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Goedel's Incompleteness Theorem


Date: 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/   
    
Associated Topics:
High School Logic

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