Re: Raatikainen's critique of Chaitin

From: Eray Ozkural exa (erayo_at_bilkent.edu.tr)
Date: 09/05/04 

Date: 4 Sep 2004 19:34:41 -0700

Torkel Franzen <torkel@sm.luth.se> wrote in message news:<vcbk6v97qlx.fsf@beta19.sm.ltu.se>...
> erayo@bilkent.edu.tr (Eray Ozkural exa) writes:
>
> > I do not, but I don't think such examples are general enough to be
> > useful.
>
> OK, so you now agree that it is an utterly trivial observation that
> there are true statements that cannot be logically deduced from
> true statements of lower complexity.

I am also saying that is not the only philosophical consequence of
Chaitin's theorems. That's not the subject of any of the
incompleteness theorems in Chaitin's work. For instance, Chaitin shows
that there is a model for Leibniz's infinite sequences of reasons.
Omega, a number which is meaningful, yet random. And his work in
randomness (his greatest contribution!) shows that a random real has
INFINITE COMPLEXITY. And then, we show that no theorem with H(Axioms)
complexity can prove more than H(Axioms) + k bits of Omega, a
particular maximally informative (every bit of it contains valuable
information about the halting problem) random real number. Moreover,
this troubling number occurs in number theory: the very cradle of
mathematics.

Chaitin says many more things, but rest assured his philosophical
claims cannot be compressed to your utterly trivial observation!

Regards, 

--
Eray Ozkural

Relevant Pages

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