Disjoint Borel Functions

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 $a \in \mathcal{M}_{1 + n}(c)$.Comment: 15 page

Disjoint Infinity Borel Functions

Consider the statement that every uncountable set of reals can be surjected onto R by a Borel function. This is implied by the statement that every uncountable set of reals has a perfect subset. It is also implied by a new statement D which we will discuss: for each real a there is a Borel function fa : RtoR and for each function g : RtoR there is a countable set G(g) of reals such that the following is true: for each a in R and for each function g : R to R, if fa is disjoint from g, then a is in G(g). We will show that D follows from ZF +AD+ whereas the negation of D follows from ZFC