Re: My claim on Omega's defn

From: r.e.s. (r.s_at_ZZmindspring.com)
Date: 02/02/05 

Date: Wed, 02 Feb 2005 22:35:25 GMT

<examachine@gmail.com> wrote ...

> 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?".

I was just reading Chaitin's recent article
_How Real are Real Numbers?_ ...
In a footnote, he says this:

    "In spite of the fact that most individual real numbers will
    forever escape us, the notion of an arbitrary real has
    beautiful mathematical properties and is a concept that helps
    us to organize and understand the real world. Individual
    concepts in a theory do not need to have concrete meaning
    on their own; it is enough if the theory as a whole can be
    compared with the results of experiments."

That last sentence is interesting, and seems to stand in contrast
to a less tolerant version of "digital philosophy".

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

Of course you meant to say "noncomputable reals".

> 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?

In some kind of digital model of the observed universe, I suppose
that question has an answer -- but I don't understand any such
model well enough to know.

--r.e.s.

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