For 10 points, name these numbers, which include both rationals and irrationals.
An estimate for certain functions on collections of these is given by Ingham's theorem. Richard's paradox arose out of attempts to define these mathematical entities in natural language. Abraham Robinson's theory extended these by creating a new system where each item consisted of one of these and a halo. These make up the only complete ordered field. One method for defining these uses Dedekind cuts. The question of whether there was another set smaller than it is the subject of the Continuum hypothesis and the size of this set was shown to be greater than one of its subsets by Cantor's diagonalization argument. For 10 points, name these numbers, which include both rationals and irrationals.
An estimate for certain functions on collections of these is given by Ingham's theorem.
Real numbers [prompt on Dirichlet series before end of "Abraham Robinson\'s theory extended these"]