Godels Incompleteness Theorem and the Limits of Mathematics

Okay. In mathematics there was this old paradox in set theory. There were two different kinds of sets, and if you said, "the sum of all of of one type of set," then you had a set that could either be the first type of set or the second type of set but wasn't either.

Along come Bertrand Russell and Alfred Whitehead and tried to fix the problem. They created hierarchical set theory. Now very few people used it, but it did appear to deal with the set theory problem.

A guy named Kurt Godel had a realization. He did a simple mapping, and showed a) that no form of set theory could overcome the paradox he had just created and b) that no form of set theory that will ever be created could ever overcome this particular problem. At least until someone makes a form of mathematics that doesn't work anything like mathematics does now. If we don't blow ourselves up, someone will probably solve this particular form of paradox, but there will probably be something new to worry about. It's a brilliant proof and well worth studying.

So what did the mathematicians say? Brilliant work Mr. Godel, but set theory is working just fine for me! Seriously. And we continue to use the old set theory with it's old paradox to this day, and studying Godel's theorem is something for specialists who research it specifically, as is hierarchical set theory.

It's math. It has to do with deduction. Physical science is inductive and works completely differently. But it is important to understand that axiomatic systems are fatally flawed, and as far as we know, permanently flawed. This hasn't stopped us from using axiomatic systems for the past 50 years with great effect, much in the way that a lot of people used philosophical techniques to great effect after Hume.