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

Am 08.10.2024 um 12:04 schrieb Alan Mackenzie:

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

On 07.10.2024 14:11, Alan Mackenzie wrote:

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

A shrinking infinite set which remains infinite has an infinite core.

He's talking about the infinitely many infinite "endsegments" (of natural numbers) here. (He always comes back to this topic.)
Say, E_n := {n, n+1, n+2, n+3, ...}  (n e IN)
Then clearly
(a) {E_n : n e IN} is infinite
(b) An e IN: E_n is infinite
(c) INTERSECTION_(n e IN) E_n = {}.
Not so in Mückenheims world. (He claims that there "have to be" E_n's which are not infinit, if INTERSECTION_(n e IN) E_n = {}. <facepalm>.)

Again, no.  There is no such thing as a "core", here.  Each of these sets
has an infinitude of elements.  No element is in all of these sets.

Right.

Try to think better. A function of sets which are losing some elements
but remain infinite, have the same infinite core.

 That is untrue.  For any element which you assert is in the "core", I
can give one of these sets which does not contain that element.  The
"core" is thus empty.

Right.

That argument is absolutely definite, a logical necessity.

Which argument? :-P