On the Cantor–Bendixson rank of a set that is searchable in Gödel’s T

Abstract

We prove that if a closed subset X of the Baire space is searchable in the sense of Escardo (Logical Methods in Computer Science 4(3) (2008), 3), and by a functional definable in Godel's T ,t henX is countable, and the Cantor-Bendixson rank of X is bounded by the ordinal e0. To this end we introduce evaluation trees, well founded decorated trees that induce operators from the full set-theoretical class N N → N to N N. We prove that when a search operator for a set X is induced from an evaluation tree T , then the Cantor-Bendixson rank of X is bounded by the Kleene-Brouwer order type of T. Further we use a theorem due to Howard (Journal of Symbolic Logic 45(3) (1980), 493-504), estimating the ordinal complexity of the reduction tree of a term in system T, to show that all functionals of type (N N → N) → N N definable in T can be computed using an evaluation tree of ordinal rank below e0.