Why does the existence or absence of a finite name affect the reality of a number? You can put every real number in a one-to-one correspondence with itself. Or a one-to-one correspondence of f(x).
This is a really interesting thought experiment, and I think in some ways reframes the problem of "exists" (which is nebulous) to "has a description which is finite (or countably infinite, but not uncountable)".
But I think if you could prove a way to countably label an arbitrary real number, you could probably inductively prove that you could do it for any. But you run face-first into the interesting number paradox and/or incompleteness.
This is a really interesting thought experiment, and I think in some ways reframes the problem of "exists" (which is nebulous) to "has a description which is finite (or countably infinite, but not uncountable)".
But I think if you could prove a way to countably label an arbitrary real number, you could probably inductively prove that you could do it for any. But you run face-first into the interesting number paradox and/or incompleteness.
https://en.wikipedia.org/wiki/Interesting_number_paradox