Disjoint Borel Functions

Abstract

For each $a\in \mathbb{R}$, we define a Borel function $f_a:\mathbb{R}\to \mathbb{R}$ which encodes $a$ in a certain sense. We show that for each Borel $g:\mathbb{R}\to \mathbb{R}$, $f_a\cap g=\varnothing$ implies $a\in {\Delta }_1^1(c)$ where $c$ is any code for $g$. We generalize this theorem for $g$ in larger pointclasses $\Gamma$. Specifically, if $\Gamma ={\mathbf{\Delta }}_2^1$, then $a\in L[c]$. Also for all $n\in \omega$, if $\Gamma ={\mathbf{\Delta }}_{3+n}^1$, then $a\in {\mathcal{M}}_{1+n}(c)$.

arXiv comment15 pages