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# Omega-Consistency and the Diamond

George Boolos
Studia Logica: An International Journal for Symbolic Logic
Vol. 39, No. 2/3 (1980), pp. 237-243
Stable URL: http://www.jstor.org/stable/20014983
Page Count: 7
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## Abstract

G is the result of adjoining the schema □ (□ A → A) → □ A to K; the axioms of $G^{\ast}$ are the theorems of G and the instances of the schema □ A → A and the sole rule of $G^{\ast}$ 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 $A^{\omega \phi}$ by: $p^{\omega \phi}=\phi (p)$ (p a sentence letter); $\perp ^{\omega \phi}=\perp$; $(A\rightarrow B)^{\omega \phi}=(A^{\omega \phi}\rightarrow B^{\omega \phi})$; and $(\square A)^{\omega \phi}$ = ω-Bew(˹$A^{\omega \phi}$˺) (˹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, $\vdash _{G}A$ iff for all φ, $\vdash _{PA}A^{\omega \phi}$; and for every modal sentence A, $\vdash _{G^{\ast}}A$ iff for all φ, $A^{\omega \phi}$ is true.

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