For 10 points, name this set containing the algebraic and transcendental numbers, contrasted with the imaginary numbers.
The Lebesgue measure assigns lengths to subsets of these, and these are both separated and formed by Dedekind cuts. Every monotone sequence of these converges if and only if it is bounded, and every sequence of these is Cauchy if and only if it is convergent. These contain no infinite or infinitesimal elements and satisfy the least upper bound property; consequently these form the only complete Archimedean ordered field. Georg Cantor used a diagonal argument to demonstrate the uncountability of these numbers that can be represented by infinite decimal expansions. For 10 points, name this set containing the algebraic and transcendental numbers, contrasted with the imaginary numbers.
The Lebesgue measure assigns lengths to subsets of these, and these are both separated and formed by Dedekind cuts.
real numbers [or reals, or R]