## Journal of Symbolic Logic

### On Strong Provability Predicates and the Associated Modal Logics

Konstantin N. Ignatiev

#### Abstract

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.

#### Article information

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

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183744189

Mathematical Reviews number (MathSciNet)
MR1217189

Zentralblatt MATH identifier
0795.03082

JSTOR