Re: Is continuum completely filled up?

>
>Well, this is kind of backwards. We can do lots of things. Some of them
>accomplish something and some don't. The real numbers are a specific
>thing. If you want to define/construct something else, it may have
>different properties. Why is that surprising?
>
All that I meant was that one could e.g. have a situation on a plane
such that the plane was totally covered both by points and by holes
concurrently (each with a fractal dimension of 2). And, possibly, that
one could construct the reals on the line such that any specified real
number existed , but that there were still holes (at unspecified
locations) not covered by defining points (also at unspecified
locations). It is probably nonsense - but, with the 2D example, if you
ask, for a given point, is it white or black in the limit, what is your
response?

I think I understand what you're saying, and it seems to be the same
thing that I think Toshiaki is saying. Namely:

1. Construct (define) the reals [0,1].

2. Construct (define) a line segment in some manner not requiring
the use of real numbers.

3. Map reals to points on the line segment.

4. Conjecture: bijection between the two sets is impossible, i.e.
the cardinality of the line is higher in Cantor's sense than
that of numbers in [0,1].

Is it something like that? I'm having trouble even envisioning
what step 2 would look like.

Speaking intuitively, I would have said that the line in[0,1] is a 1-dimensional space and it is up to somebody to demonstrate that you can cover it with (an uncountable infinity of) points of zero dimension. But fractals do do things like that ...

I find the whole business of real numbers a bit Zen-like (I am not a mathematician). So, between any two reals you can map the whole real line, and proceed ad infinitum, cascade after cascade of recursion without end. No real is "adjacent" to any other real, because you can map the continuum into the intervening space.

So this fractal recursion is what Douglas Adams would have been pleased to have invented as the ultimate form of procrastination - it defers to infinity, never to be obtained, the question of whether a zero dimensional microbe can walk from 0 to 1 if he insists on only moving if there are no gaps between his current real number location and the next one....

My point about the variant of the space-filling curve and possibly the real line is that possibly one can have a real line filled with holes, one or n per actual point, all of which were at undefined locations (if you knew where a whole actually was, you would call it a real and fill it in), and the fact that the reals were at undefined locations doesn't prevent their use as a basis for some Cauchy construct of any real number that you wish.

Doublethink, a location can be a hole or not as you choose; no more Zen like than a statement that the line is continuous but any point has no adjacent point. I can hear 1 hand clapping as I write ...

--
Andy Smith
.

Relevant Pages

  • Re: Dedekind Cuts, Fundamental Sequences: why?
    ... construction of the reals is designed to patch all of the holes. ... Then that proves that "these holes exist", ... Can a general expression ... example of such a polynomial equation. ...
    (sci.math)
  • Re: Is continuum completely filled up?
    ... such that the plane was totally covered both by points and by holes ... Construct the reals. ... cover it with (an uncountable infinity of) points of zero dimension. ... of the existence of gaps. ...
    (sci.math)
  • Re: Peanos space-filling curve
    ... > because if these 'holes' cause me a problem, ... > assures us that sets of reals can divide into two without ... Without Dedekind cuts (or Cauchy sequences, ...
    (sci.fractals)
  • Re: Peanos space-filling curve
    ... > because if these 'holes' cause me a problem, ... > assures us that sets of reals can divide into two without ... Without Dedekind cuts (or Cauchy sequences, ...
    (sci.math)
  • Re: Real numbers , what are they good for ?
    ... continuous time-space fabric, 'without holes'. ... numbers can't serve such purpose (e.g., we can think of some sequence ... for Dedekind cut also by giving it also a line or two. ... I knew it is not in standard reals, and I'm swimming against the tide ...
    (sci.math)