Re: does sqrt(2) exist in CM?

From: David C. Ullrich (ullrich_at_math.okstate.edu)
Date: 02/11/05 

Date: Fri, 11 Feb 2005 06:42:33 -0600

On 11 Feb 2005 00:53:46 -0800, examachine@gmail.com wrote:

>Jesse F. Hughes wrote:
>> examachine@gmail.com writes:
>>
>> > Anyway, the problem is, if we swallow what I say above we have a
>> > paradox: it is not possible to define a "truly random" real by
>infinite
>> > application of a fair coin! Which seems a little counter-intuitive.
>How
>> > do we explain that?
>>
>> Why not explain it the obvious way? Chaitin randomness isn't the
>same
>> thing as random.
>
>If you think so, then you will have to conclude that Martin-Lof
>randomness and Solovay randomness aren't the same thing as random,
>either. What should that make us believe in?

For heaven's sake, this is like you haven't been paying any attention
at all.

None of those notions of randomness you mention are the same
as "random" as in probability. That's been explained many times
in this thread, in great detail. For example, 1/2 is certainly
not a random number in any of those AIT senses, but if you
pick a number between 0 and 1 "at random" as in probability
there's no reason it can't be 1/2 (if it can't be 1/2 then
it can't be any other number either).

>That Borel's paradox of
>the undefinability of randomness was correct?
>
>Please, let's make the discussion concrete. So, you don't think Solovay
>randomness, which is provably equivalent to strong Chaitin randomness
>defines randomness. In other words, Solovay randomness is just wrong,
>there are other random numbers than Solovay random numbers. I require
>an example, or a proof to proceed with such a thought. Because that is
>ultimately an admission of the paradox I'm hinting at, rather than a
>refutation of it.
>
>Maybe that is something everybody agrees on, but I don't see how that
>is obvious.
>
>Regards,

************************

David C. Ullrich

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