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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 22:10:43 +0100
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Am 21.03.2025 um 20:51 schrieb Moebius:
> Am 21.03.2025 um 20:46 schrieb Moebius:
>> 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/."
> 
> Well, what can we say? Set theory in its infancy.

Moreover, Cantor wasn't THAT good as a philosopher of mathematics. Frege 
was MUCH better.

>>> Hint: WM is all about words.