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

On 10.10.2024 13:28, joes wrote:

Am Wed, 09 Oct 2024 15:29:17 +0200 schrieb WM:

On 09.10.2024 14:29, joes wrote:

Am Wed, 09 Oct 2024 11:41:31 +0200 schrieb WM:

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:

 

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.

WDYM, all numbers in the segments are indices.

All numbers n get indices of endsegments E(n).

Never can all numbers be used.
 

But what about the intersection between all infinitely many segments?

It is empty.

What about the core?
 

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.

What does this mean for the infinite intersection?

It is empty because all numbers are becoming indices and then get lost,
one by one.

It cannot be empty because there are always numbers remaining.

Then never all fractions can be indexed because always numbers are remaining in 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 2/4, 3/3, 4/2, 5/1, 1/6, 2/5, 3/4, 4/3, 5/2, 6/1,  ...

An endsegment is a set. All elements must exist. That requires actual
infinity. In potential infinity numbers come into being - and never all.

Do you mean there are no potentially infinite sets?

That is a matter of defiition. In my books I use only potetial infinity. I never mention actual infinity. But I talk about sets of numbers like ℕ, 𝔾, ℚ, ℝ.
But set theorists claim that sets must be unchangeable.
When in potential infinity "all" natural numbers are doubled, then greater, i.e., new natural numbers are created. Therefore the set ℕ has changed.
Regards, WM