Lets say that a real r is finitely definable iff there is a predicate P that is describable by a Finitary formula that is uniquely satisfied by r.
r is finitely definable <-> Exist P for all y. (P(y) <-> y=r)
where of course P(y) is a Finitary formula.
Of course NOT all reals are finitely definable in the above manner. This is an obvious corollary of Cantor's arguments of uncountability of reals.
Also it is obvious that we have only COUNTABLY many finitely definable reals.
Other kinds of reals can be "infinitely" definable, this can be achieved in a language that encounters infinitely long strings of symbols, and many known first order languages are of that sort and they are proven to be consistent and even supportive of a proof system.
However one must understand that when we say that we have countably many finitely definable reals then we are accepting the existence of a bijection between the naturals and the finitely definable reals and that this bijection is itself not finitely definable!This is also a corollary of Cantor's arguments. Also the diagonal on the list of all finitely definable reals IS also non finitely definable real! since it is defined after the bijection between the naturals and the set of all finitely definable reals, and that bijection as said above is not finitely definable.
Finitely definable reals are definitely very interesting kinds of reals, they are superior to those that are non finitely definable of course, but however that doesn't mean that the later ones do not exist, nor does it mean that the later ones cannot be spoken about, we can still speak of those kinds of reals by using formulas that do not uniquely hold of one of them, and still those sentences can illustrate interesting pieces of mathematics that might possibly find some application one day. However it is expected of course that finitely definable reals would be of more importance no doubt and therefore they would have the leading stance among reals.