TY - GEN
T1 - Synthetic undecidability and incompleteness of first-order axiom systems in Coq
AU - Kirst, Dominik
AU - Hermes, Marc
N1 - Publisher Copyright:
© Dominik Kirst and Marc Hermes.
PY - 2021/6/1
Y1 - 2021/6/1
N2 - 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) and Zermelo-Fraenkel set theory (ZF), 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 ZF are prepared 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 ZF formulated using only membership and no equality symbol, we obtain the undecidability of first-order logic with a single binary relation.
AB - 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) and Zermelo-Fraenkel set theory (ZF), 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 ZF are prepared 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 ZF formulated using only membership and no equality symbol, we obtain the undecidability of first-order logic with a single binary relation.
KW - Constructive type theory
KW - Coq
KW - First-order logic
KW - Incompleteness
KW - Peano arithmetic
KW - Synthetic computability
KW - Undecidability
KW - ZF set theory
UR - http://www.scopus.com/inward/record.url?scp=85106403832&partnerID=8YFLogxK
U2 - 10.4230/LIPIcs.ITP.2021.23
DO - 10.4230/LIPIcs.ITP.2021.23
M3 - Conference contribution
AN - SCOPUS:85106403832
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - 12th International Conference on Interactive Theorem Proving, ITP 2021
A2 - Cohen, Liron
A2 - Kaliszyk, Cezary
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 12th International Conference on Interactive Theorem Proving, ITP 2021
Y2 - 29 June 2021 through 1 July 2021
ER -