SERGEI N. ARTEMOV & LEV D. BEKLEMISHEV PROVABILITY LOGIC Summary: SERGEI N. ARTEMOV & LEV D. BEKLEMISHEV PROVABILITY LOGIC 1 INTRODUCTION The idea of provability logic seems to originate in a short paper [G¨odel, 1933]. K. G¨odel was motivated by the question of providing Brouwer's intuitionistic logic, as formalized by Heyting, with an adequate semantics. According to Brouwer, intuitionistic truth means provability. Here is a summary from Constructivism in Mathematics ([Troelstra and van Dalen, 1988], p. 4): "A statement is true if we have a proof of it, and false if we can show that the assumption that there is a proof for the statement leads to a contradiction." An axiom system for intuitionistic logic was introduced by Heyting in 1930; its full description may be found in fundamental monographs [Kleene, 1952; Troelstra and van Dalen, 1988]. In 1931­34 A. Heyting and A.N. Kol- mogorov made Brouwer's definition of intuitionistic truth explicit, though informal, by introducing what is now known as the Brouwer­Heyting­Kol- mogorov (BHK) semantics ([Heyting, 1931; Heyting, 1934; Kolmogoroff, 1932]). BHK semantics suggests that a formula is called true if it has a proof. Further, a proof of a compound statement is described in terms of Collections: Computer Technologies and Information Sciences