Re: The infintely small number b

On Nov 20, 12:43 am, Venkat Reddy <vred...@xxxxxxxxx> wrote:
On Nov 20, 9:34 am, William Hughes <wpihug...@xxxxxxxxxxx> wrote:



On Nov 19, 11:06 pm, Venkat Reddy <vred...@xxxxxxxxx> wrote:

On Nov 20, 8:57 am, William Hughes <wpihug...@xxxxxxxxxxx> wrote:

On Nov 19, 10:44 pm, Venkat Reddy <vred...@xxxxxxxxx> wrote:

On Nov 20, 6:22 am, Venkat Reddy <vred...@xxxxxxxxx> wrote:

On Nov 19, 10:54 pm, Venkat Reddy <vred...@xxxxxxxxx> wrote:

On Nov 19, 11:42 am, Joe Blow <justarandomi...@xxxxxxxxx> wrote:

The object with the properties you describe, whatever it is, would necessitate that the space in which it is contained be non-Hausdorff. Much of mathematics depends explicitly on the fact that the real line has the Hausdorff property, and when you throw it out you're left with an object that isn't terribly useful.

So, while you might be able to define a space which has some of the properties you want, the question is why would you want to? What would you be able to study on this awkward space that you can't adapt the well-studied space of real numbers to do?

The introduction of b and its strange operations might have led you to
believe that I'm trying to introduce some non-standard numbers.
However, b is not the core part of my thoughts. I think I have seen a
core problem which is manifesting itself as different issues in
different areas.

The core issue: Assumption that a line segment is made of distinct
points.

Its various forms:

- Zero, which stands for non-existence of something, can build up a
finite extent, when added to itself infinite times.

- A line segment or an interval of a continuum can be modeled using a
set of distinct real numbers.

- In the above scenario, real numbers are seen as points instead of
extents.

- Boundaries can exist as standalone objects independent of regions.

- The intersection point of two lines is *part* of those lines.

- A point can lie in one of the three areas - inside a region, on the
boundary, outside the region.

- The greatest and smallest numbers may or may not exist in a set of
ordered real numbers representing a finite interval, depending on
whether we chose to include its boundary or not, though it doesn't
make any difference to the interval itself. When you say that boundary
is not a member of an open interval, then what is largest number which
now serves as the boundary of this set? If you don't even know it's
inclusive boundary, have you actually modeled the interval?

- We can accept an infinitely large positive number but can't accept
infinitely small positive number.

- A single real number can have zero or more decimal representations.
We knew every real number has dual decimal representations, but what
is the largest x<1? I think it doesn't have any decimal
representation.

To add more forms of the same issue:

- There is no "next" number and we are comfortable in being not able
to move from a point to the next one, while the motion of a body along
a line is continuous and quite imaginable in physics.

- Likelihood of existence of most real numbers within a finite
interval and points inside a finite region becomes as unfathomable as
infinity, because from any known real number or point, all its
neighbors are as unreachable as infinity. So most real numbers and
points are unrepresentable and unreachable respectively, while they
all still claim to be part of a finite region or interval.

- A line segment is divided into two line segments when cut at a
point. If the cut is loss-less, joining the two parts back should form
the original line segment. However the point at the cut belonged to
both segments after the cut, creating a new point. And when joined
again the extra new point vanishes. Since it is possible to have a
point as intersection of infinite number of lines, it can create
infinite number of new points, but still it doesn't create a new non-
zero extent. This proves that the infinite number of points is not a
sufficient condition to build a non-zero extent. There is something
more to an extent other than points.

To the last point one might say that the point belongs to only one
segment depending how you define the sets that modeled these line
segments. But I'm not talking about line segments and points. Not
sets. Let me see if I can explain it using plain geometry. Line x=2
cuts the line y=3 at (2,3) dividing the second line into two segments
x<=2, y=3 and x>=2, y=3. Now my question is where did the point (2,3)
go?

The line is a set of points. Each segment is a set of points.
The point (2,3) has to go into one set or the other.

Why should I favor one segment over the other for inclusion of the
point?

There is no reason to favour one set over the other, you just
have to choose one set.

Yes. The set model is asking me to choose a segment to put the point
on. But the process of "cutting", however you define it, is symmetric
to both segments and I can't understand the need to make it asymmetric
in the model, except to satisfy the faulty notion that a point is
standalone object.

No. Define the process of "cutting" to be the following

-pick a point, x
-form two sets, A, the set of points greater than
x and, B, the set of points less than x
-add the point x to A or B.

This is not symmetric to both A and B, so your claim that
'"cutting" however you define it is
symmetric to both segments'

is false.

No. It is not false. You came to this conclusion by using sets which
is forcing this asymmetry. You used sets because thats the only way or
best way to define cutting. Did Euclid use sets to define intersection
of line segments? Why shuld you use a different model now?

(You want "cutting" to be symmetric to both
segments, but in this as in many other things
the real world does not care.)

However, if you want you can define the process of cutting
to be

-pick a point, x
-form two sets, A, the set of points greater than
x and, B, the set of points less than x
-add the point x to A and B

This is symmetric with respect to A and B. Yes, when you
make a cut you duplicate a point. So what?

It matters a lot. As I have explained in few possts above, in the case
of infinite number of lines intersecting at a point, this amounts to
creation of infinite number of new points which do not create a new
non-zero extent. This leads to conclusion that infinite number of
points alone do not contribute to extent, and hence a line segment
with finite extent is not just a set of points.

You're not going to hear me, but I'm going to
say this anyway.

The fundamental problem here is that you are not
used to clear thinking. Rather than proceeding from
a clear set of definitions and a clear series of
logical deductions, you write posts full of ill-
defined words and "conclusions" based on no
rule but the way you think things ought to be.
Every time you write the word "therefore" it is
a flag that what you are about to say does not
actually follow from the premise.

But because you are unused to clear thinking, you
are unable to recognize your unclearness, though it
is painfully obvious to all of us.

OK, editorial over. Now I'm going to address your
misconceptions about intersection.

Here is a set of objects in a small universe:
Q = {red square, blue circle, red triangle, purple
square}

Here is a subset, the set of red objects:

R = {red square, red triangle}

Here is another subset, the set of square objects:

S = {red square, purple square}

Now here is the intersection of R and S:

R intersect S = {red square}

The fact that R intersect S is non-empty,
does not mean that I have duplicated the red
squares. There aren't two of them, just one,
which is a member of both R and S.

- Randy
.

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  • Re: The infintely small number b
    ... core problem which is manifesting itself as different issues in ... when added to itself infinite times. ... both segments after the cut, ... But the process of "cutting", however you define it, is symmetric ...
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  • Re: The infintely small number b
    ... core problem which is manifesting itself as different issues in ... when added to itself infinite times. ... all still claim to be part of a finite region or interval. ... both segments after the cut, ...
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