{"title":"当基数确定权力集时:内部模型和Härtig量词逻辑","authors":"Jouko Väänänen, Philip D. Welch","doi":"10.1002/malq.202200030","DOIUrl":null,"url":null,"abstract":"<p>We show that the predicate “<i>x</i> <i>is the power set of</i> <i>y</i>” is <math>\n <semantics>\n <mrow>\n <msub>\n <mi>Σ</mi>\n <mn>1</mn>\n </msub>\n <mrow>\n <mo>(</mo>\n <mo>Card</mo>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$\\Sigma _1(\\operatorname{Card})$</annotation>\n </semantics></math>-definable, if V = L[E] is an extender model constructed from a coherent sequences of extenders, provided that there is no inner model with a Woodin cardinal. Here <math>\n <semantics>\n <mo>Card</mo>\n <annotation>$\\operatorname{Card}$</annotation>\n </semantics></math> is a predicate true of just the infinite cardinals. From this we conclude: the validities of second order logic are reducible to <math>\n <semantics>\n <msub>\n <mi>V</mi>\n <mi>I</mi>\n </msub>\n <annotation>$V_I$</annotation>\n </semantics></math>, the set of validities of the Härtig quantifier logic. Further we show that if no L[E] model has a cardinal strong up to one of its ℵ-fixed points, and <math>\n <semantics>\n <msub>\n <mi>ℓ</mi>\n <mi>I</mi>\n </msub>\n <annotation>$\\ell _{I}$</annotation>\n </semantics></math>, the Löwenheim number of this logic, is less than the least weakly inaccessible δ, then (i) <math>\n <semantics>\n <msub>\n <mi>ℓ</mi>\n <mi>I</mi>\n </msub>\n <annotation>$\\ell _I$</annotation>\n </semantics></math> is a limit of measurable cardinals of K, and (ii) the Weak Covering Lemma holds at δ.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202200030","citationCount":"0","resultStr":"{\"title\":\"When cardinals determine the power set: inner models and Härtig quantifier logic\",\"authors\":\"Jouko Väänänen, Philip D. Welch\",\"doi\":\"10.1002/malq.202200030\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We show that the predicate “<i>x</i> <i>is the power set of</i> <i>y</i>” is <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>Σ</mi>\\n <mn>1</mn>\\n </msub>\\n <mrow>\\n <mo>(</mo>\\n <mo>Card</mo>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation>$\\\\Sigma _1(\\\\operatorname{Card})$</annotation>\\n </semantics></math>-definable, if V = L[E] is an extender model constructed from a coherent sequences of extenders, provided that there is no inner model with a Woodin cardinal. Here <math>\\n <semantics>\\n <mo>Card</mo>\\n <annotation>$\\\\operatorname{Card}$</annotation>\\n </semantics></math> is a predicate true of just the infinite cardinals. From this we conclude: the validities of second order logic are reducible to <math>\\n <semantics>\\n <msub>\\n <mi>V</mi>\\n <mi>I</mi>\\n </msub>\\n <annotation>$V_I$</annotation>\\n </semantics></math>, the set of validities of the Härtig quantifier logic. Further we show that if no L[E] model has a cardinal strong up to one of its ℵ-fixed points, and <math>\\n <semantics>\\n <msub>\\n <mi>ℓ</mi>\\n <mi>I</mi>\\n </msub>\\n <annotation>$\\\\ell _{I}$</annotation>\\n </semantics></math>, the Löwenheim number of this logic, is less than the least weakly inaccessible δ, then (i) <math>\\n <semantics>\\n <msub>\\n <mi>ℓ</mi>\\n <mi>I</mi>\\n </msub>\\n <annotation>$\\\\ell _I$</annotation>\\n </semantics></math> is a limit of measurable cardinals of K, and (ii) the Weak Covering Lemma holds at δ.</p>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2023-09-11\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202200030\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/malq.202200030\",\"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.202200030","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
When cardinals determine the power set: inner models and Härtig quantifier logic
We show that the predicate “xis the power set ofy” is -definable, if V = L[E] is an extender model constructed from a coherent sequences of extenders, provided that there is no inner model with a Woodin cardinal. Here is a predicate true of just the infinite cardinals. From this we conclude: the validities of second order logic are reducible to , the set of validities of the Härtig quantifier logic. Further we show that if no L[E] model has a cardinal strong up to one of its ℵ-fixed points, and , the Löwenheim number of this logic, is less than the least weakly inaccessible δ, then (i) is a limit of measurable cardinals of K, and (ii) the Weak Covering Lemma holds at δ.