We mechanise the undecidability of various first-order axiom systems in Coq, employing the synthetic approach to computability underlying the growing Coq Library of Undecidability Proofs. Concretely, we cover both semantic and deductive entailment in fragments of Peano arithmetic (PA) as well as ZF and related finitary set theories, with their undecidability established by many-one reductions from solvability of Diophantine equations, i.e. Hilbert’s tenth problem (H10), and the Post correspondence problem (PCP), respectively. In the synthetic setting based on the computability of all functions definable in a constructive foundation, such as Coq’s type theory, it suffices to define these reductions as meta-level functions with no need for further encoding in a formalised model of computation. The concrete cases of PA and the considered set theories are supplemented by a general synthetic theory of undecidable axiomatisations, focusing on well-known connections to consistency and incompleteness. Specifically, our reductions rely on the existence of standard models, necessitating additional assumptions in the case of full ZF, and all axiomatic extensions still justified by such standard models are shown incomplete. As a by-product of the undecidability of set theories formulated using only membership and no equality symbol, we obtain the undecidability of first-order logic with a single binary relation.

我们在 Coq 中机械化各种一阶公理系统的不可判定性，采用合成方法来计算不断增长的 Coq 不可判定性证明库。具体来说，我们涵盖了 Peano 算术 (PA) 以及 ZF 和相关有限集理论的片段中的语义和演绎蕴涵，它们的不可判定性是由丢番图方程的可解性的多一约简建立的，即希尔伯特第十问题 (H10)，和邮政通信问题（PCP），分别。在基于构造性基础中可定义的所有函数的可计算性的综合设置中，例如 Coq 的类型理论，将这些归约定义为元级函数就足够了，不需要在形式化计算模型中进一步编码。PA 的具体案例和所考虑的集合论由不可判定公理化的一般综合理论补充，重点关注与一致性和不完整性的众所周知的联系。具体来说，我们的归约依赖于标准模型的存在，在完全 ZF 的情况下需要额外的假设，并且所有仍然由此类标准模型证明合理的公理化扩展都显示为不完整。作为仅使用隶属度而不使用相等符号制定的集合论的不可判定性的副产品，我们获得了具有单个二元关系的一阶逻辑的不可判定性。在完全 ZF 的情况下需要额外的假设，并且所有仍然由此类标准模型证明合理的公理化扩展都显示为不完整。作为仅使用隶属度而不使用相等符号制定的集合论的不可判定性的副产品，我们获得了具有单个二元关系的一阶逻辑的不可判定性。在完全 ZF 的情况下需要额外的假设，并且所有仍然由此类标准模型证明合理的公理化扩展都显示为不完整。作为仅使用隶属度而不使用相等符号制定的集合论的不可判定性的副产品，我们获得了具有单个二元关系的一阶逻辑的不可判定性。