This theorem essentially states that rational thought can never penetrate to the final truth.

A few quotations from a document I read are below. I’d like to know if you guys think this theorem holds any water. Personally, I think it does.

"How can you figure out if you are sane? … Once you begin to question your own sanity, you get trapped in an ever-tighter vortex of self-fulfilling prophecies, though the process is by no means inevitable. Everyone knows that the insane interpret the world via their own peculiarly consistent logic; how can you tell if your own logic is "peculiar’ or not, given that you have only your own logic to judge itself? I don’t see any answer. I am reminded of GÃ¶del’s second theorem, which implies that the only versions of formal number theory which assert their own consistency are inconsistent. "

“Kurt GÃ¶del demonstrated that within any given branch of mathematics, there would always be some propositions that couldn’t be proven either true or false using the rules and axioms … of that mathematical branch itself. You might be able to prove every conceivable statement about numbers within a system by going outside the system in order to come up with new rules and axioms, but by doing so you’ll only create a larger system with its own unprovable statements”

“The implication is that all logical system of any complexity are, by definition, incomplete; each of them contains, at any given time, more true statements than it can possibly prove according to its own defining set of rules.”

The proof of GÃ¶del’s Incompleteness Theorem is so simple, and so sneaky, that it is almost embarassing to relate. His basic procedure is as follows:

Someone introduces GÃ¶del to a UTM, a machine that is supposed to be a Universal Truth Machine, capable of correctly answering any question at all.

GÃ¶del asks for the program and the circuit design of the UTM. The program may be complicated, but it can only be finitely long. Call the program P(UTM) for Program of the Universal Truth Machine.

Smiling a little, GÃ¶del writes out the following sentence: “The machine constructed on the basis of the program P(UTM) will never say that this sentence is true.” Call this sentence G for GÃ¶del. Note that G is equivalent to: “UTM will never say G is true.”

Now GÃ¶del laughs his high laugh and asks UTM whether G is true or not.

If UTM says G is true, then “UTM will never say G is true” is false. If “UTM will never say G is true” is false, then G is false (since G = “UTM will never say G is true”). So if UTM says G is true, then G is in fact false, and UTM has made a false statement. So UTM will never say that G is true, since UTM makes only true statements.

We have established that UTM will never say G is true. So “UTM will never say G is true” is in fact a true statement. So G is true (since G = “UTM will never say G is true”).

“I know a truth that UTM can never utter,” GÃ¶del says. “I know that G is true. UTM is not truly universal.”