Random reals are not computable!

examachine_at_gmail.com
Date: 01/25/05 

Date: 25 Jan 2005 05:35:23 -0800

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 computable. You say it's
uncomputable but not random. There are no such numbers. There is a
theorem for that!!!!

That a random number is definable in some mathematical framework like
information theory does not mean that it is not random.

That a random number is semi-computable (to use the terminology of an
expert in Kolmogorov complexity: Paul Vitanyi) does not mean that it is
not random.

Omega is random by any criterion of randomness, e.g. Solovay
randomness, etc.

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

Regards, 

--
Eray OZkural

Relevant Pages

  • Re: A theoretical physics FAQ
    ... " The probability that someone in risk group A will die of cancer is 1/3 ... i.e., a measurable function on the set Omega of possible experiments, ... high confidence level certain statistical tests valid for such ...
    (sci.physics.research)
  • Re: Integration and metric space
    ... radius ball in M? ... you could talk about probability densities ... You're talking about random variables. ... A is a sigma-algebra on Omega and P is a probability ...
    (sci.math)
  • Re: Integration and metric space
    ... You're talking about random variables. ... A is a sigma-algebra on Omega and P is a probability ... A random variable is a measurable function on Omega. ... predefined measure on M, nor an predefined sigma-algebra. ...
    (sci.math)
  • Re: Integration and metric space
    ... You're talking about random variables. ... A is a sigma-algebra on Omega and P is a probability ... A random variable is a measurable function on Omega. ... probability distribution function. ...
    (sci.math)
  • Random reals are not computable!
    ... The Ghost In The Machine wrote: ... >> Well, Omega is a random number by definition like any probability, ... To be pedantic, Omega *is* random. ... information theory does not mean that it is not random. ...
    (comp.theory)