A Gödel Machine is an approach to Artificial General Intelligence that uses a Recursive Self-Improvement architecture proposed by Jürgen Schmidhuber based of the mathematical theories of Kurt Gödel.
Schmidhuber’s design uses axioms to systematically search through its own code to find useful re-writes (which are deemed useful if they result in greater utility for the AI)
The axioms include a detailed formal description of the machine's software and hardware, including both the components interacting with the environment and those dealing with the formal proofs, and a possibly partial description of the environment.
According to Schmidhuber this approach is globally optimal as the search space is systematically tested in such a way the code has to prove that it is not useful to continue the proof search for alternative self-rewrites.