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Goedel's thesis initially about number theory but now found applicable to all formal systems that include the arithmetic of natural numbers: "any consistent axiomatic system does include propositions whose truth is undecidable within that system and its consistency is, hence, not provable within that system". The self-reference involved invokes the paradox: "a formal system of some complexity cannot be both consistent and decidable at the same time". The theorem rendered Frege, Russell and Whitehead's ideals of finding a few axions of mathematics from which all and only true statements can be deduced non-achievable. It has profound implications for theories of human cognition, computational linguistics and limits artificial intelligence in particular.
References for studying Goedel's proof are :
-- Principia Mathematica, Bertrand Russel & Alfred North-Whitehead -- Goedel's Theorem in Focus, Ed. S.G. Shanker, ISBN 0-415-04575-4 -- The Emperor's New Mind, Roger Penrose, ISBN 0-09-97710-6, p.129-192 -- Goedel's Proof, Ernest Nagel & James R. Newman, ISBN 0-415-04040-X -- Goedel, Escher, Bach : An Eternal Golden Braid, Douglas A. Hofstadter.
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Last modified 22/May/97