Further Reflections on Godel

Zachary Ernst, writer of the blog Inklings, has a nice post on the major theorem of Kurt Godel, in which he shows that arithmetic is "incomplete."  A technical summary of Godel's Incompleteness Theorem runs along these lines, "For any system of arithmetic, there are true propositions that are necessarily unprovable."  Ernst explains what this means in layman's terms. 

There is another result of Godel's that I like to mention in my logic classes.  Say I have three rules:  1) "Whenever X is true,  squares have four sides."  2) "Whenever X is true, squares do not have four sides." and 3) "X is true."  From these three rules I can deduce "squares have four sides" and "squares don't have four sides."  I can deduce two propositions that contradict, thereby showing my rules are inconsistent: they can't all be true at the same time.  Both theoretically and practically, I would not want to use this system.   Instead, I'd want a system that was consistent, where the rules can all be true at the same time.

With arithmetic, we think that it's consistent, that the rules of arithmetic can all be true.  Godel showed that it is impossible to prove that arithmetic is consistent!  This doesn't mean it's inconsistent, but just that if it's consistent, we'll never be able to know it.  Quite an amazing theorem, I would say.