Lower and upper bounds for the provability of Herbrand consistency in weak arithmetics
We prove that for i ≥ 1, the arithmetic does not prove a variant of its own Herbrand consistency restricted to the terms of depth in , where ε is an arbitrarily small constant greater than zero. On the other hand, the provability holds for the set of terms of depths in .
On rational solution of the state equation of a finite automaton.
Sequent Calculus, Derivability, Provability. Gödel's Completeness Theorem
Fifth of a series of articles laying down the bases for classical first order model theory. This paper presents multiple themes: first it introduces sequents, rules and sets of rules for a first order language L as L-dependent types. Then defines derivability and provability according to a set of rules, and gives several technical lemmas binding all those concepts. Following that, it introduces a fixed set D of derivation rules, and proceeds to convert them to Mizar functorial cluster registrations...
Subalgebras of diagonalizable algebras of theories containing arithmetic [Book]
The Gödel Completeness Theorem for Uncountable Languages
This article is the second in a series of two Mizar articles constituting a formal proof of the Gödel Completeness theorem [15] for uncountably large languages. We follow the proof given in [16]. The present article contains the techniques required to expand a theory such that the expanded theory contains witnesses and is negation faithful. Then the completeness theorem follows immediately.
Transition of Consistency and Satisfiability under Language Extensions
This article is the first in a series of two Mizar articles constituting a formal proof of the Gödel Completeness theorem [17] for uncountably large languages. We follow the proof given in [18]. The present article contains the techniques required to expand formal languages. We prove that consistent or satisfiable theories retain these properties under changes to the language they are formulated in.
Континуум нормальных расширений модальной логики доказуемости с интерполяционным свойством.
Неразрешимые свойства расширений логики доказуемости. II
О двух расширениях логики доказуемости GL