Nyarly's Journal: ToC: An Appendix

Two things that terrify and fascinate me that relate to ToC:

1. Goedel Incompleteness. Basically, Goedel demonstrated that any logical structure either cannot discuss it's own consistancy or it can demonstrate it's own inconsistancy. This upsets me because of all that I've invested in the virtues of logic and reason. At least arthmetic is consistent.
2. Chaitin's Construction. Often called Chaitin's Constant and designated by a capital omega, this is one scary number. It's the probability that a given Turing Machine will halt for any given string. But we already decided that the Halting Problem was undecidable, right? Well, omega is uncomputable. And random, even though it's constructed in terms of a polynomial. The best you can get is that the first N digits are ones (if you express it in binary). The N+1st digit might be zero. Or not. It disturbs me greatly that you can't prove that for any given Turing machine, you can't prove that omega is not an endless string of ones - which would be unity. With the Halting Problem, at least you had the solace that some string would cause the machine to halt.

(and it really bugs me that Slashdot eats the O-umlaut element in "Goedel")

Wow.

[cioxx]

That is some really nerdy stuff.

Congratts for making my head explode.

btw, Hawking, Weinberg and others strongly disagree on the given issue.