Re: Is continuum completely filled up?

In article <SF5X+KDka4tFFwFs@xxxxxxxxxxxxxxxxxxxxxxxxxx>,
Andy Smith <Andy@xxxxxxxxxxxxxxxxxxxx> wrote:


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

One can do this by removing a dense subset from the reals, ,in which
case one hat between any two remaining points infinitely many points AND
infinitely many holes.


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.

If any point were to have an adjacent point with no points in between,
then there would be a gap between them, and a lack of coninuity because
of the existence of gaps. Which provides something against which the one
hand can clap.

I can hear 1 hand clapping as I write ...
.

Relevant Pages

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  • Re: Peanos space-filling curve
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