After all, for any specific size of mantissa and exponent, there will be a finite number of floats of that size, and there are a countable number of options for mantissa and exponent (corresponds to N²), thus the number of floating point values is countable.
Alternatively, each of them corresponds to an arbitrary length bitstring, in addition to a pair of numbers defining the mantissa and exponent, which would put them in bijection with N³ and thus also be countable.
EDIT: to add to that, I believe that one cannot in any meaningful way encode uncomputable values (in the sense that even if one introduces distinguished bitstrings intended to "encode" a specific¹ uncomputable value one can't do anything other than treat is as a distinguished value, and especially one can't perform arithmetic on it or print its digits or similar), so you'll still be limited to a countable set of floating point numbers even with other tricks.
¹ If that term even has meaning when dealing with uncomputable numbers...