Applying Generic Coding with Help to Uniformizations

Abstract

This is a follow up to a paper by the author where the disjointness relation for (the graphs of) definable functions from ${}^{\omega }\omega$ to ${}^{\omega }\omega$ is analyzed. In that paper, for each $a\in {}^{\omega }\omega$ we defined a Baire class one function $f_a^{GC}:{}^{\omega }\omega \to {}^{\omega }\omega$ which encoded $a$ in a certain sense. Given $g:{}^{\omega }\omega \to {}^{\omega }\omega$, let $\Psi (g)$ be the statement that $g$ is disjoint from at most countably many of the functions $f_a^{GC}$. We show the consistency strength of $(\forall g)\Psi (g)$ is at most one inaccessible cardinal. We show that $\mbox{AD}^+$ implies $(\forall g)\Psi (g)$. Finally, we show that assuming large cardinals, $(\forall g)\Psi (g)$ holds in models of the form $L(\mathbb{R}[\mathcal{U}]$ where $\mathcal{U}$ is a selective ultrafilter on $\omega$.

arXiv comment27 pages