Forcing more 𝖣𝖢 over the Chang model using the Thorn sequence

IF 0.8 3区 数学 Q2 MATHEMATICS
James Holland, Grigor Sargsyan
{"title":"Forcing more 𝖣𝖢 over the Chang model using the Thorn sequence","authors":"James Holland, Grigor Sargsyan","doi":"10.1090/proc/16700","DOIUrl":null,"url":null,"abstract":"<p>In the context of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"sans-serif upper Z sans-serif upper F plus sans-serif upper D sans-serif upper C\">\n <mml:semantics>\n <mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"sans-serif\">Z</mml:mi>\n <mml:mi mathvariant=\"sans-serif\">F</mml:mi>\n </mml:mrow>\n <mml:mo>+</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"sans-serif\">D</mml:mi>\n <mml:mi mathvariant=\"sans-serif\">C</mml:mi>\n </mml:mrow>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathsf {ZF}+\\mathsf {DC}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, we force <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"sans-serif upper D sans-serif upper C Subscript kappa\">\n <mml:semantics>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"sans-serif\">D</mml:mi>\n <mml:mi mathvariant=\"sans-serif\">C</mml:mi>\n </mml:mrow>\n <mml:mi>κ</mml:mi>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">\\mathsf {DC}_\\kappa</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> for relations on <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper P left-parenthesis kappa right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">P</mml:mi>\n </mml:mrow>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>κ</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathcal {P}(\\kappa )</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> for arbitrarily large <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"kappa greater-than normal alef Subscript omega\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>κ</mml:mi>\n <mml:mo>></mml:mo>\n <mml:msub>\n <mml:mi mathvariant=\"normal\">ℵ</mml:mi>\n <mml:mi>ω</mml:mi>\n </mml:msub>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\kappa >\\aleph _\\omega</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> over the Chang model <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper L left-parenthesis normal upper O normal r normal d Superscript omega Baseline right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"normal\">L</mml:mi>\n </mml:mrow>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:msup>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"normal\">O</mml:mi>\n <mml:mi mathvariant=\"normal\">r</mml:mi>\n <mml:mi mathvariant=\"normal\">d</mml:mi>\n </mml:mrow>\n <mml:mi>ω</mml:mi>\n </mml:msup>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathrm {L}(\\mathrm {Ord}^\\omega )</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> making some assumptions on the thorn sequence defined by <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"Þ Subscript 0 Baseline equals omega\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo>Þ</mml:mo>\n </mml:mrow>\n <mml:mn>0</mml:mn>\n </mml:msub>\n <mml:mo>=</mml:mo>\n <mml:mi>ω</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">Þ_0=\\omega</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"Þ Subscript alpha plus 1 Baseline\">\n <mml:semantics>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo>Þ</mml:mo>\n </mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>α</mml:mi>\n <mml:mo>+</mml:mo>\n <mml:mn>1</mml:mn>\n </mml:mrow>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">Þ_{\\alpha +1}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> as the least ordinal not a surjective image of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"Þ Subscript alpha Superscript omega Baseline\">\n <mml:semantics>\n <mml:msubsup>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo>Þ</mml:mo>\n </mml:mrow>\n <mml:mi>α</mml:mi>\n <mml:mi>ω</mml:mi>\n </mml:msubsup>\n <mml:annotation encoding=\"application/x-tex\">Þ_\\alpha ^\\omega</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"Þ Subscript gamma Baseline equals sup Underscript alpha greater-than gamma Endscripts Þ\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo>Þ</mml:mo>\n </mml:mrow>\n <mml:mi>γ</mml:mi>\n </mml:msub>\n <mml:mo>=</mml:mo>\n <mml:munder>\n <mml:mo movablelimits=\"true\" form=\"prefix\">sup</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>α</mml:mi>\n <mml:mo>></mml:mo>\n <mml:mi>γ</mml:mi>\n </mml:mrow>\n </mml:munder>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo>Þ</mml:mo>\n </mml:mrow>\n <mml:mi>α</mml:mi>\n </mml:msub>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">Þ_\\gamma =\\sup _{\\alpha >\\gamma }Þ_\\alpha</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> for limit <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"gamma\">\n <mml:semantics>\n <mml:mi>γ</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">\\gamma</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. These assumptions are motivated from results about <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Theta\">\n <mml:semantics>\n <mml:mi mathvariant=\"normal\">Θ</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">\\Theta</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> in the context of determinacy, and could be reasonable ways of thinking about the Chang model. Explicitly, we assume successor points <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"lamda\">\n <mml:semantics>\n <mml:mi>λ</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">\\lambda</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> on the thorn sequence are strongly regular—meaning regular and functions <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f colon kappa Superscript greater-than kappa Baseline right-arrow lamda\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>f</mml:mi>\n <mml:mo>:</mml:mo>\n <mml:msup>\n <mml:mi>κ</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo>></mml:mo>\n <mml:mi>κ</mml:mi>\n </mml:mrow>\n </mml:msup>\n <mml:mo stretchy=\"false\">→</mml:mo>\n <mml:mi>λ</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">f:\\kappa ^{>\\kappa }\\rightarrow \\lambda</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> are bounded whenever <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"kappa greater-than lamda\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>κ</mml:mi>\n <mml:mo>></mml:mo>\n <mml:mi>λ</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\kappa >\\lambda</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is on the thorn sequence—and justified—meaning <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper P left-parenthesis kappa Superscript omega Baseline right-parenthesis intersection normal upper L left-parenthesis normal upper O normal r normal d Superscript omega Baseline right-parenthesis subset-of-or-equal-to normal upper L Subscript lamda Baseline left-parenthesis lamda Superscript omega Baseline comma upper X right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">P</mml:mi>\n </mml:mrow>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:msup>\n <mml:mi>κ</mml:mi>\n <mml:mi>ω</mml:mi>\n </mml:msup>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>∩</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"normal\">L</mml:mi>\n </mml:mrow>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:msup>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"normal\">O</mml:mi>\n <mml:mi mathvariant=\"normal\">r</mml:mi>\n <mml:mi mathvariant=\"normal\">d</mml:mi>\n </mml:mrow>\n <mml:mi>ω</mml:mi>\n </mml:msup>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>⊆</mml:mo>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"normal\">L</mml:mi>\n </mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>λ</mml:mi>\n </mml:mrow>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:msup>\n <mml:mi>λ</mml:mi>\n <mml:mi>ω</mml:mi>\n </mml:msup>\n ","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2024-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the American Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/proc/16700","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

