If you followed all that, you can now say with reasonable confidence that you "understand Gödel's theorem." That is, you understand why no formal mathematical system can ever hope to represent all statements about natural numbers.
As I see it, there are three directions you can go from here. The first direction is down, to a more mathematical level. The explanation I have given is very "high-level," and would not satisfy a real mathematician for an instant. By learning more about the math involved, you can work the proof to ever finer levels of detail, and make it ever more rigorous and bullet-proof.
The other way to go is up, to a more philosophical level. There are many people who believe that the human mind, based on neurons and physical principles, is just a very sophisticated formal system. Does Gödel's theorem imply the existence of facts that must be true, but that our minds can never prove? Or even stronger, that our minds can never believe—or strongest yet, ever conceive?
The third direction you can go is sideways, to lunch. Who wants to spend his whole life worrying about abstract mathematical theorems?
Monday, October 11, 2010
Gödel's Theorem: The Very End of the Proof
Explanation of the theorem