König's paradox

Also known as the Zermelo– König paradox. There are non-denumerably many real numbers, but only denumerably many of them are finitely definable. Given Zermelo's proof that the reals can be well-ordered, the set of reals that are not finitely definable must have a first member. But this is itself a finite definition of that real. The paradox is similar to those of Richard and Berry, although König himself thought it turned into a proof that the reals cannot be well-ordered.