Bounding 2d functions by products of 1d functions

IF 0.4 4区数学 Q4 LOGIC

Mathematical Logic Quarterly 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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来源期刊

Mathematical Logic Quarterly 数学-数学

CiteScore

0.60

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： Mathematical Logic Quarterly publishes original contributions on mathematical logic and foundations of mathematics and related areas, such as general logic, model theory, recursion theory, set theory, proof theory and constructive mathematics, algebraic logic, nonstandard models, and logical aspects of theoretical computer science.