Re: Infinite Binary Strings: A Question

On Fri, 12 Sep 2008 16:39:14 +0200, Herman Jurjus <hjurjus@xxxxxxxxx>
wrote:

Leon Street wrote:
Given a line segment AB, and a point P arbitrarily chosen upon it,
one can ask which half of AB P lies on, left or right, then having selected
the half interval P lies on we can ask which half of that interval P lies
upon, and so on repeatedly. If we happen to have chosen a point P such that
AP is incommensurable with AB, the point P will never lie exactly at the
end of any half interval. (It will never lie at the end of any fractional
interval of the line segment.) So the point P produces an infinite, and
aperiodic, infinite string eg LRRLLLR......

Is the converse true? That is, does an infinite, aperiodic binary
string pick out a precise point on AB? Comon sense, perhaps, would tell us
that you cannot get to a point by this repeated narrowing down -- it's
intervals all the way down. Mathematics seems to be telling us that, by
somehow treating the infinite sequence of narrowings down as a whole, an
infinite binary string would indeed determine a precise point on AB. My
question is: which bit of mathematics is it, exactly, that is telling us
that this is so?

To put it a bit simplistically:
At the end of the 19th century, it was Dedekind that wanted this 'to be
so', and he was thrilled when he heard about a guy called Cantor, who
had a theory providing just that: set theory.

Nowadays, it's a necessary consequence of the way the real number line
is -defined- using set theory (either via Dedekind cuts or via Cauchy
sequences, see for example
http://en.wikipedia.org/wiki/Construction_of_real_numbers).

In short, the answer to your question is: the standard -definition- of
the real number set is the bit of mathematics that you're looking for.

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?

I also have in mind an argument which I couldn't lay out in a
couple of paragraphs but I can summarize here. (I might set it out in a
separate thread if I could get more confident about it.) I think there is a
difficulty with the notion of an arbitray infinite binary string,
understood as a point in a uniform combinatorial space of 2^infinity
possibilities. I believe an infinite binary string has to come from
somewhere, has to have something that produces it. In the present context
that is the arbitrary point P. If we had chosen the particular point P such
that AB/AP = pi, then the string we produce for this ratio by applying a
fromula for pi has its anchorage in the definite position this point has on
the line, so that geometry serves a kind of GPS for this string. (It is not
necessary to believe that pi is a geometrical notion -- it is sufficient
that it is believed to be a precise ratio, which can then certainly be
represented as a point on a line.) In short, I don't believe that the
infinite binary string associated with an arbitrary point on the line
segment has an independent existence apart from that point. On that basis,
it cannot determine anything. If that's not complete bollocks, I'd be glad
to say more. In any case, again many thanks for a thoughtful response.


Leon
.

Relevant Pages

  • Re: Infinite Binary Strings: A Question
    ... one can ask which half of AB P lies on, left or right, then having selected ... infinite binary string would indeed determine a precise point on AB. ... My own intuition of the line is that it has no "holes" and that it is ... the axiom of choice is obviously true. ...
    (sci.math)
  • Re: Infinite Binary Strings: THE ANSWER : 3 VALUED LOGIC
    ... one can ask which half of AB P lies on, ... What is an infinite binary string? ... And nonsense only by the nonsensical. ... Very well said Virgil. ...
    (sci.math)
  • Re: Infinite Binary Strings: THE ANSWER : 3 VALUED LOGIC
    ... the half interval P lies on we can ask which half of ... which bit of mathematics is it, exactly, ... What is an infinite binary string? ... And nonsense only by the nonsensical. ...
    (sci.math)
  • Re: Infinite Binary Strings: THE ANSWER : 3 VALUED LOGIC
    ... the half interval P lies on we can ask which half of ... narrowing down -- it's ... which bit of mathematics is it, exactly, ... What is an infinite binary string? ...
    (sci.math)
  • Re: Infinite Binary Strings: A Question
    ... k) than to compute the string up to the k-th digit, ... It could be that we have two models of ZFC, ... difficulty with the notion of an arbitray infinite binary string, ... I believe an infinite binary string has to come from ...
    (sci.math)