There is a perfectly good description of the real numbers that is not ugly. Namely, the real numbers are a complete Archimedean ordered field.
To actually construct them, I think using (Cauchy) convergent sequences of rational numbers would be much less ugly than using Dedekind cuts.
Also, the Löwenheim–Skolem theorem only applies to first-order logic, not second-order logic. Why are you constraining me to use only first-order logic? You have to explain that first.
Johnicholas:
I agree with your sentiment, however:
There is a perfectly good description of the real numbers that is not ugly. Namely, the real numbers are a complete Archimedean ordered field.
To actually construct them, I think using (Cauchy) convergent sequences of rational numbers would be much less ugly than using Dedekind cuts.
Also, the Löwenheim–Skolem theorem only applies to first-order logic, not second-order logic. Why are you constraining me to use only first-order logic? You have to explain that first.