In this paper we study 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 essentially hereditarily undecidable theories due to Hanf and an example of a rather natural essentially hereditarily undecidable theory strictly below R. We discuss the (non-)interaction of essential hereditary undecidability with recursive boolean isomorphism. We develop a reduction relation essential tolerance, or, in the converse direction, 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 hereditarily undecidable theory.

在本文中，我们研究本质遗传性不可判定性。具有这种性质的理论是证明其他理论的不可判定性的便捷工具。本文阐述了有关本质上遗传性不可判定性的基本事实，并提供了显着的例子，例如 Hanf 提出的本质上遗传性不可判定性理论的构造，以及严格低于R的相当自然的本质上遗传性不可判定性理论的例子。我们讨论基本遗传不可判定性与递归布尔同构的（非）相互作用。我们开发了一种还原关系本质宽容，或者，在相反的方向上，宽松的可解释性与本质的遗传不可判定性以良好的方式相互作用。我们引入\(\Sigma ^0_1\)类友好理论，并表明\(\Sigma ^0_1\) -友好性对于本质的遗传不可判定性来说是足够的，但不是必需的。最后，我们采用 Pakhomov、Murwanashyaka 和 Visser 的论点来表明，不存在可解释性最小的本质上遗传性不可判定的理论。