Essential Incompleteness of Arithmetic Verified by Coq

Abstract

Abstract. A constructive proof of the Gödel-Rosser incompleteness theorem [9] has been completed using the Coq proof assistant. Some theory of classical first-order logic over an arbitrary language is formalized.

A formal proof of the independence of the continuum hypothesis

“…Harrison gives a deeply-embedded implementation of firstorder logic in HOL Light [19] and a proof-search style account of the completeness theorem in [20]. Other formalizations of first-order logic can be found in Isabelle/HOL ( [36], [37], [5]) and Coq ( [24], [31]).…”

A Formally Verified Abstract Account of Gödel’s Incompleteness Theorems

“…Proof automation (often powered by fully automatic provers [18,28]), makes complete, fully rigorous proofs feasible. And indeed, researchers have successfully met the challenge of mechanizing IT 1 [15,25,27,35] and recently IT 2 [27]. Besides reassurance, these verification tours de force have brought superior technical insight into the theorems.…”

“…In the realm of mechanical proofs, the earliest substantial development was due to Sieg [36], who used a prover based on TEM (Theory of Elementary Meta-Mathematics) to formalize parts of the proofs of both IT 1 and IT 2 . But the first full proof of IT 1 was achieved by Shankar [35] in the Boyer-Moore prover, followed by Harrison in HOL Light [15] and O'Connor in Coq [25]. IT 2 has only been fully proved recently-by Paulson in Isabelle/HOL [26,27] (who also proved IT 1 ).…”

The First-Order Syntax of Variadic Functions

“…In performing this partial mechanization, we were influenced by R. O'Connor's mechanization of ordinary first-order logic [11].…”