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 $\omega$.