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{"title":"二维函数的边界是一维函数的乘积","authors":"François Dorais, Dan Hathaway","doi":"10.1002/malq.202000008","DOIUrl":null,"url":null,"abstract":"<p>Given sets <math>\n <semantics>\n <mrow>\n <mi>X</mi>\n <mo>,</mo>\n <mi>Y</mi>\n </mrow>\n <annotation>$X,Y$</annotation>\n </semantics></math> and a regular cardinal μ, let <math>\n <semantics>\n <mrow>\n <mi>Φ</mi>\n <mo>(</mo>\n <mi>X</mi>\n <mo>,</mo>\n <mi>Y</mi>\n <mo>,</mo>\n <mi>μ</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$\\Phi (X,Y,\\mu )$</annotation>\n </semantics></math> be the statement that for any function <math>\n <semantics>\n <mrow>\n <mi>f</mi>\n <mo>:</mo>\n <mi>X</mi>\n <mo>×</mo>\n <mi>Y</mi>\n <mo>→</mo>\n <mi>μ</mi>\n </mrow>\n <annotation>$f : X \\times Y \\rightarrow \\mu$</annotation>\n </semantics></math>, there are functions <math>\n <semantics>\n <mrow>\n <msub>\n <mi>g</mi>\n <mn>1</mn>\n </msub>\n <mo>:</mo>\n <mi>X</mi>\n <mo>→</mo>\n <mi>μ</mi>\n </mrow>\n <annotation>$g_1 : X \\rightarrow \\mu$</annotation>\n </semantics></math> and <math>\n <semantics>\n <mrow>\n <msub>\n <mi>g</mi>\n <mn>2</mn>\n </msub>\n <mo>:</mo>\n <mi>Y</mi>\n <mo>→</mo>\n <mi>μ</mi>\n </mrow>\n <annotation>$g_2 : Y \\rightarrow \\mu$</annotation>\n </semantics></math> such that for all <math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <mi>x</mi>\n <mo>,</mo>\n <mi>y</mi>\n <mo>)</mo>\n <mo>∈</mo>\n <mi>X</mi>\n <mo>×</mo>\n <mi>Y</mi>\n </mrow>\n <annotation>$(x,y) \\in X \\times Y$</annotation>\n </semantics></math>, <math>\n <semantics>\n <mrow>\n <mi>f</mi>\n <mrow>\n <mo>(</mo>\n <mi>x</mi>\n <mo>,</mo>\n <mi>y</mi>\n <mo>)</mo>\n </mrow>\n <mo>≤</mo>\n <mi>max</mi>\n <mo>{</mo>\n <msub>\n <mi>g</mi>\n <mn>1</mn>\n </msub>\n <mrow>\n <mo>(</mo>\n <mi>x</mi>\n <mo>)</mo>\n </mrow>\n <mo>,</mo>\n <msub>\n <mi>g</mi>\n <mn>2</mn>\n </msub>\n <mrow>\n <mo>(</mo>\n <mi>y</mi>\n <mo>)</mo>\n </mrow>\n <mo>}</mo>\n </mrow>\n <annotation>$f(x,y) \\le \\max \\lbrace g_1(x), g_2(y) \\rbrace$</annotation>\n </semantics></math>. In <math>\n <semantics>\n <mi>ZFC</mi>\n <annotation>$\\mathsf {ZFC}$</annotation>\n </semantics></math>, the statement <math>\n <semantics>\n <mrow>\n <mi>Φ</mi>\n <mo>(</mo>\n <msub>\n <mi>ω</mi>\n <mn>1</mn>\n </msub>\n <mo>,</mo>\n <msub>\n <mi>ω</mi>\n <mn>1</mn>\n </msub>\n <mo>,</mo>\n <mi>ω</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$\\Phi (\\omega _1, \\omega _1, \\omega )$</annotation>\n </semantics></math> is false. However, we show the theory <math>\n <semantics>\n <mrow>\n <mi>ZF</mi>\n <mo>+</mo>\n <mtext>“the</mtext>\n <mspace></mspace>\n <mtext>club</mtext>\n <mspace></mspace>\n <mtext>filter</mtext>\n <mspace></mspace>\n <mtext>on</mtext>\n <mspace></mspace>\n <msub>\n <mi>ω</mi>\n <mn>1</mn>\n </msub>\n <mspace></mspace>\n <mtext>is</mtext>\n <mspace></mspace>\n <mtext>normal”</mtext>\n <mo>+</mo>\n <mi>Φ</mi>\n <mo>(</mo>\n <msub>\n <mi>ω</mi>\n <mn>1</mn>\n </msub>\n <mo>,</mo>\n <msub>\n <mi>ω</mi>\n <mn>1</mn>\n </msub>\n <mo>,</mo>\n <mi>ω</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$\\mathsf {ZF}+ \\text{``the club filter on $\\omega _1$ is normal''} + \\Phi (\\omega _1, \\omega _1, \\omega )$</annotation>\n </semantics></math> (which is implied by <math>\n <semantics>\n <mrow>\n <mi>ZF</mi>\n <mo>+</mo>\n <mi>DC</mi>\n </mrow>\n <annotation>$\\mathsf {ZF}+ \\mathsf {DC}$</annotation>\n </semantics></math>+ “<math>\n <semantics>\n <mrow>\n <mi>V</mi>\n <mo>=</mo>\n <mi>L</mi>\n <mo>(</mo>\n <mi>R</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$V = L(\\mathbb {R})$</annotation>\n </semantics></math>” + “ω<sub>1</sub> is measurable”) implies that for every <math>\n <semantics>\n <mrow>\n <mi>α</mi>\n <mo><</mo>\n <msub>\n <mi>ω</mi>\n <mn>1</mn>\n </msub>\n </mrow>\n <annotation>$\\alpha < \\omega _1$</annotation>\n </semantics></math> there is a <math>\n <semantics>\n <mrow>\n <mi>κ</mi>\n <mo>∈</mo>\n <mo>(</mo>\n <mi>α</mi>\n <mo>,</mo>\n <msub>\n <mi>ω</mi>\n <mn>1</mn>\n </msub>\n <mo>)</mo>\n </mrow>\n <annotation>$\\kappa \\in (\\alpha ,\\omega _1)$</annotation>\n </semantics></math> such that in some inner model, κ is measurable with Mitchell order <math>\n <semantics>\n <mrow>\n <mo>≥</mo>\n <mi>α</mi>\n </mrow>\n <annotation>$\\ge \\alpha$</annotation>\n </semantics></math>.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"68 2","pages":"202-212"},"PeriodicalIF":0.4000,"publicationDate":"2022-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"Bounding 2d functions by products of 1d functions\",\"authors\":\"François Dorais, Dan Hathaway\",\"doi\":\"10.1002/malq.202000008\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Given sets <math>\\n <semantics>\\n <mrow>\\n <mi>X</mi>\\n <mo>,</mo>\\n <mi>Y</mi>\\n </mrow>\\n <annotation>$X,Y$</annotation>\\n </semantics></math> and a regular cardinal μ, let <math>\\n <semantics>\\n <mrow>\\n <mi>Φ</mi>\\n <mo>(</mo>\\n <mi>X</mi>\\n <mo>,</mo>\\n <mi>Y</mi>\\n <mo>,</mo>\\n <mi>μ</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$\\\\Phi (X,Y,\\\\mu )$</annotation>\\n </semantics></math> be the statement that for any function <math>\\n <semantics>\\n <mrow>\\n <mi>f</mi>\\n <mo>:</mo>\\n <mi>X</mi>\\n <mo>×</mo>\\n <mi>Y</mi>\\n <mo>→</mo>\\n <mi>μ</mi>\\n </mrow>\\n <annotation>$f : X \\\\times Y \\\\rightarrow \\\\mu$</annotation>\\n </semantics></math>, there are functions <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>g</mi>\\n <mn>1</mn>\\n </msub>\\n <mo>:</mo>\\n <mi>X</mi>\\n <mo>→</mo>\\n <mi>μ</mi>\\n </mrow>\\n <annotation>$g_1 : X \\\\rightarrow \\\\mu$</annotation>\\n </semantics></math> and <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>g</mi>\\n <mn>2</mn>\\n </msub>\\n <mo>:</mo>\\n <mi>Y</mi>\\n <mo>→</mo>\\n <mi>μ</mi>\\n </mrow>\\n <annotation>$g_2 : Y \\\\rightarrow \\\\mu$</annotation>\\n </semantics></math> such that for all <math>\\n <semantics>\\n <mrow>\\n <mo>(</mo>\\n <mi>x</mi>\\n <mo>,</mo>\\n <mi>y</mi>\\n <mo>)</mo>\\n <mo>∈</mo>\\n <mi>X</mi>\\n <mo>×</mo>\\n <mi>Y</mi>\\n </mrow>\\n <annotation>$(x,y) \\\\in X \\\\times Y$</annotation>\\n </semantics></math>, <math>\\n <semantics>\\n <mrow>\\n <mi>f</mi>\\n <mrow>\\n <mo>(</mo>\\n <mi>x</mi>\\n <mo>,</mo>\\n <mi>y</mi>\\n <mo>)</mo>\\n </mrow>\\n <mo>≤</mo>\\n <mi>max</mi>\\n <mo>{</mo>\\n <msub>\\n <mi>g</mi>\\n <mn>1</mn>\\n </msub>\\n <mrow>\\n <mo>(</mo>\\n <mi>x</mi>\\n <mo>)</mo>\\n </mrow>\\n <mo>,</mo>\\n <msub>\\n <mi>g</mi>\\n <mn>2</mn>\\n </msub>\\n <mrow>\\n <mo>(</mo>\\n <mi>y</mi>\\n <mo>)</mo>\\n </mrow>\\n <mo>}</mo>\\n </mrow>\\n <annotation>$f(x,y) \\\\le \\\\max \\\\lbrace g_1(x), g_2(y) \\\\rbrace$</annotation>\\n </semantics></math>. In <math>\\n <semantics>\\n <mi>ZFC</mi>\\n <annotation>$\\\\mathsf {ZFC}$</annotation>\\n </semantics></math>, the statement <math>\\n <semantics>\\n <mrow>\\n <mi>Φ</mi>\\n <mo>(</mo>\\n <msub>\\n <mi>ω</mi>\\n <mn>1</mn>\\n </msub>\\n <mo>,</mo>\\n <msub>\\n <mi>ω</mi>\\n <mn>1</mn>\\n </msub>\\n <mo>,</mo>\\n <mi>ω</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$\\\\Phi (\\\\omega _1, \\\\omega _1, \\\\omega )$</annotation>\\n </semantics></math> is false. However, we show the theory <math>\\n <semantics>\\n <mrow>\\n <mi>ZF</mi>\\n <mo>+</mo>\\n <mtext>“the</mtext>\\n <mspace></mspace>\\n <mtext>club</mtext>\\n <mspace></mspace>\\n <mtext>filter</mtext>\\n <mspace></mspace>\\n <mtext>on</mtext>\\n <mspace></mspace>\\n <msub>\\n <mi>ω</mi>\\n <mn>1</mn>\\n </msub>\\n <mspace></mspace>\\n <mtext>is</mtext>\\n <mspace></mspace>\\n <mtext>normal”</mtext>\\n <mo>+</mo>\\n <mi>Φ</mi>\\n <mo>(</mo>\\n <msub>\\n <mi>ω</mi>\\n <mn>1</mn>\\n </msub>\\n <mo>,</mo>\\n <msub>\\n <mi>ω</mi>\\n <mn>1</mn>\\n </msub>\\n <mo>,</mo>\\n <mi>ω</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$\\\\mathsf {ZF}+ \\\\text{``the club filter on $\\\\omega _1$ is normal''} + \\\\Phi (\\\\omega _1, \\\\omega _1, \\\\omega )$</annotation>\\n </semantics></math> (which is implied by <math>\\n <semantics>\\n <mrow>\\n <mi>ZF</mi>\\n <mo>+</mo>\\n <mi>DC</mi>\\n </mrow>\\n <annotation>$\\\\mathsf {ZF}+ \\\\mathsf {DC}$</annotation>\\n </semantics></math>+ “<math>\\n <semantics>\\n <mrow>\\n <mi>V</mi>\\n <mo>=</mo>\\n <mi>L</mi>\\n <mo>(</mo>\\n <mi>R</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$V = L(\\\\mathbb {R})$</annotation>\\n </semantics></math>” + “ω<sub>1</sub> is measurable”) implies that for every <math>\\n <semantics>\\n <mrow>\\n <mi>α</mi>\\n <mo><</mo>\\n <msub>\\n <mi>ω</mi>\\n <mn>1</mn>\\n </msub>\\n </mrow>\\n <annotation>$\\\\alpha < \\\\omega _1$</annotation>\\n </semantics></math> there is a <math>\\n <semantics>\\n <mrow>\\n <mi>κ</mi>\\n <mo>∈</mo>\\n <mo>(</mo>\\n <mi>α</mi>\\n <mo>,</mo>\\n <msub>\\n <mi>ω</mi>\\n <mn>1</mn>\\n </msub>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$\\\\kappa \\\\in (\\\\alpha ,\\\\omega _1)$</annotation>\\n </semantics></math> such that in some inner model, κ is measurable with Mitchell order <math>\\n <semantics>\\n <mrow>\\n <mo>≥</mo>\\n <mi>α</mi>\\n </mrow>\\n <annotation>$\\\\ge \\\\alpha$</annotation>\\n </semantics></math>.</p>\",\"PeriodicalId\":49864,\"journal\":{\"name\":\"Mathematical Logic Quarterly\",\"volume\":\"68 2\",\"pages\":\"202-212\"},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2022-03-30\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematical Logic Quarterly\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/malq.202000008\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"LOGIC\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Logic Quarterly","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/malq.202000008","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"LOGIC","Score":null,"Total":0}
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摘要
给定集合X, Y $X,Y$和正则基数μ，令Φ (X, Y，μ) $\Phi (X,Y,\mu )$对于任意函数f:X × Y→μ $f : X \times Y \rightarrow \mu$，有函数g1:X→μ $g_1 : X \rightarrow \mu$和g2:Y→μ $g_2 : Y \rightarrow \mu$使得对于所有(x, Y)∈x × Y $(x,y) \in X \times Y$，F (x, y)≤{Max g1 (x)，g2 (y)}$f(x,y) \le \max \lbrace g_1(x), g_2(y) \rbrace$。在ZFC $\mathsf {ZFC}$中，语句Φ (ω 1， ω 1， ω) $\Phi (\omega _1, \omega _1, \omega )$为假。然而，我们展示了理论ZF +“ω 1上的俱乐部滤波器是正常的”+ Φ (ω 1 ω 1，ω) $\mathsf {ZF}+ \text{``the club filter on $\omega ＿1 $ is normal''} + \Phi (\omega _1, \omega _1, \omega )$(由ZF + DC $\mathsf {ZF}+ \mathsf {DC}$ +“V = L (R) $V = L(\mathbb {R})$隐含“+”ω1是可测量的”)意味着对于每一个α ＆lt;ω 1 $\alpha < \omega _1$存在一个κ∈(α， ω 1) $\kappa \in (\alpha ,\omega _1)$，使得在某个内部模型中，κ可测，Mitchell阶≥α $\ge \alpha$。
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