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 84½. Kolmogorov was not only one of the greatest mathematicians of the 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) the foundations of mathematics, and (3) the theory of algorithms.

[1]  Vladimir A. Uspensky,et al.  What are the gains of the theory of algorithms: Basis developments connected with the concept of algorithm and with its application in mathematics , 1979, Algorithms in Modern Mathematics and Computer Science.

[2]  A. N. Kolmogorov Combinatorial foundations of information theory and the calculus of probabilities , 1983 .

[3]  A. Heyting Mathematische Grundlagenforschung : Intuitionismus, Beweistheorie , 1935 .

[4]  Per Martin-Löf,et al.  The Definition of Random Sequences , 1966, Inf. Control..

[5]  D. Loveland A New Interpretation of the von Mises' Concept of Random Sequence† , 1966 .

[6]  Michiel van Lambalgen,et al.  Von Mises' Definition of Random Sequences Reconsidered , 1987, J. Symb. Log..

[7]  Ray J. Solomonoff,et al.  A Formal Theory of Inductive Inference. Part I , 1964, Inf. Control..

[8]  Gregory J. Chaitin,et al.  Algorithmic Information Theory , 1987, IBM J. Res. Dev..

[9]  S. C. Kleene,et al.  Introduction to Metamathematics , 1952 .

[10]  A. Sh. Shen Tables of Random Numbers , 1993 .

[11]  K. Jacobs Turing-Maschinen und zufällige 0–1-Folgen , 1970 .

[12]  C. Schnorr A Survey of the Theory of Random Sequences , 1977 .

[13]  A. Shiryayev On Tables of Random Numbers , 1993 .

[14]  A. Kolmogoroff Zur Deutung der intuitionistischen Logik , 1932 .

[15]  Arnold Schönhage Storage Modification Machines , 1980, SIAM J. Comput..

[16]  D. C. Cooper,et al.  Theory of Recursive Functions and Effective Computability , 1969, The Mathematical Gazette.

[17]  Donald Ervin Knuth,et al.  The Art of Computer Programming, Volume II: Seminumerical Algorithms , 1970 .

[18]  D. Hilbert Die logischen Grundlagen der Mathematik , 1922 .

[19]  A. N. Shiryayev On the Notion of Algorithm , 1993 .

[20]  A. Heyting Die formalen Regeln der intuitionistischen Logik , 1930 .

[21]  R. Mises,et al.  Wahrscheinlichkeit, Statistik und Wahrheit. , 1936 .

[22]  A. Kolmogorov,et al.  ALGORITHMS AND RANDOMNESS , 1988 .

[23]  Claus-Peter Schnorr,et al.  Process complexity and effective random tests , 1973 .

[24]  Jaak Peetre,et al.  Preface to The English Translation , 1991 .

[25]  Andrei N. Kolmogorov,et al.  Logical basis for information theory and probability theory , 1968, IEEE Trans. Inf. Theory.

[26]  Paul G. Stecher,et al.  Translation Editor's Preface , 1965 .

[27]  Ming Li,et al.  Kolmogorov Complexity and its Applications , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[28]  A. Slisenko Complexity problems in computational theory , 1981 .

[29]  A. Church On the concept of a random sequence , 1940 .

[30]  Donald E. Knuth,et al.  The art of computer programming. Vol.2: Seminumerical algorithms , 1981 .

[31]  Donald Ervin Knuth,et al.  The Art of Computer Programming , 1968 .

[32]  A. Heyting Die intuitionistische Grundlegung der Mathematik , 1931 .

[33]  Ray J. Solomonoff,et al.  A Formal Theory of Inductive Inference. Part II , 1964, Inf. Control..

[34]  Jr. Hartley Rogers Theory of Recursive Functions and Effective Computability , 1969 .

[35]  Alonzo Church,et al.  Introduction to Mathematical Logic , 1991 .

[36]  A N Kolmogorov,et al.  Letters of A. N. Kolmogorov to A. Heyting , 1988 .

[37]  Claus-Peter Schnorr,et al.  The process complexity and effective random tests. , 1972, STOC.

[38]  J. Heijenoort From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931 , 1967 .

[39]  W. V. Quine,et al.  Der Minimalkalkul, ein Reduzierter Intutionistischer Formalismus. , 1937 .

[40]  Yu. I. Manin,et al.  Course in mathematical logic , 1977, Graduate texts in mathematics.