[alex_d]

oh, boy, not godel...

godel isn't wrong per se, but everyone's interpretation of his proof IS.

First of all, Godel's statement isn't "I am a liar." That's just a plain old self-contradictory statement.

Rather, a Godel statement is something like "A person who says only truths will never say this sentence." Will a person who says only truths say the aforementioned sentence? If the sentence is true, he might say it. But then the sentence becomes false. If the sentence is false, he won't say it. But then the sentence becomes true.

What happens in the end? Big surprise: in the end the guy doesn't say it. The sentence is true.

However, the truth teller wouldn't have told you that. The truth teller can't even tell you what's true about his own behavior. So much for his wisdom.


At least, that's everyone's interpretation: "There are things that are true about a logical system like math which the system itself can't prove." Ie, "you can't use math to know everything that's true about math." And then people go "whoa... Incredible!"

That interpretation is as wrong as it seems. It's just as wrong as is Zeno's paradox. It is based on the false presumption that infinities are impossible.

In fact, if you sit by the man and listen to every truth that has has to say, ad infinity, and note that the sentence "the truth-sayer will never say this sentence" is never uttered by him, it IS a proof that the sentence is true. The truth-teller does, in fact, prove it all on his own. And like a buddhist monk, he makes his point with silence.

Zeno of ancient Greece argued that motion is impossible because any movement would require passing through an infinity of distances. You can't move through an infinity of distances, so you can't move! Ooo! He even had a nice story to illustrate his point starring the legendary Achilles and a crappy old turtle that had a head-start. Every time Achilles would catch up to where the turtle was, the turtle would have already moved a little further. When Achilles reached that new point, the turtle would move yet another millimeter ahead. An infinity of catching up would be required. That story was passed around by sophisticated folk for about two thousand years seemingly proving that Achilles could never best the crappy old turtle.

Godel essentially argued that some mathematical statements will take an infinity of steps to prove. You can't have an infinity of steps, therefore some statements that are true about math can't be proved by math! Ooo!


The truth is that infinities are perfectly possible and valid. You can move through them, you can (theoretically) wait them. Thus Godel's theoreom must actually be "some things about every logical system can't be proved in a finite amount of time/steps. But, given infinite time, math will be able to prove everything that's true about it." This sounds much more reasonable and less mystical, doesn't it? And a lot less fun, eh?


In fact, I would argue that acknowledging that infinites are valid mathematical constructs, and that infinite proofs still count as proofs, gives many more "whoa" moments and moves mathematics ahead far more than the defeatist 20th century interpretation of Godel.

Ok, so... let's now please stop going around saying Zeno was right.