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

On 08.10.2024 21:17, joes wrote:

Am Tue, 08 Oct 2024 17:40:50 +0200 schrieb WM:

On 08.10.2024 15:36, joes wrote:

Am Tue, 08 Oct 2024 12:40:26 +0200 schrieb WM:

On 08.10.2024 12:04, Alan Mackenzie wrote:

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

>

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.

But not the point inbetween?

If it exists then this point is next to zero.

Ah, then the former point wasn’t the one next to zero. Same goes for this
one. There are always infinitely many points between any two reals.
 

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.

These are infinite sets: {2, 3, 4, …}, {3, 4, 5, …}, {4, 5, 6, …}.
They contain all naturals larger than a given one, and nothing else.
Every natural is part of a finite number of these sets (namely, its own
value is that number). The set {n+1, n+2, …} does not contain n and is
still infinite; there are (trivially) infinitely many further such
sets. All of them differ.

All of them differ by a finite set of numbers (which is irrelevant) but
contain an infinite set of numbers in common.

Every *finite* intersection.

>
As long as infinitely many numbers are captivated in endsegments, only finitely many indices are available, and the intersection is between finitely many infinite endsegments.
>

Think about it this way: we are taking the limit of N\{0, 1, 2, …}.

>
In the limit not a single natural number remains, let alone infinitely many.