> If it is sufficiently impossible to describe the number, how did it get in the set.

It got added along with infinitely many others!

That's the thing, it's no problem dealing with "hairy" reals as long as you deal with infinitely many of them at a time. You can say, "all the numbers between zero and one", and the fact that it will include numbers whose decimal expansion can't be described by any rule shouldn't cause any trouble.

But if I ask you, "show me one of these numbers whose decimal expansion can't be described by any rule", then by definition, you can't.

> You can’t say you can’t chose an item from a set because you can’t describe the item.

Or rather: the whole point of the axiom is giving you such a description.

How did God create universe?

Does your (non) answer imply that Universe doesn't exist?