An uncountable countable set
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From: Virgil on 25 Jun 2006 15:40 In article <1151227367.193817.313740(a)b68g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > mueckenh(a)rz.fh-augsburg.de schrieb: > > > Virgil schrieb: > > > > > > > If an almighty God cold not arrange to have the sun appear to stand > > > still in the sky without disastrous side effects, such a God would > > > hardly be almighty. > > > > Of course not. He would not even be able to construct a stone so heavy > > that he was unable to lift it. > > > > If such a God first says: It is not good for the man to stay alone, so > > we will create a women for him. > > Oh, HE created only one woman, of course. > Is "mueckenh" now reduced to answering his own posts?
From: Dik T. Winter on 25 Jun 2006 22:24 In article <1151079612.527394.67300(a)i40g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > If actual infinity does exist, then also an actually infinite set of > transpositions must exist. Cantor knew that. > > G E O R G C A N T O R: GESAMMELTE ABHANDLUNGEN MATHEMATISCHEN UND > PHILOSOPHISCHEN INHALTS Mit erl=E4uternden Anmerkungen sowie > mitmErg=E4nzungen aus dem Briefwechsel Cantor - Dedekind Herausgegeben > von ERNST ZERMELO Nebst einem Lebenslauf Cantors von ADOLF FRAENKEL1966 > GEORG OLMS VERLAGSBUCHHANDLUNG HILDESHEIM > > p=2E 214: "Die Frage, durch welche Umformungen einer wohlgeordneten Menge > ihre Anzahl ge=E4ndert wird, durch welche nicht, l=E4=DFt sich einfach so > beantworten, da=DF diejenigen und nur diejenigen Umformungen die Anzahl > unge=E4ndert lassen, welche sich zur=FCckf=FChren lassen auf eine endliche > oder unendliche Menge von Transpositionen, d. h. von Vertauschungen je > zweier Elemente." > Cantor admitted infinitely many transpositions. I think he meant > countably many. But more aren't needed. No he did not. Pray read again. He was talking about the kind of transformations of well-ordered sets for which the *number* of elements would change. Or is well-orderedness part of the number of elements? I think not. And indeed any number of interchanges can be performed on a set without changing the number of elemens. I think what he is arguing here (about a field that needed new exploration) that at least for well-ordered sets any number of transpositions (and in fact any finite or infinite permutation) would not change the number of elements. Apparently it was not clear to him if that was also the case for sets that could not be well-ordered. That is the best I can make out of your quote. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 25 Jun 2006 22:44 In article <1151079833.849194.73370(a)c74g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > A *surjective* mapping does exist, but it is not f. There exists a > > > surjective mapping g -> M(f). > > > > Indeed. But your short list does not show anything at all, I wonder what > > I have to do with it. I think you intend to show something about the > > diagonal number of a list of reals. But we were talking about a mapping > > from N to P(N). So please keep to the subject. > > Don't you recognize that K(f) takes exactly the same function as the > diagonal number D in Cantor's second argument? f enumerates the list > numbers. If f can arbitrarily be replaced by g, then this proof is > invalid and shows only what it does show in fact, namely the > countability of all list numbers including all diagonal numbers which > can be constructed. I have diffculty making sense from this. But let me try, and the answer is (in my opinion) no. In the case of K(f) and M(f) I start with a bijection between N and S, where S is the set of finite subsets of N. You state: but K(f) is not in it. I answer: no obviously not, K(f) is not in S, and I constructed f as a bijection between N and S. Not you state: so M(f) is not countable, and I say, but it is, and I construct a bijection (g) between N and M(f). So you utter, but K(g) is not in it, and I answer, no obviously, g was a bijection between N and M(f) and K(g) is not an element of M(f). Next you state, but g is not a bijection between N and M(g), and my answer is: yes, that is obvious, because g is not even a mapping between N and M(g) because K(f) (which is in the image of g) is not an element of M(g). Let's be generous and let you have said that it was not a bijection between N and M(f, g) (where M(f, g) is appropriately defined). And I state, of course not, that was not the requirement, the requirement was a bijection between N and M(f). So each time I come with a bijection you give a different set to which it is not a bijection. That is obvious. Now consider the case of a bijection between N and P(N). You state: f is a bijection, and I state: no, it is not a bijection because H(f) (H standing for Hessenberg set) is not in the image, while it is in P(N). OK you state, lets reformulate to g such that H(f) is in the mapping, and I say, no, now H(g) is not in the image. You can continue, but everytime I can state a set that is not in the image and should be there. Note that in this case the target does *not* change. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 25 Jun 2006 22:53 In article <1151080005.054210.206040(a)r2g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > > Because, in case f should be surjective, a number k need be contained > > > in a set in which it must not be contained. > > > > Yes, and that shows that whatever mapping f you give, it is simply not > > surjective. You like selective quoting? This was about a surjective mapping N to P(N). > And if I map all natural numbers on a set with only two elements, the > empty set and that single set K? What single set K? This makes no sense, especially when we were talking about mappings from N to P(N). > The mapping is not surjective? Why > should it not be *surjective*, if there are far more elements in the > source than in the target? The mapping is impredicably defined. I would say the target is ill-defined. Give source and target first, and than we can talk about mappings. When the target depends on the mapping it is not a set, and there is no way to talk about mappings. In the case N -> P(N) the target is well-defined and independent of any possible mapping. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 25 Jun 2006 23:09
In article <vmhjr2-F4EEE8.16301223062006(a)news.usenetmonster.com> Virgil <vmhjr2(a)comcast.net> writes: > In article <e7hk0j01rlj(a)drn.newsguy.com>, > stevendaryl3016(a)yahoo.com (Daryl McCullough) wrote: > > mueckenh(a)rz.fh-augsburg.de says... > > >Virgil schrieb: > > > > > >> > Cantor said: A well-ordered set remains well-ordered, if finitely many > > >> > or infinitely many transpositions are executed. .... > > >If actual infinity does exist, then also an actually infinite set of > > >transpositions must exist. Cantor knew that. > > > > The issue is whether "A well-ordered set remains well-ordered, if ... > > infinitely many transpositions are executed." That's a provably false > > statement, and I doubt that Cantor ever said it. > > Actually, even if it IS true, it would not allow transmutation of a set > with a first element into one without a first element, at least if the > transpositions are applied sequentially, as they must be. Mueckenheim's statement was a complete misreading of the source he quoted in another article. I abviously can make jokes about my ability to read German and his ability to do so, but the quote shows (without a doubt) that Cantor wrote that an arbitrary number of interchanges of a well-ordered set would not change *the number of elements* in such a set. It does *not* state that well-orderedness is preserved. What I think this means (and I am not very deep in set-theory) is that it is possible to compare the number of elements in well-orderable sets, and so also possible to assign cardinal numbers to such sets. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |