Re: Is a line segment composed of points?

On Nov 11, 7:15 pm, Venkat Reddy <vred...@xxxxxxxxx> wrote:
On Nov 11, 9:51 pm, William Hughes <wpihug...@xxxxxxxxxxx> wrote:



On Nov 11, 11:15 am, vred...@xxxxxxxxx wrote:

On Nov 11, 2:04 pm, Dave Seaman <dsea...@xxxxxxxxxxxx> wrote:

On Sun, 11 Nov 2007 00:29:30 -0800, vred...@xxxxxxxxx wrote:
On Nov 5, 2:54 am, Phil Carmody <thefatphil_demun...@xxxxxxxxxxx>
wrote:
William Hughes <wpihug...@xxxxxxxxxxx> writes:
On Nov 4, 3:29 pm, Adam <n...@xxxxxxxx> wrote:
On Sun, 04 Nov 2007 09:48:48 -0800, William Hughes wrote:
On Nov 4, 10:04 am, Phil Carmody <thefatphil_demun...@xxxxxxxxxxx>
wrote:
William Hughes <wpihug...@xxxxxxxxxxx> writes:

On Nov 4, 5:54 am, vred...@xxxxxxxxx wrote:

If you keep cutting a line at different places, will it be cut into
pieces of zero length? It is self contradictory, because when the
piece if of zero length, then it is not a result of cutting at
different places.

Incorrect. Start with [0,1]; at step n cut at 1/n;
Do this for every n. You end up with an infinite number of pieces,
one of which is a single point of length zero. (Note there
is no last cut.)

"end up with" clashes with "no last".

As Dave Seaman pointed out, "end up with" means after all the cuts.
This does not imply a last cut.

There is a completed set in which all the cuts have been completed.
But does that set consist of "line segments" or "points"?

A line segment is a set of points.

After all cuts (omega cuts, one for each element of N)
have been completed we have
omega+1 sets of points. One set of points is
the singleton {0}. Whether you call this a line
segment is a matter of how you define line segment.
All other sets of points are
line segments.

Yup, the one at 0 is of length 0, but how long's the one next to it?
No, it is not of zero length, but of infinitesimal length. The next
one also is of almost same length.

The one at 0 has exactly zero length, even if we are dealing with the
nonstandard reals.

I don't see the need for non-standard numbers.

Because in that case, there is a cut at 1/n for each
n in *N. For every e > 0 (even if infinitesimal), there is a nonstandard
integer n in *N such that 1/n < e.

By same logic, for every N (however large it may be) there is a
standard infinitesimal e such that 1/e > N. So, you can never make
that huge jump from infinitesimal to zero using division process,

Correct. Note, however, that the length of the
first segment is not reached by a division process.

Cutting a line segment at its non-boundary points is a division
process, I think. This what we have been doing all along this thread.


Correct. Each of these cuts takes place at a non-zero
point. And the result of *any* *one* of these cuts
(indeed of any finite number of these cuts)
is not zero. However, this does not tell all what the
result of *all* the cuts is.

A division process can only reach points that are the result
of *a* division. The length of the first segment is not
the result of a division.

The division process is used to decide
where to make the cuts. The division process cannot
reach zero so all cuts are made at
non-zero points.

Correct.

Despite this fact, if we write the first
segment as [0,b], we have that b is less than every
infinitesimal.

Yes.

Since b is not defined by a cut,

It is defined by a cut. We have been busy cutting the line segment,
remember?


Yes. However b is not defined by *a* cut.
b is the result of a process that involves
cuts, but it is not the result of any one cut.


the fact that all cuts are non-zero does not mean
that b is non-zero.

Why not?

Because b is not the result of *a* cut. We know that
the result of *a* cut is non-zero. However, this
does not tell us what the result of a whole set
of cuts is.

There was no significant change in our cutting process - have
been using same knife or same for loop. Why is it difficult to see a
infinitely small number is just as much legal citizen as an infinitely
large number?

An infinite small number *is* just as much a legal citizen as an
infinitely
large number. However, the reason that b is not an infinitely small
number has nothing to do with the question of whether infinitely small
numbers exist.

- William Hughes

.

Relevant Pages

  • Re: Is a line segment composed of points?
    ... piece if of zero length, then it is not a result of cutting at ... You end up with an infinite number of pieces, ... segment is a matter of how you define line segment. ... first segment is not reached by a division process. ...
    (sci.math)
  • Re: Is a line segment composed of points?
    ... piece if of zero length, then it is not a result of cutting at ... You end up with an infinite number of pieces, ... segment is a matter of how you define line segment. ... first segment is not reached by a division process. ...
    (sci.math)
  • Re: Is a line segment composed of points?
    ... piece if of zero length, then it is not a result of cutting at ... You end up with an infinite number of pieces, ... segment is a matter of how you define line segment. ... Cutting a line segment at its non-boundary points is a division ...
    (sci.math)
  • Re: Is a line segment composed of points?
    ... piece if of zero length, then it is not a result of cutting at ... You end up with an infinite number of pieces, ... segment is a matter of how you define line segment. ... first segment is not reached by a division process. ...
    (sci.math)
  • Re: Is a line segment composed of points?
    ... It is self contradictory, because when the ... piece if of zero length, then it is not a result of cutting at ... segment is a matter of how you define line segment. ... Given any 0<r<1 there are an infinite number of segments between ...
    (sci.math)