Re: Is one-to-one mapping valid for comparing infinite-sized sets?

In article
<89deac13-e0bd-4649-b712-e00f0af7016b@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
venkat.6123@xxxxxxxxx wrote:

On Sep 20, 3:43 am, Virgil <Vir...@xxxxxxxxx> wrote:
In article
<026b833f-1718-4fc3-8691-84ac2923a...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,

 venkat.6...@xxxxxxxxx wrote:
On Sep 20, 2:56 am, Virgil <Vir...@xxxxxxxxx> wrote:
"If the extent around points were ever filled,
 then some points in R would be adjacent."

Is that true according to you?

Yes, if "filled" means "filled by points".

Do you mean that the midpoint between two points need not always be
a
point?

No. I don't mean it. You can always identify more points anywhere
inside an extent.

Can you tell us what an "extent" is, or at least why we should pay any
attention to "extents"?
So far, they seem to be totally useless, at least as far as any
mathematical issues are concerned.



Since the extent around a point is always
unfilled, then ... what could come next?

Its the extent !!! right next to every point.

In what direction(s) from the point does an extent extend?





What comes next AFTER that extent? The same extent again, repeating
itself endlessly? That is what you seem to be saying.

No. Whenever you imagine or identify a point, you are creating one
more new extents around it.

Just how far does one of your alleged 'extents' extend?

If that distance is greater than zero, there is another point within
that same extent. if the distance is not greater than zero, then your
alleged 'extents' do not extend at all.

The magnitude of an extent is always greater than zero. That is the
essence of an extent. Yes, you can imagine any more points inside it,
possibly breaking the extent into more extents.

Of what mathematical use are your "extents"?

The usefulness of extents is in building more natural and robust model
of division of continuum as a fractal that doesn't change structure
with change in scale. This fractal structure is exposed only when the
continuum is being cut into parts.

Since the fractal nature of "the continuum" as the set of reals is
easily deduced without any reference to "extents", for example by the
bijectability of any two real open intervals, regardless of length,
there is no mathematical or philosophical need for those "extents".

The current model (set of points) changes it structure while we reduce
the scale.

On the contrary, its geometric "structure" is totally independent of
scale, as the Archimedean property of the reals demonstrates.


The line segments and points converge into only points
according this model. There is no explanation or reason for this
change in structure except hiding behind infinite and hoping its
properties would convert a line segment into a point just by cutting
the segment.

Those, like you, who would prefer a "line segment"(closed real
interval) of 0 length to contain more than one point will have to
explain how that is to work in a complete Archimedean field.

Is there any mathematical task that cannot be as easily completed
without them? If so, give us an example of some mathematical task which
is made easier by considering your "extents".

I don't know of the tasks. But I knew that the set theoritic model of
continuum is unnatural and asymmetric.

Only in the sense that all mathematics is "unnatural" by being
abstracted from but not part of the natural world.


My fractal model should provide
a better means for the any tasks that are currently done using the set
model. Do you have any specific tasks in mind?

Try proving a proof of Rolle's theorem using your model.


So far, they have been both mathematically evanescent, and
mathematically purposeless.

I just can't imagine continuum is a set of points.

I can't imagine it as anything else.

Except, of course, as a set of numbers. There is is the existence of
least upper and greatest lower bounds for bounded sets which provides
the continuity of that continuum.


For me, it is like
saying a region is a set of borders. I get lot of other questions -
what is a plane? a set of lines or set of points? If it is a set of
points, what sort of difference in arrangement of these points made it
a plane instead of line? If points can have different arrangement, are
they still dimensionless? When we pile up points, sometimes it chooses
to become a plane and some other times a line? Are we building line
with points or dividing an existing line into points? Are they
reversible processes? I mean, if you break up a line into points, then
you can build a line with points? Then do you build a plane
differently than a line? etc.

All of these questions become trivialities once you learn about real
vector spaces and affine spaces.
.

Relevant Pages

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    (sci.math)
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