Re: The reality of sets, on a scale of 1 to 10

Moebius <invalid@example.invalid> wrote:

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)

What was the context of this quote?  Was it a letter to a fellow
mathematician, or a in a published work, or what?

It seems Cantor was fumbling around, trying to get a handle on some
concept, and called it Realität for want of a better word.  With the
benefit of over a century of development, we can see there is no need for
this concept, which might not even be consistent.

OK, Cantor said/wrote this.  It is still rubbish from a modern point of
view.  It is only to be expected that the pioneers of a new field, along
with valid developments, also make mistakes.  "Potential infinity" was
another such mistake.  We should accept that there were falsehoods among
these original ideas, and disregard those falsehoods today.  The Boeing
aircraft company does not rely any more on calculations made by the
Wright brothers way back when.  Neither should we pretend that Cantor was
an infallible god when it comes to set theory.

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.

Yes.

--
Alan Mackenzie (Nuremberg, Germany).