Goedel's Incompleteness TheoremDate: 08/08/97 at 14:36:30 From: Hari Seldon Subject: Logic How did Goedel prove that any nontrivial logical system cannot be proven to be inconsistent? Date: 08/14/97 at 16:12:05 From: Doctor Rob Subject: Re: Logic Essentially Goedel created a statement about integers which is true if and only if it is unprovable in the axiom system of the integers. Of course if it is false, then it is provable, which means that it is true if the axioms are consistent. Given the consistency of the axioms, then this would be a contradiction, and the statement must be true. But then the statement must be unprovable. The fallout is that in every system with consistent axioms rich enough to include the integers, there are statements which are true, but cannot be proved. Thus no consistent axiom system can be devised from which a proof of every true statement in that system exists. This is Goedel's Incompleteness Theorem. There are several books on this subject: Nagel and Newman, _Goedel's Proof_ Douglas R. Hofstadter, _Goedel, Escher, Bach_ Martin Davis, _Computability and Decidability_ Raymond Smullyon, _Goedel's Incompleteness Theorem_ At least the first two should be available at larger public or college libraries, and are written for laymen. -Doctor Rob, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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