Bounding 2d functions by products of 1d functions

Pub Date : 2022-03-30 DOI:10.1002/malq.202000008

François Dorais, Dan Hathaway

{"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":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/malq.202000008","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}

引用次数: 2

Abstract

Given sets $X, Y$ and a regular cardinal μ, let $Φ (X, Y, μ)$ be the statement that for any function $f : X \times Y \to μ$ , there are functions $g_{1} : X \to μ$ and $g_{2} : Y \to μ$ such that for all $(x, y) \in X \times Y$ , $f (x, y) \leq \max {g_{1} (x), g_{2} (y)}$ . In $ZFC$ , the statement $Φ (ω_{1}, ω_{1}, ω)$ is false. However, we show the theory $ZF + “the club filter on ω_{1} is normal” + Φ (ω_{1}, ω_{1}, ω)$ (which is implied by $ZF + DC$ + “ $V = L (R)$ ” + “ω₁ is measurable”) implies that for every $α < ω_{1}$ there is a $κ \in (α, ω_{1})$ such that in some inner model, κ is measurable with Mitchell order $\geq α$ .

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二维函数的边界是一维函数的乘积

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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