求助PDF
{"title":"使用索恩序列在张模型上施加更多的 𝖣𝖢","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":"{\"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}","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}
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