Kolmogorov and Mathematical Logic

There are human beings whose intellectual power exceeds that of ordinary men. In my life, in my personal experience, there were three such men, and one of them was Andrei Nikolaevich Kolmogorov. I was lucky enough to be his immediate pupil. He invited me to be his pupil at the third year of my being student at the Moscow University. This talk is my tribute, my homage to my great teacher. Andrei Nikolaevich Kolmogorov was born on April 25, 1903. He graduated from Moscow University in 1925, finished his post-graduate education at the same University in 1929, and since then without any interruption worked at Moscow University till his death on October 20, 1987, at the age 84-. Kolmogorov was not only one of the greatest mathematicians ofthe twentieth century. By the width of his scientific interests and results he reminds one of the titans of the Renaissance. Indeed, he made prominent contributions to various fields from the theory of shooting to the theory of versification, from hydrodynamics to set theory. In this talk I should like to expound his contributions to mathematical logic. Here the term "mathematical logic" is understood in a broad sense. In this sense it, like Gallia in Caesarian times, is divided into three parts: (1) mathematical logic in the strict sense, i.e. the theory of formalized languages including deduction theory,2 (2) the foundations of mathematics, and (3) the theory of algorithms.