Re: Infinite Binary Strings: A Question

On Fri, 12 Sep 2008 13:56:59 -0600, Virgil <Virgil@xxxxxxxxx> wrote:

In article <jlelc4h9uqr26tviolqkj2qibu5dibs9n2@xxxxxxx>, leon street
wrote:


Many thanks for that, and I understand (I think) your answer.
But isn't what's at stake here more general than the issue of the
definition of the real number? And I'm still struck by the feeling that the
idea of this binary sequence of lefts and rights determining a point is
unconvincing. Of course I need to say why.

Part of what I have in mind is that an aperiodic string is in
general, if not always, chaotic or unpredictable. By that I suppose I
should mean something specific to the effect that there is no significantly
shorter way to determine the k-th digit (for some arbitrary, perhaps large
k) than to compute the string up to the k-th digit, or something like that.
At any rate, the last digit computed so far flips erratically between 0 and
1 as the string is explicated. Where is the precision in such a concept?

If there is some rule, however complex, by which, for any given positive
natural n, the nth digit can be determined to be a 0 or a 1, then the
expansion is "computable', which is satisfactory for the existence of
the the number represented by that string for all mathematics, including
constructionist mathematics.

It can be proved indirectly that there "are" infinite sequences of 0's
and 1's for which no such rule can exist. Constructionists reject the
"existence" of such numbers, but in standard mathematics they are
regarded as existing but inaccessible.

Thanks for the clarification. I want to address another poster's
remarks about arbitrary infinite binary strings, which may have a bearing
on this.

leon
.

Relevant Pages

  • Re: writing get_script as an external routine callable by C
    ... I _STRONGLY_ suggest to attend a Perl class or a self-study course where ... If you mean "begins with a digit" as you said above ... Case 2 matches if the beginning of the string is followed by 1 or more ... If you put the two Bushs together in their over seven years of their two ...
    (comp.lang.perl.misc)
  • Re: abundance of irrationals
    ... >>digits of pi and the same string, where the last one was exchanged ... >>by another digit) then, in fact, they would not exist. ... >>to mathematics too. ... At which position does he get the smalles positive fraction? ...
    (sci.math)
  • Re: Sorted Fixed Length String
    ... >I have a String of Numerical Digits Created Using Concatenate. ... >The Strings could be from 6 Characters in Length to 11 Characters in ... >The Least Characters in a String with a Digit GREATER than 1 can ONLY be ... Dim str As String ...
    (microsoft.public.excel.programming)
  • Re: Pi as the Mother Number
    ... For example if we find our 100-digit string starting at the ... 37th digit of pi, we can just say, go to the 37th digit of pi and print the ... problem with using this principle to compress numbers is the problem of the ... Foundation of Mathematics, first to understand the difference between ...
    (sci.math)
  • Re: writing get_script as an external routine callable by C
    ... I _STRONGLY_ suggest to attend a Perl class or a self-study course where ... If you mean "begins with a digit" as you said above ... used as delimiters, therefore it doesn't show up in the FAQ. ... Case 2 matches if the beginning of the string is followed by 1 or more ...
    (comp.lang.perl.misc)