Journal of Symbolic Logic

On Strong Provability Predicates and the Associated Modal Logics

Konstantin N. Ignatiev

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PA is Peano Arithmetic. $\mathrm{Pr}(x)$ is the usual $\Sigma_1$-formula representing provability in PA. A strong provability predicate is a formula which has the same properties as $Pr(\cdot)$ but is not $\Sigma_1$. An example: $Q$ is $\omega$-provable if $\mathrm{PA} + \neg Q$ is $\omega$-inconsistent (Boolos [4]). In [5] Dzhaparidze introduced a joint provability logic for iterated $\omega$-provability and obtained its arithmetical completeness. In this paper we prove some further modal properties of Dzhaparidze's logic, e.g., the fixed point property and the Craig interpolation lemma. We also consider other examples of the strong provability predicates and their applications.

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J. Symbolic Logic, Volume 58, Issue 1 (1993), 249-290.

First available in Project Euclid: 6 July 2007

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Ignatiev, Konstantin N. On Strong Provability Predicates and the Associated Modal Logics. J. Symbolic Logic 58 (1993), no. 1, 249--290.

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