{"title":"大心形层次结构的结构反映模式","authors":"Joan Bagaria, Philipp Lücke","doi":"10.1090/tran/9120","DOIUrl":null,"url":null,"abstract":"<p>We unveil new patterns of Structural Reflection in the large-cardinal hierarchy below the first measurable cardinal. Namely, we give two different characterizations of strongly unfoldable and subtle cardinals in terms of a weak form of the principle of Structural Reflection, and also in terms of weak product structural reflection. Our analysis prompts the introduction of the new notion of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C Superscript left-parenthesis n right-parenthesis\"> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding=\"application/x-tex\">C^{(n)}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-strongly unfoldable cardinal for every natural number <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=\"application/x-tex\">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and we show that these cardinals form a natural hierarchy between strong unfoldable and subtle cardinals analogous to the known hierarchies of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C Superscript left-parenthesis n right-parenthesis\"> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding=\"application/x-tex\">C^{(n)}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-extendible and <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Sigma Subscript n\"> <mml:semantics> <mml:msub> <mml:mi mathvariant=\"normal\">Σ<!-- Σ --></mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding=\"application/x-tex\">\\Sigma _n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-strong cardinals. These results show that the relatively low region of the large-cardinal hierarchy comprised between the first strongly unfoldable and the first subtle cardinals is completely analogous to the much higher region between the first strong and the first Woodin cardinals, and also to the much further upper region of the hierarchy ranging between the first supercompact and the first Vopěnka cardinals.</p>","PeriodicalId":23209,"journal":{"name":"Transactions of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.2000,"publicationDate":"2024-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Patterns of structural reflection in the large-cardinal hierarchy\",\"authors\":\"Joan Bagaria, Philipp Lücke\",\"doi\":\"10.1090/tran/9120\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We unveil new patterns of Structural Reflection in the large-cardinal hierarchy below the first measurable cardinal. Namely, we give two different characterizations of strongly unfoldable and subtle cardinals in terms of a weak form of the principle of Structural Reflection, and also in terms of weak product structural reflection. Our analysis prompts the introduction of the new notion of <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper C Superscript left-parenthesis n right-parenthesis\\\"> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=\\\"false\\\">)</mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding=\\\"application/x-tex\\\">C^{(n)}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-strongly unfoldable cardinal for every natural number <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"n\\\"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and we show that these cardinals form a natural hierarchy between strong unfoldable and subtle cardinals analogous to the known hierarchies of <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper C Superscript left-parenthesis n right-parenthesis\\\"> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=\\\"false\\\">)</mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding=\\\"application/x-tex\\\">C^{(n)}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-extendible and <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"normal upper Sigma Subscript n\\\"> <mml:semantics> <mml:msub> <mml:mi mathvariant=\\\"normal\\\">Σ<!-- Σ --></mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\Sigma _n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-strong cardinals. These results show that the relatively low region of the large-cardinal hierarchy comprised between the first strongly unfoldable and the first subtle cardinals is completely analogous to the much higher region between the first strong and the first Woodin cardinals, and also to the much further upper region of the hierarchy ranging between the first supercompact and the first Vopěnka cardinals.</p>\",\"PeriodicalId\":23209,\"journal\":{\"name\":\"Transactions of the American Mathematical Society\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2024-03-09\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Transactions of the American Mathematical Society\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1090/tran/9120\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Transactions of the American Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/tran/9120","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
我们揭示了第一个可测红心以下大红心层次中结构反射的新模式。也就是说，我们根据结构反射原理的弱形式，以及弱积结构反射，给出了强可展开和微妙红心的两种不同特征。我们的分析促使我们为每一个自然数 n n 引入了 C ( n ) C^{(n)} -强可展开红心的新概念，并且我们证明了这些红心在强可展开红心和微妙红心之间形成了一个自然的等级体系，类似于已知的 C ( n ) C^{(n)} -可展开红心和 Σ n \Sigma _n -强红心的等级体系。这些结果表明，大红心层次结构中介于第一个强可展开红心和第一个微妙红心之间的相对较低区域，完全类似于介于第一个强红心和第一个伍丁红心之间的更高区域，也类似于介于第一个超紧密红心和第一个沃潘卡红心之间的更高区域。
Patterns of structural reflection in the large-cardinal hierarchy
We unveil new patterns of Structural Reflection in the large-cardinal hierarchy below the first measurable cardinal. Namely, we give two different characterizations of strongly unfoldable and subtle cardinals in terms of a weak form of the principle of Structural Reflection, and also in terms of weak product structural reflection. Our analysis prompts the introduction of the new notion of C(n)C^{(n)}-strongly unfoldable cardinal for every natural number nn, and we show that these cardinals form a natural hierarchy between strong unfoldable and subtle cardinals analogous to the known hierarchies of C(n)C^{(n)}-extendible and Σn\Sigma _n-strong cardinals. These results show that the relatively low region of the large-cardinal hierarchy comprised between the first strongly unfoldable and the first subtle cardinals is completely analogous to the much higher region between the first strong and the first Woodin cardinals, and also to the much further upper region of the hierarchy ranging between the first supercompact and the first Vopěnka cardinals.
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