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

In article
<026b833f-1718-4fc3-8691-84ac2923a926@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
venkat.6123@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"?
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".

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

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