Self-embeddings of computable trees

TL;DR

This paper classifies infinite computable trees into three types based on the complexity of their nontrivial self-embeddings, establishing optimal results and exploring their complexity and connections to reverse mathematics.

Contribution

It provides a classification of computable trees by the complexity of their self-embeddings and establishes optimal bounds for these complexities.

Findings

For the first two types, 0' computes a nontrivial self-embedding.

For the third type, 0'' computes a nontrivial self-embedding.

Every infinite computable tree has either an infinite computable chain or an infinite Pi^0_1 antichain.

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 $\Pi^0_1$ antichain. This result is optimal and has connections to the program of reverse mathematics.