Disjoint Borel Functions

TL;DR

This paper constructs Borel functions encoding real numbers and explores their intersections with other Borel functions, revealing deep connections with descriptive set theory and inner model theory.

Contribution

It introduces a family of Borel functions encoding reals and characterizes their intersection properties with functions in larger pointclasses, linking to inner models and definability.

Findings

If $f_a$ and $g$ are disjoint, then $a$ is in $ ext{Δ}^1_1(c)$.

For $g$ in $ ext{Δ}^1_2$, then $a$ is in $L[c]$.

For higher pointclasses, $a$ belongs to corresponding inner models $ ext{M}_{1+n}(c)$.

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 $a \in \mathcal{M}_{1 + n}(c)$.