Essential Hereditary Undecidability

TL;DR

This paper explores the concept of essential hereditary undecidability in logical theories, providing foundational results, examples, and examining its interactions with other properties, ultimately showing no minimal such theory exists.

Contribution

It develops the basic theory of essential hereditary undecidability, introduces new concepts like essential tolerance, and demonstrates the non-existence of a minimal essentially hereditarily undecidable theory.

Findings

Constructs examples of essentially hereditarily undecidable theories.

Shows the relationship between essential hereditary undecidability and other properties.

Proves there is no interpretability minimal essentially hereditarily undecidable theory.

Abstract

In this paper we study \emph{essential hereditary undecidability}. Theories with this property are a convenient tool to prove undecidability of other theories. The paper develops the basic facts concerning essentially hereditary undecidability and provides salient examples, like a construction of \ehu\ theories due to Hanf and an example of a rather natural essentially hereditarily undecidable theory strictly below {\sf R}. We discuss the (non-)interaction of essential hereditary undecidability with recursive boolean isomorphism. We develop a reduction relation \emph{essential tolerance}, or, in the converse direction, \emph{lax interpretability} that interacts in a good way with essential hereditary undecidability. We introduce the class of $\Sigma^0_1$-friendly theories and show that $\Sigma^0_1$-friendliness is sufficient but not necessary for essential hereditary undecidability. Finally, we adapt an argument due to Pakhomov, Murwanashyaka and Visser to show that there is no interpretability minimal essentially hereditarilyundecidable theory.