Here's a variant which is slightly different, and perhaps stronger since it also allows some operations with "infinite accuracy".

In order to decide who won the bet, we need a referee. A natural choice is to say that the referee is a Blum-Shub-Smale machine, i.e. a program that gets a single real number x∈[0,1] as input, and whose operations are: loading real number constants; (exact) addition, substraction, multiplication and division; and branching on whether a≤b (exactly).

Say you win if the machine accepts x in a finite number of steps. Now, I think it's always the case that the set of numbers which are accepted after n steps is a finite union of (closed or open) intervals. So then the set of numbers that get accepted after any finite number of steps is a countable union of finite unions of intervals, hence Borel.