Omega-consistency and the diamond

Abstract

G is the result of adjoining the schema □ (□qA→A)→□qA to K; the axioms of G* are the theorems of G and the instances of the schema □qA→A and the sole rule of G* is modus ponens. A sentence is ω-provable if it is provable in P(eano) A(rithmetic) by one application of the ω-rule; equivalently, if its negation is ω-inconsistent in PA. Let ω-Bew(x) be the natural formalization of the notion of ω-provability. For any modal sentence A and function ϕ mapping sentence letters to sentences of PA, inductively define A ^ωϕ by: p ^ωϕ = ϕ(p) (p a sentence letter); ⊥^ωϕ= ⊥; (A→B)suωϕ}= (A ^ωϕ→B^ωϕ); and (□qA)^ωϕ= ω-Bew(⌜A ^ωϕ⌝)(⌜S⌝) is the numeral for the Gödel number of the sentence S). Then, applying techniques of Solovay (Israel Journal of Mathematics 25, pp. 287–304), we prove that for every modal sentence A,⊢ _G A iff for all ϕ, ⊢ _PA A ^ωϕ; and for every modal sentence A, ⊢ _G* A iff for all ϕ, A ^ωϕ is true.