Re: My claim on Omega's defn

examachine_at_gmail.com
Date: 02/02/05 

Date: 2 Feb 2005 11:45:13 -0800

Daryl McCullough wrote:
> r.e.s. says...
> >
> >"Daryl McCullough" <stevendaryl3016@yahoo.com> wrote ...
> >
> >> Finally, define Omega to be the limit as n-->infinity of S(n).
> >> This is the same as
> >> Omega = sum over all valid bit strings p of 2^{-length(p)}
>
> You're right that interpreting Omega as a probability requires
> the consideration of infinite bit strings. However, the definition
> of Omega can be made in terms limits of finite bit strings.

I was going to say exactly this to r.e.s. This would be exactly the
same thing Cauchy did with calculus.

>>From a computationalist point of view, that is why Omega has any
relevance to our world, because it does not depend on a purely
metaphysical (impossible!) idea like the continuum.

So, the limit itself becomes uncomputable, e.g. _unreal_, however as
always approximations are possible, which avoids the conceptual duality
in Chaitin's monograph.

Consider this like the correct conception of a black-hole. The
event-horizon does _not_ exist, it is only approximated, you can only
get this close to such physical limits. (Maybe this was a grossly
inappropriate way of making the analogy. Please correct my silly
errors) I'd heard that russian physicists were careful enough to make
the distinction between a limit and something that actually exists.

In fact, _everything_ in Chaitin's theory is independent from Cantor's
naive set theory or axiomatic set theory, which is why I think it is
relevant to foundational "thinking". In a sense, I think the language
of AIT and Turing computation comes before, or encompasses more than
set theory. Reading his work with this frame of mind is a lot of fun
"can I write this whole thing in a constructivist language?".

Now, I must make the additional claim that, within all computable
reals, only Omegas would be random. I find that kind of significant.

As a fun question, I must ask you, given the current estimates for the
bounds and particle count of our universe, how long do you think is
actual Omega of our universe? How many actually random bits are there
here?

Regards, 

--
Eray Ozkural

Relevant Pages

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