Omega, probability, Godel's Incompleteness Theorem

this page seems really interesting.
Click to expand...

Yes. It might be the most interesting site on the Internet.

Of course, there can't be a least interesting site -- as soon as you identified such a site, it would become interesting for that very reason. I guess that means there can't be a most interesting site either...in fact, I guess it means all sites must be equally interesting...hmmm, that doesn't seem right. What am I doing wrong here? Wait...if the least interesting sites were all equally uninteresting, that would work.

Anyway.

What he describes as the Halting Probability (Omega) looked at first like simply a fresh approach to explaining the principle of Turing computability, but there seems to be something more here that I haven't quite grasped yet.

I'm particularly fascinated by this:

"We can also describe this irreducibility non-technically, but very forcefully, as follows: Whether each bit of Omega is a 0 or a 1 is a mathematical fact that is true for no reason, it's true by accident!"

I may need to gnaw on that for a while. I wonder what Roger Penrose might have to say about it.
 
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