Cantor Confusion
in [Math]
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From: Virgil on 17 Feb 2007 13:41 In article <1171702351.590890.177460(a)h3g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 16 Feb., 15:58, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > In article <1171615110.930410.270...(a)s48g2000cws.googlegroups.com> > > mueck...(a)rz.fh-augsburg.de writes: > > > > > On 15 Feb., 14:05, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > ... > > > > Indeed, my error. So your comment: > > > > > When seen as a set of curly brackets it has 3 at the left sinde and > > > > > 3 > > > > > at the right. > > > > was actually completely irrelevant. Let's get on with the actual > > > > representation of 3: {{{{}}}}. =20 > > > This is not a representation of 3 other than in a perverted system, > > > which calls 0 the first number, 1 the second and so on. Of course > > > {{{{}}}}, or better and easier {{{{, denotes the fourth number which > > > is 4 and not 3. > > > > Can you tell me a form of set theory where 0 is *not* the first ordinal > > or cardinal number? If so, how many elements does the empty set have > > in such a system? > > 0 may be the first (or better the zeroest) ordinal or cardinal number > (if you wish to have the empty set in the theory). Nevertheless it is > not the first natural number and not a natural number at all. WM has not the authority to determine what are and want are not allowed to be natural numbers, however much he may thing he does. > > > > So how many natural numbers precede the first natural numbers? We are > > counting natural numbers, so it should be a natural number? > > Natural numbers are counting the elements of natural sets, i.e., of > sets which exist in reality (in nature, as Cantor woud have said). The empty set exists quite naturally in every set theory. > An unnatural set cannot have a natural number of elements. Since {} is quite naturally the natural intersection of any two disjoint sets, it is quite natural in every set theory, so must have a natural number of members.
From: G. Frege on 17 Feb 2007 20:47 On 17 Feb 2007 00:52:31 -0800, mueckenh(a)rz.fh-augsburg.de wrote: > > 0 may be the first [...] ordinal or cardinal number (if you wish to have > the empty set in the theory). Nevertheless it is not the first natural > number and not a natural number at all. > That's fascinating stuff, man! Where did you get that from? :-o > > Natural numbers are counting the elements of natural sets, > *lol* Another M�ckenheim invention: /natural sets/! :-) > > i.e., of sets which exist in reality ([i.e] in nature [...]). > Wow. Sets exist in reality, in nature? With other words, sets are physical entities? Interesting. The empty set... > > ...a set which "also streng genommen als solche gar nicht vorhanden > ist." (Cantor) > > ...a set which "ist also verm�ge der Definition von S. 4 gar keine > Menge." (Fraenkel) > Cantor is pre-axiomatic. And Fraenkel is referring to Cantor's pre-axiomatic "definition" of the notion of /set/. But *we* (well except you, I guess) are talking about _axiomatic set theory_ here. > > An unnatural set cannot have a natural number of elements. > Is this an axiom (or a basic principle) of your M�ckenmathics? F. -- E-mail: info<at>simple-line<dot>de
From: G. Frege on 17 Feb 2007 20:53 On Sat, 17 Feb 2007 11:41:13 -0700, Virgil <virgil(a)comcast.net> wrote: > > The empty set exists quite naturally in every set theory. > Well, actually we might exclude the empty set from our theory. (Of course, one might ask WHY we should want to do that. Just to make thinks more complicate?) > > Since {} is quite naturally the natural intersection of any two disjoint > sets, it is quite natural in every set theory [...] > Right. In a set theory without an empty set disjoint sets would not have an intersection. (With other words, there wouldn't be an inter- section of disjoint sets.) F. -- E-mail: info<at>simple-line<dot>de
From: Dik T. Winter on 17 Feb 2007 20:51 In article <1171702351.590890.177460(a)h3g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > On 16 Feb., 15:58, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: .... > > > This is not a representation of 3 other than in a perverted system, > > > which calls 0 the first number, 1 the second and so on. Of course > > > {{{{}}}}, or better and easier {{{{, denotes the fourth number which > > > is 4 and not 3. > > > > Can you tell me a form of set theory where 0 is *not* the first ordinal > > or cardinal number? If so, how many elements does the empty set have > > in such a system? > > 0 may be the first (or better the zeroest) ordinal or cardinal number > (if you wish to have the empty set in the theory). Nevertheless it is > not the first natural number and not a natural number at all. But you said "perverted system which calls 0 the first number". The only place where it is called the first *natural* number is in Bourbaki and its followers, but that is only a simple renaming of the term "natural number". But following your reasoning, {{{}}} is the third number, which is 3. BTW, I can quote you as saying: > > > > {{{}}} > > > It was page 93 of my book. > > I have seen it earlier than that. > By the way, above is only number 2 given. So earlier you said it is 2. What is it? > > So how many natural numbers precede the first natural numbers? We are > > counting natural numbers, so it should be a natural number? > > Natural numbers are counting the elements of natural sets, i.e., of > sets which exist in reality (in nature, as Cantor woud have said). If my house contains no dogs, in what way does the set of dogs in my house not exist in reality? > > > |{1,2,3,...}| = ... i.e. potentially infinite, not fixed, capable of > > > growing without bound, denoted by oo but not by a fixed number omega. > > > > And |{}| = ? > > A set which "also streng genommen als solche gar nicht vorhanden > ist" (Cantor) > A set which "ist also verm�ge der Definition von S. 4 gar keine > Menge" (Fraenkel) > An unnatural set cannot have a natural number of elements. If you think {} to be an unnatural set, so be it (that is not mathematics, because there is no mathematical definition of natural set). But if somebody asks how many coins I have in my purse, he is asking for the cardinality of the number of coins in my purse. And I can correctly answer 0 at some times. In that case the set is the set of coins in my purse, and that can be empty (and is quite often in reality). According to current definitions and axioms in set theory, {} *is* a set. The distinction "natural" vs. "unnatural" set is not known. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: G. Frege on 17 Feb 2007 22:34
On Sun, 18 Feb 2007 01:51:16 GMT, "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote: > > The distinction "natural" vs. "unnatural" set is not known. > At least in modern mathematics. Though it seems to play a certain role in WM's M�ckenmathics. F. -- E-mail: info<at>simple-line<dot>de |