Self-embeddings of computable trees

Abstract

We divide the class of infinite computable trees into three types. For the first and second types, 0′ computes a nontrivial self-embedding while for the third type 0″ computes a nontrivial self-embedding. These results are optimal and we obtain partial results concerning the complexity of nontrivial self-embeddings of infinite computable trees considered up to isomorphism. We show that every infinite computable tree must have either an infinite computable chain or an infinite Φ01 antichain. This result is optimal and has connections to the program of reverse mathematics. © 2007 by University of Notre Dame.