Citat:
Isak666:
Nista ja ne bih tu dodao.
The problem is this: given any formal system F that we might want to take as a foundation for mathematics (for example, Peano Arithmetic or Zermelo-Fraenkel set theory), Gödel tells us that there are Turing machines that run forever, but that can’t be proved to run forever in F. An example is a Turing machine M that enumerates all the proofs in F one by one, and that halts if it ever encounters a proof of 0=1.
The claim that M doesn’t halt is equivalent to the claim that F is consistent—but if F is indeed consistent, then the Second Incompleteness Theorem says that it can’t prove its own consistency.
http://www.scottaaronson.com/blog/?cat=32