow>\n <mi>κ</mi>\n </msup>\n <mi>μ</mi>\n </mrow>\n <annotation>${}^\\kappa \\mu$</annotation>\n </semantics></math> quotiented by the ideal <math>\n <semantics>\n <msup>\n <mrow>\n <mo>[</mo>\n <mi>κ</mi>\n <mo>]</mo>\n </mrow>\n <mrow>\n <mo><</mo>\n <mi>θ</mi>\n </mrow>\n </msup>\n <annotation>$[\\kappa ]^{< \\theta }$</annotation>\n </semantics></math> where <math>\n <semantics>\n <mrow>\n <mi>θ</mi>\n <mo><</mo>\n <mi>κ</mi>\n </mrow>\n <annotation>$\\theta < \\kappa$</annotation>\n </semantics></math> are infinite cardinals.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"69 1","pages":"286-301"},"PeriodicalIF":0.8000,"publicationDate":"2023-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12183","citationCount":"1","resultStr":"{\"title\":\"On strong chains of sets and functions\",\"authors\":\"Tanmay C. Inamdar\",\"doi\":\"10.1112/mtk.12183\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Shelah has shown that there are no chains of length ω<sub>3</sub> increasing modulo finite in <math>\\n <semantics>\\n <mrow>\\n <msup>\\n <mrow></mrow>\\n <msub>\\n <mi>ω</mi>\\n <mn>2</mn>\\n </msub>\\n </msup>\\n <msub>\\n <mi>ω</mi>\\n <mn>2</mn>\\n </msub>\\n </mrow>\\n <annotation>${}^{\\\\omega _2}\\\\omega _2$</annotation>\\n </semantics></math>. We improve this result to sets. That is, we show that there are no chains of length ω<sub>3</sub> in <math>\\n <semantics>\\n <msup>\\n <mrow>\\n <mo>[</mo>\\n <msub>\\n <mi>ω</mi>\\n <mn>2</mn>\\n </msub>\\n <mo>]</mo>\\n </mrow>\\n <msub>\\n <mi>ℵ</mi>\\n <mn>2</mn>\\n </msub>\\n </msup>\\n <annotation>$[\\\\omega _2]^{\\\\aleph _2}$</annotation>\\n </semantics></math> increasing modulo finite. This contrasts with results of Koszmider who has shown that there are, consistently, chains of length ω<sub>2</sub> increasing modulo finite in <math>\\n <semantics>\\n <msup>\\n <mrow>\\n <mo>[</mo>\\n <msub>\\n <mi>ω</mi>\\n <mn>1</mn>\\n </msub>\\n <mo>]</mo>\\n </mrow>\\n <msub>\\n <mi>ℵ</mi>\\n <mn>1</mn>\\n </msub>\\n </msup>\\n <annotation>$[\\\\omega _1]^{\\\\aleph _1}$</annotation>\\n </semantics></math> as well as in <math>\\n <semantics>\\n <mrow>\\n <msup>\\n <mrow></mrow>\\n <msub>\\n <mi>ω</mi>\\n <mn>1</mn>\\n </msub>\\n </msup>\\n <msub>\\n <mi>ω</mi>\\n <mn>1</mn>\\n </msub>\\n </mrow>\\n <annotation>${}^{\\\\omega _1}\\\\omega _1$</annotation>\\n </semantics></math>. More generally, we study the depth of function spaces <math>\\n <semantics>\\n <mrow>\\n <msup>\\n <mrow></mrow>\\n <mi>κ</mi>\\n </msup>\\n <mi>μ</mi>\\n </mrow>\\n <annotation>${}^\\\\kappa \\\\mu$</annotation>\\n </semantics></math> quotiented by the ideal <math>\\n <semantics>\\n <msup>\\n <mrow>\\n <mo>[</mo>\\n <mi>κ</mi>\\n <mo>]</mo>\\n </mrow>\\n <mrow>\\n <mo><</mo>\\n <mi>θ</mi>\\n </mrow>\\n </msup>\\n <annotation>$[\\\\kappa ]^{< \\\\theta }$</annotation>\\n </semantics></math> where <math>\\n <semantics>\\n <mrow>\\n <mi>θ</mi>\\n <mo><</mo>\\n <mi>κ</mi>\\n </mrow>\\n <annotation>$\\\\theta < \\\\kappa$</annotation>\\n </semantics></math> are infinite cardinals.</p>\",\"PeriodicalId\":18463,\"journal\":{\"name\":\"Mathematika\",\"volume\":\"69 1\",\"pages\":\"286-301\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2023-01-03\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12183\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematika\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1112/mtk.12183\",\"RegionNum\":3,\"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":"Mathematika","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/mtk.12183","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
Shelah has shown that there are no chains of length ω3 increasing modulo finite in . We improve this result to sets. That is, we show that there are no chains of length ω3 in increasing modulo finite. This contrasts with results of Koszmider who has shown that there are, consistently, chains of length ω2 increasing modulo finite in as well as in . More generally, we study the depth of function spaces quotiented by the ideal where are infinite cardinals.
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Mathematika publishes both pure and applied mathematical articles and has done so continuously since its founding by Harold Davenport in the 1950s. The traditional emphasis has been towards the purer side of mathematics but applied mathematics and articles addressing both aspects are equally welcome. The journal is published by the London Mathematical Society, on behalf of its owner University College London, and will continue to publish research papers of the highest mathematical quality.
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