Essential Hereditary Undecidability

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.