Information-theoretic incompleteness (Q1200218)

In this interesting note the author continues his many investigations of the incompleteness of basic topological systems. Several fundamental incompleteness theorems in his book on algorithmic information theory [Algorithmic information theory (1987; Zbl 0655.68003)] are reformulated based upon an improved definition of the complexity of a formal axiomatic system. The value added comes from measuring the complexity of a formal system in terms of the program-size complexity of enumerating its infinite set of theorems rather than in terms of the program-size complexity of the finite strings of axioms. Included is an incompleteness theorem for diophantine equations.