[seanhunter]

> We've never used any of them in all history.

You just used them yourself in your previous post to make your argument that computable numbers are dense in incomputable numbers.[1] So presumably that makes you the first person in all of maths history to use them. Congratulations I guess? The other possibility is they get used a lot and we just don’t make a fuss about it because most of the time it’s exactly like you used them - to express an argument where it doesn’t matter whether they are computable or not.

I have no idea why you think they are absurd but as I say it’s an alternative axiomatic perspective. It just makes a huge amount of maths very inconvenient without really yielding (as far as I can see) much of anything.

I don’t have a perspective on your questions about the universe. Non-computable numbers don’t really trouble my world view. We invented the calculus and the language of continuous functions which gave rise to the rigorous construction of the real numbers precisely to better describe the universe, so it seems that they are pretty useful in that context.

[1] Which is interesting right, because this is a trivial argument in the sense that in standard first year analysis you learn that the rationals are dense in the irrationals and rationals are obviously computable so it must be that rationals are dense in the non-computable numbers. But this is a very basic argument that we can only make because we permit the Reals to be complete. It’s not possible to express the argument in the restriction to computable numbers because there is no ambient space outside the computable numbers for them to be dense into.