Re: How many different unit fractions are lessorequal than all unit fractions? (infinitary)

Am Wed, 09 Oct 2024 18:56:29 +0200 schrieb WM:

Am 09.10.2024 um 18:12 schrieb Alan Mackenzie:

WM <wolfgang.mueckenheim@tha.de> wrote:

 

Dark numbers don't exist, or at least they're not natural numbers.
There is no number in each and every end segment of N.

True. But those endsegments which have lost only finitely many numbers
and yet contain infinitely many, have an infinite intersection.

End segments don't "lose" anything.  They are what they are, namely
well defined sets.  Note that your "True" in your last paragraph,
agrees that the intersection of all end segments is empty, which you
immediately contradict by asserting it is not empty.

You do not understand the least! The intersection of all endsegments is
empty. The intersection of infinite endsegments is infinite.

What does this --------------^ specify exactly that distinguishes it from
the preceding sentence? Especially since both all segments are infinite,
and there are infinitely many of them.

Note: The shrinking endsegments cannot acquire new numbers.

An end segment is what it is.  It doesn't change.

But the terms of the sequence do. Here is a simple finite example:
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
{2, 3, 4, 5, 6, 7, 8, 9, 10}
{3, 4, 5, 6, 7, 8, 9, 10}
{4, 5, 6, 7, 8, 9, 10}
{5, 6, 7, 8, 9, 10}
{6, 7, 8, 9, 10}
{7, 8, 9, 10}
{8, 9, 10}
{9, 10}
{10}
{ } .
Theorem: Every set that contains at least 3 numbers (call it TN-set)
holds these numbers in common with all TN-sets.
Now complete all sets by the natural numbers > 10 and complete the
sequence.

Then you get different sets, which weren't the ones you were trying to
reason about.

The completion of the above sets does not change the principle:
Non-empty inclusion-monotonic sets like infinite endsegments have a
non-empty intersection. All endsegments have an empty intersection.

You should really be more careful with your phrasing. Intersection with
what?

--
Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.