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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 Gödel’s 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 Turing’s 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
- Natural number, Alan Turing, Greatest common divisor