PA is Peano Arithmetic. Pr(x) is the usual Σ1-formula representing provability in PA. A strong provability predicate is a formula which has the same properties as Pr(·) but is not Σ1. An example: Q is ω-provable if PA + ¬ Q is ω-inconsistent (Boolos [4]). In [5] Dzhaparidze introduced a joint provability logic for iterated ω-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. (shrink)

The author elucidates the ontological basis of elementary mathematical theories and thereby assesses their certainty as a foundation for the more complex theories of modern mathematics, such as mathematical analysis and set theory. He adduces arguments in favor of the position of Frege, who held that geometry can provide a sufficiently broad and certain foundation for mathematics.

We say that two arithmetical formulas A, B have the Σ1-interpolation property if they have an ‘interpolant’ σ, i.e., a Σ1 formula such that the formulas A→σ and σ→B are provable in Peano Arithmetic PA. The Σ1-interpolability predicate is just a formalization of this property in the language of arithmetic.Using a standard idea of Gödel, we can associate with this predicate its provability logic, which is the set of all formulas that express arithmetically valid principles in the modal language with (...) two modal operators, □ for provability in PA and for Σ1-interpolability. In this paper we determine this provability logic .It turns out that this problem appears to be quite similar to the problem of such modal description of relative interpretability, a problem, solved by Berarducci and Shavrukov. However, the present case is sometimes easier. In particular, we prove in this paper the fixed point property and the Craig interpolation property for our modal system ELH. (shrink)

t. 1. Problemy vospitanii︠a︡ vsestoronne razvitoĭ lichnosti. Dukhovnyĭ mir shkolʹnika. Metodika vospitanii︠a︡ kollektiva.--t. 2. Formirovanie Kommunisticheskikh ubezhdeniĭ molodogo pokolenii︠a︡. Kak vospitat nastoi︠a︡shchego cheloveka. Sto sovetov uchiteli︠u︡.--t. 3. Serdt︠s︡e otdai︠u︡ deti︠a︡m. Rozhdenie grazhdanina. Pisʹma k synu.--t. 4. Pavlyshskai︠a︡ sredni︠a︡i︠a︡ shkola. Razgovor s molodym direktorom shkoly.--t. 5. Statʹi.

t. 1. Serdt︠s︡e otdai︠u︡ deti︠a︡m. Rozhdenie grazhdanina.--t. 2. Pavlyshskai︠a︡ sredni︠a︡i︠a︡ shkola.

t. 1. Problemy vykhovanni︠a︡ vsebichno rozvynenoï osobystosti. Dukhovnyĭ svit shkoli︠a︡ra. Metodika vykhovanni︠a︡ kolektyvu -- t. 2. Formuvanni︠a︡ komunistychnykh perekonanʹ molodoho pokolinni︠a︡. I︠A︡k vykhovaty spravz︠h︡ni︠u︡ li︠u︡dynu. Sto porad uchytelevi -- t. 3. Sert︠s︡e viddai︠u︡ diti︠a︡m. Narodz︠h︡enni︠a︡ hromadi︠a︡nyna. Lysty do syna -- t. 4. Pavlysʹka seredni︠a︡ shkola. Rozmova z molodym dyrektorom -- t. 5. Statti.