Bounding 2d functions by products of 1d functions

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

François Dorais, Dan Hathaway

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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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