> Yet among the computable numbers, what our computers can
> represent in any language is but a small subset of them.
This is only true because of physical limitations of the machine (say, it has only finite memory). In the same way not all Turing machines cannot be implemented on actual computers. This is not a restriction of the languages we use. There is nothing stopping an actual programming language representing all computable real numbers.
> (Although computable numbers are equinumerous with the naturals.)
This is only true in classical mathematics. In constructive mathematics we are free to assume that all real numbers are computable (And hence, not equinumerous with the naturals, per cantor’s argument).
> represent in any language is but a small subset of them.
This is only true because of physical limitations of the machine (say, it has only finite memory). In the same way not all Turing machines cannot be implemented on actual computers. This is not a restriction of the languages we use. There is nothing stopping an actual programming language representing all computable real numbers.
> (Although computable numbers are equinumerous with the naturals.)
This is only true in classical mathematics. In constructive mathematics we are free to assume that all real numbers are computable (And hence, not equinumerous with the naturals, per cantor’s argument).