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"