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

On 03/21/2025 03:15 PM, Moebius wrote:

Am 21.03.2025 um 22:25 schrieb Alan Mackenzie:

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?

>
I really don't know (can't tell).
>

It seems Cantor was fumbling around, trying to get a handle on some
concept, and called it Realität for want of a better word.

>
Indeed. As I said: "Set theory in its infancy."
>
"Substance" might have worked too, imho. :-P
>

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.

>
Right.
>

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.

>
Again: "Moreover, Cantor wasn't THAT good as a philosopher of
mathematics. Frege was MUCH better."
>
BUT: The notion of "potential infinity" was not invented by Cantor or
his contemporaries, actually it goes back to _Aristotle_. Cantor just
voted for the (existence of the) actual infinite as represented by his
infinite sets.
>
"Cantor's work was well received by some of the prominent mathematicians
of his day, such as Richard Dedekind. But his willingness to regard
infinite sets as objects to be treated in much the same way as finite
sets was bitterly attacked by others, particularly Kronecker. There was
no objection to a 'potential infinity' in the form of an unending
process, but an 'actual infinity' in the form of a completed infinite
set was harder to accept."
>
[H.B. Enderton: "Elements of set theory", Academic Press, New York
(1977) p. 14f]

"Set Theory" is simply any of a small variety of theories of one relation.
The other usual particular is "Ordering Theory", and helps
explain why "counting" and "numbering" are _two different things_.
Aristotle had actual infinities, and Zeno had motion.
"Descriptive Set Theory" is quite usual.
It's called model theory, and it's equi-interpretable with proof theory.
If you have a theory, ....