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This essential companion volume to CHAITIN s highly successful books The Unknowable and The Limits of Mathematics, also published by Springer, presents the technical core of his theory of program-size complexity, also known as algorithmic information theory. (The two previous volumes are more concerned with applications to meta-mathematics.) LISP is used to present the key algorithms and to enable computer users to interact with the author s proofs and discover for themselves how they work. The LISP code for this book is available at the author s Web site together with a Java applet LISP interpreter: http://www cs.auckland.ac.nz/CDMTCS/chaitin/ait/"No one has looked deeper and farther into the abyss of randomness and its role in mathematics than Greg Chaitin. This book tells you everything he s seen. Don t miss it."John Casti, Santa Fe Institute, Author of "Goedel: A Life of Logic"