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Kolmogorov was born at Tambov in Russia and educated at Moscow State University, graduating in 1925. He became a research associate at the university, later a professor (1931), and in 1933 was appointed director of the Institute of Mathematics there. He made distinguished contributions to a wide variety of mathematical topics. He is best known for his work on the theoretical foundations of probability, but also made lasting contributions to such diverse subjects as Fourier analysis, automata theory, and intuitionism.

In 1933 Kolmogorov published his major treatise on probability, translated into English in 1950 as The Foundations of the Theory of Probability. The book is a landmark in the development of the theory, for in it he presented the first fully axiomatic treatment of the subject. It also contains the first full realization of the basic and underivable nature of the so-called ‘additivity assumption’ about probability, first put forward by Jakob I Bernoulli. This claims simply that if an event can be realized in any one of an infinite number of mutually exclusive ways, the probability of the event is simply the sum of the probabilities of each of these ways. This assumption is fundamental to the whole measure-theoretic study of probability.

Kolmogorov's interest in Luitzen Brouwer's intuitionism led him to prove that intuitionistic arithmetic, as formalized by Brouwer's disciple Arend Heyting, is consistent if and only if classical arithmetic is. In 1936 Kolmogorov settled a key problem in Fourier analysis when he constructed a function that is (Lebesgue) integrable, but whose Fourier series diverges at every point. In 1939, the same year in which he was elected an academician of the Soviet Academy of Sciences, he published a paper on the extrapolation of time series. This was later taken much further by Norbert Wiener and became known as ‘single-series prediction’.