Intuitively, Theorem 1.1 says that if a first order theory has arbitrarily complicated countable models then it has countable models of every sufficiently complicated kind. Cones in Borel orderings. This uses D.A. Martin's cone theorem for Turing degrees and my work on the metamathematics of the cone theorem for Turing degrees and other degrees. Cones with real functions. Let K be a set of unary functions on the reals, R, which is closed under composition.
R is the set of all real numbers.
Furthermore, this result holds if R is replaced by the closed unit interval, the Cantor set, or even any uncountable complete separable metric space.