Re: The reality of sets, on a scale of 1 to 10 [Was: The non-existence of "dark numbers"]

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:

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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!

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I doubt very much Cantor said such rubbish.

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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.)

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Her's the original quote:
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"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)
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Google Translator:
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"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.

Hint: WM is all about words.