The uncomputable real numbers always seemed strange to me. I can understand a convergent sequence of rationals, or the idea of a program that outputs a number to arbitrary precision, but something that cannot be computed at all is a very bizarre object. I think NJ Wildberger has some interesting ideas in this area, although I’m not sure I agree with his finititist interpretation in all circumstances. Specifically I don’t think comparisons to the number of atoms in the universe or information theoretic limits on storage based on the volume of the observable universe are interesting considerations here.

To me at least, if you can write down a finite procedure that can produce a number to arbitrary precision, I think it is fair to say the number at that limit exists.

This made me think of a possible numerical library where rather than storing numbers as arbitrary precision rationals, you could store them as the combination of inputs and functions that generate that number, and compute values to arbitrary precision.