Re: Why we cannot compute omega

examachine_at_gmail.com
Date: 01/25/05 

Date: 25 Jan 2005 12:12:39 -0800

An edited post follows. Thanks to Mike Oliver and Daryl McCullough for
the corrections. Good food for thought.

The Ghost In The Machine wrote:
> In sci.logic, examachine@gmail.com
> <examachine@gmail.com>
> wrote
> on 24 Jan 2005 06:22:07 -0800
> <1106576527.401408.268510@c13g2000cwb.googlegroups.com>:
> > Well, Omega is a random number by definition like any probability,
e.g.
> > it's in (0,1). If you believe that real numbers exist, then it
wouldn't
> > be so hard to believe that Omega, too, exists :)
>
> Pedant point: Omega is not random. It's just not computable. :-)
> Though we should be able to get close via various heuristics.

To be pedantic, Omega *is* random.

A real number is random iff it is not compressible. You say it's
uncomputable but not random. It might not matter that it is
uncomputable (because as others pointed out there are random but
compressible reals). What matters is that there is absolutely no
redundancy in Omega. Then, it is uncomputable, incompressible and
random at the same time.

That a random number is definable in some mathematical framework like
information theory does not mean that it is not random. (Although the
total set of "definable" or "nameable" numbers itself would be
countable according to Chaitin)

That a random number is semi-computable (to use the terminology of an
expert in Kolmogorov complexity: Paul Vitanyi) like Omega does not mean
that it is not random. [If you somehow had in mind the fact that it can
be approximated from below]

Omega is random by any criterion of randomness, e.g. Solovay
randomness, etc. There is a theorem for that.

If you don't believe me, read Chaitin's monograph and find out the
proofs yourself.

Best Regards, 

--
Eray

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