Abstract
We give a nontechnical account of the mathematical theory of randomness. The theory of randomness is founded on computability theory, and it is nowadays often referred to as algorithmic randomness. It comes in two varieties: A theory of finite objects, that emerged in the 1960s through the work of Solomonoff, Kolmogorov, Chaitin and others, and a theory of infinite objects (starting with von Mises in the early 20th century, culminating in the notions introduced by Martin-Lof and Schnorr in the 1960s and 1970s) and there are many deep and beautiful connections between the two. Research in algorithmic randomness connects computability and complexity theory with mathematical logic, proof theory, probability and measure theory, analysis, computer science, and philosophy. It also has surprising applications in a variety of fields, including biology, physics, and linguistics. Founded on the theory of computation, the study of randomness has itself profoundly influenced computability theory in recent years.
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