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

WM wrote on 10/8/2024 :

On 08.10.2024 12:04, Alan Mackenzie wrote:

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

>

All unit fractions are points with uncounably many points between each
pair.

 Yes, OK.
 

Hence all must be visible including the point next to zero, but they
are not.

 There is no point next to zero.

>
Points either are or are not. The points that are include one point next to zero.

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

 

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.

 

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.

>
Of course, the core is dark.
>

 The
"core" is thus empty.

>
The infinite sets contain what? No natural numbers? Natural numbers dancing around, sometimes being in a set, sometimes not? An empty intersection requires that the infinite sets have different elements.

An infinite intersection requires that all elements are in each set. The intersection is empty because not all elements are in every set. Each successive set in your sequence of sets removes another element from consideration - until they are all gone.