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.