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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: How many different unit fractions are lessorequal than all unit
 fractions?
Date: Sat, 5 Oct 2024 11:39:15 +0200
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On 05.10.2024 10:38, Moebius wrote:
> Am 05.10.2024 um 10:10 schrieb joes:
>> Am Fri, 04 Oct 2024 20:25:35 +0200 schrieb WM:

> Hint: There is no ordinal number o such that o + 1 = ω. Hence there is 
> especially no ordinal number denoted by "ω-1" such that ω-1 + 1 = ω.

May be. But in actual infinity there is no gap before ω.

> 
>>> I understand that the sum 1+2+3+... > 1. 

> Die gewöhnliche Summe von "1 + 2 + 3 + ... " IM 
> KONTEXT DER REELLEN ZAHLEN ist NICHT DEFINIERT,

Jedenfalls ist sie nicht negativ.

> Allerdings kann man IR auch etwas erweitern, nämlich um die beiden 
> Elemente {-oo, oo} um auf diese Weise die "extended reals" IR* zu erhalten.
> 
> IN DIESEM Kontext kann man nun tatsächlich schreiben:
> 
>           "1 + 2 +  3 + ... = oo"
> 
> mit oo > 1 .

In actual infinity ω-2 is a natural number and 3 + ω-2 = ω+1.

> 
> IN DIESEM KONTEXT kann man also
> 
>           "1 + 2 +  3 + ... > 1"
> 
> tatsächlich hinschreiben und es ist dann sinnvoll und korrekt.

Na also.

Gruß, WM