Re: "A random real number will be on a computables list to an infinite number of digits"

From: |-|erc (h_at_r.c)
Date: 01/20/05 

Date: Thu, 20 Jan 2005 16:32:34 +1000

"george" <greeneg@cs.unc.edu> wrote in message
> |-|erc wrote:
> > "All finite subsequences of a random real number will be on a
> computables list" True / False / Other
>
> Other.
> You don't know what you mean by "random".
> What you REALLY mean here is "arbitrary", NOT "random".
> The point is that you are asking the question about
> ABSOLUTELY EACH AND EVERY real number. It goes without
> saying that the numbers that are NOT "random" -- i.e., are
> computable -- must be on the list of all computable numbers.

No I don't mean arbitrary. I mean 'non-computable irrational', arbitrary
has a different interpretation, arbitrary could be 0.10000000..
in which case a single row matches oo amount of the digits, which
is 'different' to every initial segment being matched. I am
examining the case of non-computable numbers, hoping to demonstrate
in a rudimentary fashion that the entire sequence, i.e. all digits of it
appear in order anyhow.

the diagonal coin sequence either has been flipped to some finite number
or it has been flipped to oo amount of flips.

there is no inbetween, finite amount per person is not middle ground.

>
> >
> > "All digits of a random real number are covered in all finite
> subsequences of that number" True / False / Other
> > ____
>
> Other, not only for mis-definition of "random" but for mis-definition
> of "covered". You have to replace "random" by "arbitrary" (or by
> "each")
> and you have to replace "covered" by "equals" or "matches" or "occurs"
> or some other verb that has well-understood application to
> string-matching.

rubbish. you have a vocabulary of 20 books George, its nothing. I even
stated 'for any suitable interpretation' to get any result.

> > "If you have the list of computables, a random real number can be on
> it to an infinite number
> > of digits, and yet not be on the list" True / False / Other

> That's just false; all the numbers under this paradigm only HAVE

Its a direct quote of John Savard. Any comment John?

> the SMALLEST infinity AS their number of digits, so "be on it"
> TO "some number of digits" is just incoherent. What you actually mean
> is that some prefix of the real is also a prefix of some element of
> the list. And since the only infinitely long prefix is the real
> itself,
> anything ELSE you MIGHT mean by "be on it to an infinite number of
> digits"
> is just irrelevant.
>
> There is a core difference between the infinite and the finite case
> that you are just willfully refusing to grasp.

No, you are willfully restricting conversations to newspeak to perpetuate
the myriads of models that accomodate 100 year old errors.

>
> Having a matching prefix of length N ENTAILS, REQUIRES, LOGICALLY
> NECESSITATES having matching prefixes of all smaller lengths.
> If 2 strings have the same first 99 characters then they necessarily
> ALSO MUST have the same first 0 characters, the same first 1 character,
> the same first 2 characters, ..., the same first 98 characters, AND
> the same first 99 characters. So you could define a predicate
> P(N,s1,s2)
> meaning that s1 and and s2 have the same first N characters.
> HOWEVER, YOU COULD ALSO define a predicate p(n,s1,s2) meaning that
> all prefixes of s1 and s2 that are of a length m STRICTLY LESS THAN n
> must match. So if s1=ABCDEFGHIJKZ and s2=ABCDEFGHqqqqqqq,
> then P(8,s1,s2) and
> p(8,s1,s2) would both be true, while
> P(9,s1,s2) would be false, but
> p(9,s1,s2) WOULD STILL BE TRUE
> because p(9,s1,s2) ONLY Depends on comparison of the 9 prefix-pairs
> of lengths 0,1,2,3,4,5,6,7, and 8. It doesn't say or care anything
> about what happens at position 9, not EVEN when 9 is the first
> argument.
>
> Your basic problem is that you are needing a predicate like p
> (one for which p(w,s1,s2) could still be meaningful despite the
> fact that the wth place of s1 and s2 DOES NOT EXIST and that therefore
> NOTHING can be meaningfully said about w-length prefixes -- the ONLY
> w-length prefix is the whole real) but talking about it and reasoning
> from it as though it were a predicate like P.
>

Exactly. All the English to talk about good old countable infinity applying to differnent
things has been removed from language. Multiple oo types is like adding HTTP jargon
to English at the expense of removing peoples names. Hi there 244.123.55.211!
This is a shared IP address, please be specific.

Herc

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