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

Am 21.03.2025 um 20:37 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.