Re: Is continuum completely filled up?


Andy Smith wrote:
David Marcus writes

Actually, that would be "constructing the reals".

yes, noted, sorry.

Following a rather chaotic line of thought re the title of this thread,
what about space filling curves - covering the plane with a curve is the
same issue as covering the line with points.

Space filling curves e.g. Hilbert, Peano are fractals; there is a
generator function - a suitably line defined in e.g. a square. At each
iteration the generator is e.g. reduced in linear scale by a factor 2,
replicated by 4 (with some defined rotations for Hilbert) and the scheme
is then iterated. As I understand it, the gist of the argument that it
is space filling is that, as a function of iteration number n, one can
show that one can choose an n such that the distance from any point is
less than any eta>0 and that the distance reduces for increasing n.

The argument is more complicated than that. What you wrote just shows
that we can find a sequence of iterations that come arbitrarily close to
a given point. However, it turns out that the limit defines a curve and
this curve completely fills the space.


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.

- Randy

.

Relevant Pages

  • Re: Is continuum completely filled up?
    ... what about space filling curves - covering the plane with a curve is the ... iteration the generator is e.g. reduced in linear scale by a factor 2, ... 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. ...
    (sci.math)
  • Re: Coordinates of closest points on a pair of skew lines
    ... There *is* a theory of floating point computation - and it does NOT predict that the points generated will lie on a plane. ... My personal take on this generic argument is that exact rational arithmetic is fine for folk who want to do combinatorics and call it geometry, but that for real geometry floating point is LESS of a concession (compared to the ideal of using real Reals) than "exact rational arithmetic". ...
    (comp.graphics.algorithms)
  • Re: discrete mathematics
    ... Proginoskes: ... Out of the plane we delete all point with common factors in the numerator and the denominator. ... If you only work with finite sets, you don't have to worry about AC. ... Thus I don't accept this construction of the reals. ...
    (sci.math)
  • Re: Metric Tensor of Flat Space-Time
    ... >> always be reduced to reals? ... the world, this plane is not measured, it is assumed. ... idT/dt (where idT is a small time interval on the tangent plane and dt ... is a small interval on the space-time surface). ...
    (sci.math)
  • Re: Metric Tensor of Flat Space-Time
    ... >> always be reduced to reals? ... the world, this plane is not measured, it is assumed. ... idT/dt (where idT is a small time interval on the tangent plane and dt ... is a small interval on the space-time surface). ...
    (sci.physics)