Unit interval and random numbers

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

Date: 4 Sep 2004 19:23:10 -0700

daryl@atc-nycorp.com (Daryl McCullough) wrote in message news:<chde2702s4d@drn.newsguy.com>...
> Jesse F. Hughes says...
>
> >It is a bit off-topic with the main thread, but I still think it's
> >worth pointing out. Chaitin's use of the term "random" has little to
> >do with the intuitive use or with basic probabilities.
>
> I don't think that's completely true. The set of reals in [0,1] that
> are "random" in the sense of Chaitin has measure 1, which means that
> an infinite sequence of coin tosses is almost certain to produce one.

Chaitin says pretty much the same thing in "Omega", but I am not sure.
The probability that a real number picked from the unit interval is
random, is 1, which means to my simple mind it is not just *almost*
certain but entirely and positively certain that it will be a random
number. It might not make immediate sense, but it goes like this.

Could the real we picked turn out to be 0.5?

No, because then in

0.5000000000000000000000000000000000000000000....

the digits do not have equal limiting frequency, that simple.

So, although it might fail our intuition, it is absolutely impossible
for computable reals to be the outcome of a random choice in the unit
interval. My interpretation: we cannot choose random numbers in the
unit interval, in the real world, it is not a good idea! This property
that Turing and others demonstrated might imply, in my opinion, that
continuum does not exist in any physical sense - but it might and
still exists in an imaginery sense. Of course, these are all very
tangential to the main point we are discussing so I changed
subject....

Regards, 

--
Eray Ozkural
"There is no perfect (continuous) circle"

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