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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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