Omega-consistency and the diamond

Studia Logica 39 (2-3):237 - 243 (1980)
Abstract
G is the result of adjoining the schema (qAA)qA to K; the axioms of G* are the theorems of G and the instances of the schema qAA and the sole rule of G* is modus ponens. A sentence is -provable if it is provable in P(eano) A(rithmetic) by one application of the -rule; equivalently, if its negation is -inconsistent in PA. Let -Bew(x) be the natural formalization of the notion of -provability. For any modal sentence A and function mapping sentence letters to sentences of PA, inductively define A by: p = (p) (p a sentence letter); = ; (AB)su}= (A B); and (qA)= -Bew(A )(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 , PA A ; and for every modal sentence A, G* A iff for all , A is true.
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    M. H. Lob (1955). Solution of a Problem of Leon Henkin. Journal of Symbolic Logic 20 (2):115-118.
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