Applying Generic Coding with Help to Uniformizations

TL;DR

The paper investigates the disjointness properties of certain definable functions from Baire class one, establishing consistency results and implications under large cardinal assumptions and determinacy axioms.

Contribution

It introduces the analysis of disjointness relations for definable functions and connects these properties to large cardinal hypotheses and determinacy axioms.

Findings

Consistency strength of the disjointness property is at most one inaccessible cardinal.

Under AD^+, the property holds for all functions.

Assuming large cardinals, the property holds in models with a selective ultrafilter.

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 $\omega$.