Godel's Incompleteness Theorem

Date: 01/18/99 at 04:07:32
From: Scott Andersen
Subject: Godel's incompleteness theorem

Does Godel's Theorem really prove arithmetic systems incomplete as a 
stand-alone system? Does his proof rely on similar ideas? 

How do people who marvel at mathematics still maintain an active 
interest in it even though it is an inconsistent and ultimately an 
unprovable system in its own right?

Date: 01/18/99 at 09:06:07
From: Doctor Floor
Subject: Re: Godel's incompleteness theorem

Hi Scott,

Thank you for sending your question to Dr. Math!

Kurt Godel proved (using meta-mathematics) in 1930 his so-called 
"incompleteness theorem." It states that there is a theorem about the 
natural numbers that can neither be proved true nor proved false. You 
can add this theorem or its negation as an axiom, and either will give 
a consistent theory of the natural numbers.

What's more, even if you add this axiom, there will be another 
undecidable theorem, and no matter how many axioms you add, there will 
always be another.

Godel did NOT prove any inconsistency! So to that extent, his proof did 
not shock mathematicians. However, Godel proves that there must be 
theorems on natural numbers that can't be proven or rejected.

Some theories can be proven complete; others can't. The study of this 
is very deep, and very interesting. If you want to know more about 
Godel's proof, you might want to read the magnificent book _Godel, 
Escher, Bach_ by Douglas Hofstadter.

I hope this clears things up!

Best regards,

- Doctor Floor, The Math Forum
  http://mathforum.org/dr.math/