Cantor Confusion
in [Math]
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From: Dik T. Winter on 9 May 2007 22:02 In article <1178722003.108344.38050(a)y80g2000hsf.googlegroups.com> WM <mueckenh(a)rz.fh-augsburg.de> writes: > On 9 Mai, 03:52, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: .... > > > Sometimes it is preferable to have a calm atmosphere. > > > > I have first to sign up, next I get probably an interface that can only be > > handled with a mouse. No, thanks, I have already too many problems with > > mouse use. > > I have got the same interface as here. I have not. I do not use a web-browser for discussions. But you apparently do not understand the principles of Usenet newsgroups. > I will answer there. I will not see it, so you clearly cut short the discussion. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Virgil on 9 May 2007 22:13 In article <1178739059.924059.128150(a)y80g2000hsf.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > We can state: There are sets which are said countable but which do > *not* allow for a bijection or one-to-one correspondence with the set > N of all natral numbers. You can state anything you want, but if you call a set countable that does not allow either an injection from it into N or a surjection from N onto it, you are in violation of any standard definition of "countable". > > > There does not exist anyting in mathematics, unless it is finitely > > > definable. > > > > Perhaps not in WM's mathematics. > > If you cannot finitely define an entity in mathematics, then it does > not exist. Speak only for yourself. You have no business trying to tell others what they are not allowed to do. > > > In particular, any infinite definition is not a definition, > > > because there is a definition of "definition" which says: Definition: > > > A definition has an end, i.e., a last word and a point. > > > > Where does that definition of 'definition' occur? Certainly not in any > > English language nor mathematical dictionary. > > Do you so gossly underestimate English textbooks that you think they > state: "A definition must not have an end, i.e., there must not be a > last word." I merely doubt that all of them say explicitly that a definitions must have an end, i.e., a last word. Does WM have any examples which do say that? > Or do you think, "definition" should not be defined at all? > Or do you disregard tertium non datur? WM seems very much to want to disregard tertium non datur in the matter of whether there can be a 'complete' set of all natural numbers. > Or do you refrain from any thinking except that set theory is correct? Wm seems to refrain from any thinking except that any set theory which does not conform to his own peculiar standards is necessarily self-contradictory, but continually fails to find any internal contradictions in them.
From: Virgil on 9 May 2007 22:15 In article <1178739230.073383.98140(a)o5g2000hsb.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 9 Mai, 20:25, Virgil <vir...(a)comcast.net> wrote: > > In article <1178716861.173911.234...(a)y5g2000hsa.googlegroups.com>, > > > WM has no power to constrain mathematics to what he thinks it should be. > > > > That it is not what WM thinks it should be is by now apparent, but that > > it is quite productive anyway is equally apparent > > Producing what? Yes, the working mathematicians, they are quite > productive, but they do not need transfinite set theory. Transfinite > set theory is the most useless science ever created. That set theory is mathematics, and therefore not science at all, seems to have escaped WM.
From: Dik T. Winter on 9 May 2007 22:10 In article <1178722674.540579.289510(a)e51g2000hsg.googlegroups.com> WM <mueckenh(a)rz.fh-augsburg.de> writes: > On 9 Mai, 04:09, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: .... > > > Even in the *infinite* tree a path cannot be distinguished from all > > > other paths. > > > > Why not? For each other path there is a finite node where it goes a > > different way. > > Obviously not. For each node there is a co-path p' which has been with > p all the way long. Where is the contradiction? > > That is simply opion. Indeed, only countably many can be described by a > > finite formula (as is clear from the work of Turing). In mathematics that > > does *not* mean that the other numbers do not exist. > > How do you address, represent or use it in any other form "in > mathematics"? By having a set of it. Why is representation needed? > > > Cantor's diagonal proof fails, because the diagonal number is never > > > distinguished from all other real numbers (if uncountably many real > > > numbers exist). > > > > But that is not the proof. The proof is that the diagonal number is > > distinguished from the real numbers in the list (which are countably many). > > That's your (and others') error. Did you ever correctly read the proof? > The proof in the tree is, that for > every path p there is another path *existing in the tree* which is not > different from p. Nonsense, if two paths are different there is a node where they differ. > And if you apply Cantor's proof in the tree, by > forming a "diagonal" by switching a bit for every path, then it is > undisputed that the constructed diagonal number is represented by a > path in the tree. I need a list of paths before I can even try to start this. The set of paths is not a list. So, first, provide me with a list of paths and I will come up with a path not in the list. > Why should this be different in Cantor's list? Because it is a list. A tree is not a list. -- 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 9 May 2007 22:20
In article <1178723452.432944.32310(a)e51g2000hsg.googlegroups.com> WM <mueckenh(a)rz.fh-augsburg.de> writes: > On 9 Mai, 15:50, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: .... > > But that is not the definition, their formal definition is: > > "A binary relation F is called a function (or mapping, correspondence), if > > aFb_1 and aFb_2 imply b_1 = b_2 for any a, b_1, and b_2." > > And how do you think the a's and b's are selected? And from what sets > do you think are they selected? The first thing you should do is find how they define "binary relation". But if you skip introductory material you can be lead to errors. > And why do you think H&J give that informal > statement? Just in order to confuse the students? I would not know. But to guess, to guide students? It is fairly standard in mathematics text books to give an informal introduction to concepts so that they can relate it to things they know. > It is ridiculous to > follow this discussion. I can assure you, if one of my students would > not know that a function is a formula (or rule or whatever) together > with a domain where it is defined and a range, then he or she would > not pass the exame. And this is the same in the better math courses in > Germany. If that is true, it tells us a lot about mathematics education in Germany. Functions in set theory are *not* the same as functions in analysis. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |