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

On 10/4/24 5:53 AM, WM wrote:

On 04.10.2024 10:29, joes wrote:

Am Thu, 03 Oct 2024 20:25:07 +0200 schrieb WM:

On 02.10.2024 22:07, Jim Burns wrote:

On 10/2/2024 7:12 AM, WM wrote:

On 01.10.2024 22:05, Jim Burns wrote:

>

For every nonzero distance d from 0 there are ℵ₀.many unit.fractions
closer than d

That means there is no d by what you can distinguish ℵ₀ unit
fractions?

You (WM) require us to guess what you mean by 'distinguish'.

You distinguish two unit fractions 1/n and 1/m if you place a point in
distance d from 0 between them: 1/n < d < 1/m.

Which is always possible.

 For a finite collection of unit fractions. For almost all unit fractions it is impossible.

So, you ADMIT that you your concept of the "Natural Numbers" isn't actually the full INFINITE set of Natural Numbers but just some (possibly indefinite) finite subset of it.
That you concept of "Infinity" isn't actually infinity, but a poor substitute for it.
Note, the term "Almost all" is defined for infinite sets, and no finte subset of an infinite set is "Almost all" of it. Almost All say that there is a FINITE set of exceptions, and thus there is a last exception, not a last member of the set.

>

There is no d between ℵ₀.many.smaller.unit.fractions and
slightly.fewer.than.ℵ₀.many.smaller.unit.fractions.

That means you cannot distinguish ℵ₀ smaller unit fractions.

Misconception: there is no „slightly fewer than ℵ₀”.

 Of course there is fewer, namely zero. If slightly fewer is not possible, then ℵ₀ unit fractions sit at one point.

Nope, Zero is not "slightly fewer" than infinite.
 From any finite point, there ARE Aleph_0 points below it that are unit fractions.

 Regards, WM