I understand that a set whose members can, in principle, be enumerated (by having a formula) can be considered as a well-defined set. Therefore, set of all even numbers, multiples of 3, and so on constitute what a well-defined set, for its members are well-defined. I personally also refer to this as an identified set. Here I am saying that it makes "sense" to identify this set i.e. to assert its definite existence (in set theory) because there is a formal method to enumerate/identify its members.
Now, what about the reals, ℝ? There is no method to enumerate its members (Cantor's diagonalization). Why is it not contradictory/absurd to speak of set R if one cannot formally identify it? We know what its members are integers, rational, irrational numbers...but relying on what we mutually understand appears more like handwaving. To support its formal existence, what concrete definitions have been provided by mathematicians/logicians?