Path: ...!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!eternal-september.org!.POSTED!not-for-mail From: Moebius 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 Organization: A noiseless patient Spider Lines: 58 Message-ID: References: <0b8644b2-7027-420e-b187-8214daaf9e3b@att.net> Reply-To: invalid@example.invalid MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8bit Injection-Date: Fri, 21 Mar 2025 22:10:44 +0100 (CET) Injection-Info: dont-email.me; posting-host="1fe1706e9f690cb83ee95849cd29f0ec"; logging-data="2504232"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX1+OZEWmIFH7x8jNE4bnN238" User-Agent: Mozilla Thunderbird Cancel-Lock: sha1:zcimg8itWzCth9zaoYXYz4XhEk0= Content-Language: de-DE In-Reply-To: Bytes: 4468 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 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.