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From: Moebius <invalid@example.invalid>
Newsgroups: sci.math
Subject: Re: The reality of sets, on a scale of 1 to 10 [Was: The
 non-existence of "dark numbers"]
Date: Fri, 21 Mar 2025 20:46:59 +0100
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Am 21.03.2025 um 20:37 schrieb Moebius:
> Am 21.03.2025 um 19:48 schrieb Alan Mackenzie:
>> WM <wolfgang.mueckenheim@tha.de> wrote:
> 
>>> Learn that [...] Cantor [once] has [uttered] that the positive 
>>> numbers have more
>>> reality than the even positive numbers. He said that is not in 
>>> conflict with the identical cardinality of both
>>> sets. And he was right!
>>

>> I doubt very much Cantor said such rubbish.

> Actually, WM is right here. But the notion of "more reality" clearly 
> wasn't meant as a technical term (by Cantor). He -Cantor- was just 
> trying to explain the mathematical fact that 2IN is a PROPER subset of 
> IN, while both sets still have the same cardinality. (I'd dare to bet 
> that this was the only time he ever used that phrase in this context.)

Her's the original quote:

"Sei M die Gesamtheit (nü) aller endlichen Zahlen nü, M' die
Gesamtheit (2nü) aller geraden Zahlen 2nü. Hier ist unbedingt richtig, daß
M seiner Entität nach /reicher/ ist, als M'; enthält doch M außer den
geraden Zahlen, aus welchen M' besteht, noch außerdem alle ungeraden
Zahlen M''. Andererseits ist ebenso unbedingt richtig, daß den beiden
Mengen M und M' nach Nr. 2 und 3 /dieselbe/ Kardinalzahl zukommt. Beides
ist sicher und keines steht dem andern im Wege, wenn man nur auf die
Distinktion von /Realität/ und /Zahl/ achtet. Man muß also sagen: /die
Menge M hat mehr Realität wie M', weil sie M' und außerdem M'' als
Bestandteile enthält; die den beiden Mengen M und M' zukommenden
Kardinalzahlen sind aber gleich/." (G. Cantor)

Google Translator:

"Let M be the totality (nu) of all finite numbers nu, and M' the 
totality (2nu) of all even numbers 2nu. Here it is absolutely true that 
M is /richer/ than M' in its essence [entity]; after all, M contains, in 
addition to the even numbers of which M' consists, all the odd numbers 
M''. On the other hand, it is equally absolutely true that the two sets 
M and M', according to no. 2 and 3, have /the same/ cardinal number. 
Both are certain, and neither precludes the other, if one only pays 
attention to the distinction between /reality/ and /number/. One must 
therefore say: /the set M has more reality than M' because it contains 
M' and, in addition, M'' as components; but the cardinal numbers 
belonging to the two sets M and M' are equal/."

> Hint: WM is all about words.