> I believe real numbers to be completely natural,

Most of real numbers are not even computable. Doesn't that give you a pause?

Why would we expect most real numbers to be computable? It's an idealized continuum. It makes perfect sense that there are way too many points in it for us to be able to compute them all.

	▲	egorelik 4 minutes ago \| parent [-]
		The idea is we can't actually prove a non-computable real number exists without purposefully having axioms that allow for deriving non-computable things. (We can't prove they don't exist either, without making some strong assumptions).