Bounding 2D Functions by Products of 1D Functions

Abstract

Given sets $X,Y$ and a regular cardinal $\mu$, let $\Phi (X,Y,\mu )$ be the statement that for any function $f:X\times Y\to \mu$, there are functions $g_1:X\to \mu$ and $g_2:Y\to \mu$ such that or all $(x,y)\in X\times Y$, $$f(x,y)\le \max \{g_1(x),g_2(y)\}.$$ In ZFC, the statement $\Phi ({\omega }_1,{\omega }_1,\omega )$ is false. However, we show the theory ZF + ``the club filter on ${\omega }_1$ is normal'' + $\Phi ({\omega }_1,{\omega }_1,\omega )$ (which is implied by ZF + AD) implies that for every $\alpha <{\omega }_1$ there is a $\kappa \in (\alpha ,{\omega }_1)$ such that in some inner model, $\kappa$ is measurable with Mitchell order $\ge \alpha$. There was an error in Welch's paper ``Characterizing Subsets of ${\omega }_1$ Constructible From a Real'', which he has retracted in a personal communication. Our paper originally referenced that paper. In this version of our paper, we are not using that result. Our consistency strength upper bound has changed accordingly.

17 pages

References in corpus (1)

• One Repeat Point Gives a Closed, Unbounded Ultrafilter on {ω_1}