Mutual Interpretability of Weak Essentially Undecidable Theories

TL;DR

This paper demonstrates mutual interpretability between a theory of binary trees and weak arithmetic theories, establishing deep connections among various weak undecidable theories of numbers, strings, and sets.

Contribution

It proves that the elementary theory of full binary trees is mutually interpretable with several weak undecidable theories, including Robinson arithmetic Q and others.

Findings

Mutual interpretability of theory T and Robinson arithmetic Q.

Introduction of a hybrid theory of strings and trees.

Establishment of interpretability among multiple weak undecidable theories.

Abstract

Kristiansen and Murwanashyaka recently proved that Robinson arithmetic Q is interpretable in an elementary theory of full binary trees, T. We prove that, conversely, T is interpretable in Q by producing a formal interpretation of T in an elementary concatenation theory, thereby also establishing mutual interpretability of T with several well-known weak essentially undecidable theories of numbers, strings and sets. We also in introduce a "hybrid" elementary theory of strings and trees and establish its mutual interpretability with Robinson's weak arithmetic R, the weak theory of binary trees WT of Kristiansen and Murwanashyaka and a weak concatenation theory of Higuchi and Hirohata.