[pherq]

Assuming "floating point" refers to things fitting the IEEE754 spec at some precision, I'm pretty sure there are still only countably many of them.

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...

[stcredzero]

Assuming "floating point" refers to things fitting the IEEE754 spec at some precision

Not my assumption!