Synthetic Undecidability and Incompleteness of First-Order Axiom Systems in Coq

Abstract

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.

An Analysis of Tennenbaum's Theorem in Constructive Type Theory

“…To make the paper self-contained, we also give an introduction to the essential features of constructive type theory, synthetic computability, and the type-theoretic specification of first-order logic in Section 2. This is continued in Section 3 by the presentation of the first-order axiomatization of PA as given in previous work [KH21,KH23].…”