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

Am Mon, 14 Oct 2024 12:26:31 +0200 schrieb WM:

On 12.10.2024 22:47, Richard Damon wrote:

On 10/12/24 1:49 PM, WM wrote:

On 10.10.2024 21:54, joes wrote:

Am Thu, 10 Oct 2024 20:53:07 +0200 schrieb WM:

On 10.10.2024 20:45, Alan Mackenzie wrote:

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

>

If all natnumbers are there and if 2n is greater than n, then the
doubled numbers do not fit into ℕ.

For any finite n greater than zero, 2n is greater than n.  The same
does not hold for infinite n.

There are no infinite n = natural numbers.

Exactly! There are furthermore no infinite doubles of naturals (2n).

But the doubles are larger. Hence after doubling the set has a smaller
density and therefore a larger extension on the real line. Hence not
all natural numbers have been doubled.

Doubling does not create a second infinity.

The doubles are larger than the element they replace, but that value
was always in the set to begin with, so it never creates a "new" term.

Doubling creates terms which were not in the doubled set.

Not with even numbers.

When doubled then 2n > n. If a set of natural numbers is doubled,
then the results cover a larger set than before..

Additionally: if n is finite, so is 2n. It cannot go beyond w.

Then there is no complete set. The doubling can be repeated and
repeated. Always new numbers are created. Potential infinity.

No, we are taking the complete, actually infinite set which reaches
to "before" w.

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Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.