Re: does sqrt(2) exist in CM?
examachine_at_gmail.com
Date: 02/07/05
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Date: 7 Feb 2005 11:27:09 -0800
tchow@lsa.umich.edu wrote:
> In article <1107787399.948658.246920@l41g2000cwc.googlegroups.com>,
> <examachine@gmail.com> wrote:
> >Martin-Lof's measure theoretic definition which you refer to is
> >obviously not information theoretic. And that is equivalent to the
> >prob. theoretical one too.
> [...]
> >Chaitin explains in Part III the _equivalence_ relations between
four
> >definitions, Martin-Lof randomness, Solovay randomness, weak and
strong
> >Chaitin randomness.
> [...]
> >I am indeed talking about a random variable as I said, and nothing
> >more.
>
> No. You're conflating two different notions of "random." Others
have
> explained this, but let me give it a shot too.
>
> The definitions by Martin-Loef, Chaitin, etc., have the property that
> some real numbers are random and others aren't. For example, 0.5 is
> definitely not random according these definitions.
>
> In contrast, when you talk about a random variable X, which takes on
> real number values according to some probability distribution, then
> asking "Is 0.5 random?" doesn't make any sense. Whatever value X
> happens to take on---and it *might* take the value 0.5, but it might
> not---is "random" in a trivial sense that it's the value of a random
> variable. However, you cannot, on the basis of this definition,
> separate real numbers into two classes, those that are random and
> those that aren't.
>
> Now, it's not an accident that the same word "random" is used for
these
> two different meanings. They are related. As you say, Martin-Loef's
> definition uses concepts from measure theory and statistics, which
are
> of course related to probability theory. And you're also right that
> a random variable X that is uniformly distributed on [0,1] will take
> on a noncomputable value with probability 1.
>
> But the fact that there are *relationships* between the two different
> notions of randomness does not mean that they are the *same* concept.
> It's important to keep the distinction between them straight.
Whatever, Tim, it should be by now clear that I've indulged in no
falsehood on the uncomputability of random reals. I'm tired of
Ullrich's ad hominem arguments.
Regards,
-- Eray
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