Here we are considering subsets $\mathcal{F}$ of $2^\omega$, which are in correspondence with families of subsets of $\omega$ (sets of "reals"). Such a family is *Borel* if it is a Borel subset of $2^\omega$ under the usual topology.


Such a family is *almost disjoint* if, for every pair $X\not=Y$ from $\mathcal{F}$, $X\cap Y$ is finite.


*Note:* the question originally only required that the symmetric difference between $X$ and $Y$ be infinite, which is substantially weaker.


Countable almost disjoint families can be constructed fairly trivially. Uncountable almost disjoint families exist, and are a standard object of study in some branches of set theory. However, the constructions I've seen do not result in a Borel set. Can this be done? Can $\mathcal{F}\subset 2^\omega$ be an uncountable Borel set which is an almost disjoint family?



Is there an uncountable Borel almost disjoint family?
Firstly, as commented by Asaf Karagila, it is routine to build a Borel uncountable family of almost disjoint subsets of $\omega$: fix a one-to-one map $f$ from the set $2^{<\omega}$ of finite binary sequences to $\omega$. It is clear that if $B$ and $B'$ are any two distinct branches of $2^{<\omega}$, then $f(B)$ and $f(C)$ are almost disjoint. This makes it evident that the collection of all subsets of $\omega$ of the form $f(B)$, where $B$ is a branch of $2^{<\omega}$, is an uncountable almost disjoint family. I will leave the verification that it is also Borel as an exercise.



> 
> On the other hand, there is no Borel *mad* family, where *mad* stands for *maximal almost disjoint.* This follows from a theorem of Mathias, who showed that no mad family can even be analytic (analytic = $\Sigma^1\_1$ = continuous image of a Borel set). 
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However, by a joint result of Miller and Kunen, it is consistent with $ZFC$ that there is a co-analytic (i.e., $\Pi^1\_1$) mad family, they showed this by building a co-analytic mad family assuming $ZF + V =L$. See [Arnie Miller's wonderful paper](http://www.math.wisc.edu/~miller/res/infcomb.pdf) for this and other gems exploring definability issues in infinite combinatorics (mad families are treated in sec.8).


