## 13.4. ` `Goedel Theorem

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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© by Hans-Peter Bischof. All Rights Reserved (1998).
Last modified: 27/July/98 (12:14)