Verifier Theory from Axioms to Unverifiability of Mathematical Proofs, Software and AI

Despite significant developments in Proof Theory, surprisingly little attention has been devoted to the concept of proof verifier. In particular, mathematical community may be interested in studying different types of proof verifiers (people, programs, oracles, communities, superintelligences, etc.) as mathematical objects, their properties, their powers and limitations (particularly in human mathematicians), minimum and maximum complexity, as well as self-verification and self-reference issues in verifiers. We propose an initial classification system for verifiers and provide some rudimentary analysis of solved and open problems in this important domain. Our main contribution is a formal introduction of the notion of unverifiability, for which the paper could serve as a general citation in domains of theorem proving, software and AI verification.