The more conventional name for what you are calling "nameable" is "computable": https://en.wikipedia.org/wiki/Computable_number https://en.wikipedia.org/wiki/Co

The more conventional name for what you are calling "nameable" is "computable": https://en.wikipedia.org/wiki/Computable_number https://en.wikipedia.org/wiki/Computable_number The computable numbers are indeed countable, because they are defined by a set of Turing Machines, which are themselves countable. They're an interesting set of numbers, but they're a complicated thing to do work with because of the "holes" in them. For instance, see the section in that article labelled "Use in place of the reals". To see how bad missing things like supremums can be, consider something like this video: https://www.youtube.com/watch?v=vV7ZuouUSfs https://www.youtube.com/watch?v=vV7ZuouUSfs That video is about the difficulty/impossibility of defining calculus over the rational numbers, which is a simpler lens to see the problems that can arise when you're unable to define some simple properties that you may be so used to you don't realize it. Imagining how that can go wrong with a different set of "holes" in the number line is not too much of a strain after that. For all their paradoxes and oddities, and the unlikelihood of them corresponding to anything actually "real" in the real world, the real numbers are popular in math for a reason.