Information-Theoretic Incompleteness

We propose an improved definition of the complexity of a formal axiomatic system: this is now taken to be the minimum size of a self-delimiting program for enumerating the set of theorems of the formal system. Using this new definition, we show (a) that no formal system of complexity n can exhibit a specific object with complexity greater than n+c, and (b) that a formal system of complexity n can determine, at most, n + c scattered bits of the halting probability @w. We also present a short, self-contained proof of (b).

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