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

Am Tue, 08 Oct 2024 12:51:03 +0200 schrieb WM:

On 07.10.2024 17:18, joes wrote:

Am Mon, 07 Oct 2024 11:08:33 +0200 schrieb WM:

On 07.10.2024 10:05, joes wrote:

Am Mon, 07 Oct 2024 09:41:23 +0200 schrieb WM:

On 06.10.2024 17:48, Alan Mackenzie wrote:
even an unbounded sequence does not get longer when shifted by one
step.

Nor does it get shorter, it stays infinite.

It keeps all its elements but not more.

„More” being a different kind of infinity, namely at least uncountable.

Nonsense.

Only according to your broken concept of cardinality, by which N u {a}
is „bigger” than N, and N\{1} u {a} can’t even be compared.

All ω+k are equally infinite.

Nonsense.

I believe it is called the order type.

Bijection is not about completeness, countability is.

Of course stopping after a finite number, which potential infinity
seems to mean, is not „complete” in that sense. Hilbert’s Hotel is
actually infinite, it already holds infinite guests.

Name them by all the natural numbers. Then no further guest can
appear.

It can, if I begin numbering with 2. The cardinality of N\{1} can’t be
finite.

Cardinality is nonsense.

Isn’t N\{1} finite? It has ω-1 elements.

All of them can at once move to the next room,

All rooms are enumerated by all the natural numbers. Hence there is no
chance to move. Only in potential infinity there is.

Huh? They are all fixed, we can move them „rigidly”.

That shows my point. Infinite sets can be moved. 0.999...999 moved gives
9.99...9990.

You have not indicated what this notation means. Where does the zero come
from?

Another point is this: [0, 1) moved gives (0, 1].

Can you generalise this?

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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.