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 $g : \mathbb{R} \to \mathbb{R}$, $f_a \cap g = \emptyset$ 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 $\Gamma = \mathbf{\Delta}^1_2$, then $a \in L[c]$. Also for all $n \in \omega$, if $\Gamma = \mathbf{\Delta}^1_{3 + n}$, then $a \in \mathcal{M}_{1 + n}(c)$.