Re: Is continuum completely filled up?



On Jan 24, 12:19 pm, Andy Smith <A...@xxxxxxxxxxxxxxxxxxxx> wrote:
Randy Poe <poespam-t...@xxxxxxxxx> writes(snip)



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.

The point of my post was that you can't even ask the
question "do the reals cover it" until "it" is defined. Any
attempt I might make to define the line (for instance
in terms of distance from 0) depends on the real numbers.

So I don't know what "it" is. If I think of it as a representatioin
of [0,1], then of course [0,1] covers it because [0,1] and
"it" are essentially the same thing. "It" has no existence (in
my scheme) outside of [0,1].

Let me be even more specific. I will define "the line segment
from P1 to P2" as the set of points P
{P : P = a*P1 + (1-a)*P2, 0 <= a <= 1}

So everything in this set is defined by a real number a
between 0 and 1. There's no such thing as a point which
is not associated with such an a, since that's my membership
test. Do you see that it's obvious there is no member of
my set not corresponding to some a?No, but in the context of all of this, how do you adjust a such that
successive points in P (there can be no such thing) are continuous ?

I'm leaving the floor open to other types of definitions, but I
don't know what they are.

You seem to be conceiving of points which have a distance
from one end or the other, but that distance is not a real
number. Is that correct?

No, I just have a problem with covering something continuous with
something point-like. But I am ignorant anyway.

Then the answer is "yes", you have difficulty believing every
point on the line has a real distance from the endpoint. There
must be points which aren't "covered" by the set of real
distances. In other words, those points do not have a real
distance.

- Randy

.

Relevant Pages

  • Re: Is continuum completely filled up?
    ... cover it with (an uncountable infinity of) points of zero dimension. ... in terms of distance from 0) depends on the real numbers. ... is not associated with such an a, since that's my membership ...
    (sci.math)
  • Re: Is continuum completely filled up?
    ... cover it with (an uncountable infinity of) points of zero dimension. ... is not associated with such an a, since that's my membership ... what does "continuous" mean in this context? ...
    (sci.math)