2024/03/18 by Sergei Artëmov, Artemov, Sergei · 3 citations
Computer Science · #03A05 #03B30 #03F03 #03F07 #03F30 #03F40 #Computability, Logic, AI Algorithms #F.3.0 #F.4.0 #F.4.1 #FOS: Mathematics #I.2.0 #I.2.3 #Logic (math.LO) #Logic, programming, and type systems #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2403.12272
openalex publication_date 2024/03/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For Hilbert, the consistency of a formal theory T is an infinite series of statements "D is free of contradictions" for each derivation D and a consistency proof is i) an operation that, given D, yields a proof that D is free of contradictions, and ii) a proof that (i) works for all inputs D. Hilbert's two-stage approach to proving consistency naturally generalizes to the notion of a finite proof of a series of sentences in a given theory. Such proofs, which we call selector proofs, have already been tacitly employed in mathematics. Selector proofs of consistency, including Hilbert's epsilon substitution method, do not aim at deriving the Gödelian consistency formula Con(T) and are thus not precluded by Gödel's second incompleteness theorem. We give a selector proof of consistency of Peano Arithmetic PA and formalize this proof in PA.