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Unformatted text preview: There was a need for a more precise definition of algorithm. Recall the Hilbert Program (consistency, completeness, and decidability). Machine Theory (cont.) Kurt Gdels proof (1931): Given an axiomatic system that is comparable in complexity to, e.g., arithmetic, then it cannot be both consistent and complete. There will always be true but formally undecidable sentences in such a system. But can we at least devise an algorithm for recognizing such undecidable sentences? Alan Turings proof (1936): No! To prove this, Turing invented a formal machine (Turing Machine) that amounted to a precise definition of an algorithm....
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- Spring '07