Abstract

In the context of Z F + D C \mathsf {ZF}+\mathsf {DC} , we force D C κ \mathsf {DC}_\kappa for relations on P ( κ ) \mathcal {P}(\kappa ) for arbitrarily large κ > ω \kappa >\aleph _\omega over the Chang model L ( O r d ω ) \mathrm {L}(\mathrm {Ord}^\omega ) making some assumptions on the thorn sequence defined by Þ 0 = ω Þ_0=\omega , Þ α + 1 Þ_{\alpha +1} as the least ordinal not a surjective image of Þ α ω Þ_\alpha ^\omega and Þ γ = sup α > γ Þ α Þ_\gamma =\sup _{\alpha >\gamma }Þ_\alpha for limit γ \gamma . These assumptions are motivated from results about Θ \Theta in the context of determinacy, and could be reasonable ways of thinking about the Chang model. Explicitly, we assume successor points λ \lambda on the thorn sequence are strongly regular—meaning regular and functions f : κ > κ λ f:\kappa ^{>\kappa }\rightarrow \lambda are bounded whenever κ > λ \kappa >\lambda is on the thorn sequence—and justified—meaning P ( κ ω ) L ( O r d ω ) L λ ( λ ω

使用索恩序列在张模型上施加更多的 𝖣𝖢
在 Z F + D C (ZF}+DC} 的背景下,我们强制 D C κ (DC}_\kappa)为 P ( κ ) 上的\mathcal{P}(\kappa)关系。 对于任意大 κ > ℵ ω \kappa >\aleph _\omega 在 Chang 模型 L ( O r d ω ) 上的关系,我们强制 D C κ\mathsf {DC}_\kappa \对由 Þ 0 = ω Þ_0=\omega 定义的刺序列做一些假设、 Þ α + 1 Þ{\alpha +1} 作为不是 Þ α ω Þ_\alpha ^\omega 的射影的最小序数,并且 Þ γ = sup α > γ Þ α Þ_\gamma =\sup _{\alpha >\gamma }Þ_\alpha 对于极限 γ \gamma 。这些假设的动机来自于确定性背景下关于 Θ \ Theta 的结果,也可能是思考 Chang 模型的合理方法。明确地说,我们假定荆棘序列上的λ \lambda 的后继点是强正则的--意思是正则的,并且只要 κ > λ \kappa > \lambda 在荆棘序列上,函数 f : κ > κ → λ f:\kappa ^{>\kappa }\rightarrow \lambda 都是有界的--意思是 P ( κ ω ) ∩ L ( O r d ω ) 是有理的。 ⊆ L λ ( λ ω )
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来源期刊
CiteScore
1.70
自引率
10.00%
发文量
207
审稿时长
2-4 weeks
期刊介绍: All articles submitted to this journal are peer-reviewed. The AMS has a single blind peer-review process in which the reviewers know who the authors of the manuscript are, but the authors do not have access to the information on who the peer reviewers are. This journal is devoted to shorter research articles (not to exceed 15 printed pages) in all areas of pure and applied mathematics. To be published in the Proceedings, a paper must be correct, new, and significant. Further, it must be well written and of interest to a substantial number of mathematicians. Piecemeal results, such as an inconclusive step toward an unproved major theorem or a minor variation on a known result, are in general not acceptable for publication. Longer papers may be submitted to the Transactions of the American Mathematical Society. Published pages are the same size as those generated in the style files provided for AMS-LaTeX.
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