{"title":"当地的选择原则","authors":"Farmer Schlutzenberg","doi":"10.1002/malq.202000089","DOIUrl":null,"url":null,"abstract":"<p>Assume <math>\n <semantics>\n <mi>ZFC</mi>\n <annotation>$\\mathsf {ZFC}$</annotation>\n </semantics></math>. Let κ be a cardinal. A <math>\n <semantics>\n <mrow>\n <mo><</mo>\n <mspace></mspace>\n <mi>κ</mi>\n </mrow>\n <annotation>${\\mathord {<}\\hspace{1.111pt}\\kappa }$</annotation>\n </semantics></math><i>-ground</i>\nis a transitive proper class <i>W</i> modelling <math>\n <semantics>\n <mi>ZFC</mi>\n <annotation>$\\mathsf {ZFC}$</annotation>\n </semantics></math> such that <i>V</i> is a generic extension of <i>W</i> via a forcing <math>\n <semantics>\n <mrow>\n <mi>P</mi>\n <mo>∈</mo>\n <mi>W</mi>\n </mrow>\n <annotation>$\\mathbb {P}\\in W$</annotation>\n </semantics></math> of cardinality <math>\n <semantics>\n <mrow>\n <mo><</mo>\n <mspace></mspace>\n <mi>κ</mi>\n </mrow>\n <annotation>${\\mathord {<}\\hspace{1.111pt}\\kappa }$</annotation>\n </semantics></math>. The κ<i>-mantle</i> <math>\n <semantics>\n <msub>\n <mi>M</mi>\n <mi>κ</mi>\n </msub>\n <annotation>$\\mathcal {M}_\\kappa$</annotation>\n </semantics></math> is the intersection of all <math>\n <semantics>\n <mrow>\n <mo><</mo>\n <mspace></mspace>\n <mi>κ</mi>\n </mrow>\n <annotation>${\\mathord {<}\\hspace{1.111pt}\\kappa }$</annotation>\n </semantics></math>-grounds. We prove that certain partial choice principles in <math>\n <semantics>\n <msub>\n <mi>M</mi>\n <mi>κ</mi>\n </msub>\n <annotation>$\\mathcal {M}_\\kappa$</annotation>\n </semantics></math> are the consequence of κ being inaccessible/weakly compact, and some other related facts.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202000089","citationCount":"2","resultStr":"{\"title\":\"Choice principles in local mantles\",\"authors\":\"Farmer Schlutzenberg\",\"doi\":\"10.1002/malq.202000089\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Assume <math>\\n <semantics>\\n <mi>ZFC</mi>\\n <annotation>$\\\\mathsf {ZFC}$</annotation>\\n </semantics></math>. Let κ be a cardinal. A <math>\\n <semantics>\\n <mrow>\\n <mo><</mo>\\n <mspace></mspace>\\n <mi>κ</mi>\\n </mrow>\\n <annotation>${\\\\mathord {<}\\\\hspace{1.111pt}\\\\kappa }$</annotation>\\n </semantics></math><i>-ground</i>\\nis a transitive proper class <i>W</i> modelling <math>\\n <semantics>\\n <mi>ZFC</mi>\\n <annotation>$\\\\mathsf {ZFC}$</annotation>\\n </semantics></math> such that <i>V</i> is a generic extension of <i>W</i> via a forcing <math>\\n <semantics>\\n <mrow>\\n <mi>P</mi>\\n <mo>∈</mo>\\n <mi>W</mi>\\n </mrow>\\n <annotation>$\\\\mathbb {P}\\\\in W$</annotation>\\n </semantics></math> of cardinality <math>\\n <semantics>\\n <mrow>\\n <mo><</mo>\\n <mspace></mspace>\\n <mi>κ</mi>\\n </mrow>\\n <annotation>${\\\\mathord {<}\\\\hspace{1.111pt}\\\\kappa }$</annotation>\\n </semantics></math>. The κ<i>-mantle</i> <math>\\n <semantics>\\n <msub>\\n <mi>M</mi>\\n <mi>κ</mi>\\n </msub>\\n <annotation>$\\\\mathcal {M}_\\\\kappa$</annotation>\\n </semantics></math> is the intersection of all <math>\\n <semantics>\\n <mrow>\\n <mo><</mo>\\n <mspace></mspace>\\n <mi>κ</mi>\\n </mrow>\\n <annotation>${\\\\mathord {<}\\\\hspace{1.111pt}\\\\kappa }$</annotation>\\n </semantics></math>-grounds. We prove that certain partial choice principles in <math>\\n <semantics>\\n <msub>\\n <mi>M</mi>\\n <mi>κ</mi>\\n </msub>\\n <annotation>$\\\\mathcal {M}_\\\\kappa$</annotation>\\n </semantics></math> are the consequence of κ being inaccessible/weakly compact, and some other related facts.</p>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2022-05-07\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202000089\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/malq.202000089\",\"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.202000089","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Assume . Let κ be a cardinal. A -ground
is a transitive proper class W modelling such that V is a generic extension of W via a forcing of cardinality . The κ-mantle is the intersection of all -grounds. We prove that certain partial choice principles in are the consequence of κ being inaccessible/weakly compact, and some other related facts.