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A Gödel machine solves general computational problems in a possibly stochastic and reactive environment. Its initial software includes an axiomatic description of (1) the Gödel machine's hardware, (2) known aspects of the environment, (3) goals and rewards to be achieved, (4) costs of actions and computations, (5) the initial software itself (no circularity involved here). It also includes a possibly sub-optimal initial problem-solving policy and a proof searcher searching the space of computable proof techniques--that is, programs whose outputs are proofs. Unlike previous approaches, the self-referential Gödel machine will rewrite any part of its software (including axioms and proof searcher) as soon as it has found a proof that this will improve its future performance. By definition, it produces optimal self-improvements, given arbitrary formalized problems and typically limited computational resources; its optimality notion is not restricted to the concept of asymptotic optimality. To initialize the proof searcher we may use the recent Optimal Ordered Problem Solver.