An uncountable countable set
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From: mueckenh on 4 Sep 2006 07:08 David R Tribble schrieb: > mueckenh wrote: > >> As long as the set of natural numbers includes only finite numbers, its > >> cardinal number is also finite. > > > > David R Tribble schrieb: > >> What is that cardinal number? Do you have a name for it? > > > > mueckenh wrote: > > The set is potentially infinite. > > That contradicts what you stated above, "... its cardinal number is > also finite". See below. > > > It has no cardinal number in the sense of set theory. > > Then it is not a set, since cardinality is a property of all sets. > > > All we can attach to it is the number of elements known > > or existing. Disregarding physical constraints ... > > I was not aware that abstract mathematical concepts (e.g., sets) > had any physical constraints. Then you should learn it. It you are unable to physically (i.e. in written form or in your mind) distinguish all the elements of a set, then the set does not exist. > > > ... we can assume that the > > elements of the set of natural numbers are counted by the largest > > natural number temporarily known. That is the finite cardinal number I talked about aove. With respect to physical constraints > > the number of elements is less than 10^100. > > So how many numbers are in this sequence?: > 1, 2, 3, ..., 10^100, 10^100+1 How many can you write down? To give you a hint: Can you determine whether the first 10^100 digits of pi make up a naturall number? Exchange the last igit of this number P by 6 and find out whether the new number is larger than P or not. > > What exactly is it about your set theory that physically constrains > your sets to being less than some arbitrarily chosen maximum size? Not arbitrarily cosen. There is no means in the universe to surpass this amount of information. > > > David R Tribble schrieb: > >> Tony calls it "alpha", the cardinality of the finite but "unbounded" > >> set of all finite naturals, and also the largest natural. It appears > >> to have the curious property of being a finite natural but having > >> no finite successor. He's never provided an estimate of how > >> large it is, though. Perhaps you have a better idea of this? > > > > mueckenh wrote: > > Sorry, there is no largest natural and no natural is infinite. > > Therefore there can be no finite cardinal for the set of naturals. That is a common error. The number P does not exist. But the number 10^100^1000 does. Regards, WM
From: Dik T. Winter on 4 Sep 2006 10:04 In article <1157367096.604428.36330(a)i42g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > But if you wish, provide a definition of "number". As far as I know, > > there is not one in mathematics. > > A number has only one of the following properties: It is larger than or > smaller than or equal to any natural number. So omega and aleph-0 are numbers. They satisfy the definition. Thanks. -- 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 4 Sep 2006 10:08 In article <1157367209.318653.75760(a)p79g2000cwp.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > Yes. I asked you about the cardinality, not what it was. What is the > > cardinality of that set? > > It has none. So you claim that set is not in bijection with the set {1, 2, 3, ..., w}? > > > Cantor defined aleph_0 in this manner: "Da aus jedem einzelnen Elemente > > > m, wenn man von seiner Beschaffenheit absieht, eine "Eins" wird, so ist > > > die Kardinalzahl [von M] selbst eine bestimmte aus lauter Einsen > > > zusammengesetzte Menge, die als intellektuelles Abbild oder Projektion > > > der gegebenen Menge M in unserm Geiste Existenz hat." But if you think > > > he was wrong, there is no need to discuss this topic. > > > > Where is he talking about addition? "when man von seiner Beschaffenheit > > absieht"; I would translate this as "when you disregard all properties". > > And so, in the transformation the ability to add is lost. And so you > > get a set (the current, proper, term is multi-set, I believe) consisting > > of only ones. > > How would you distinguish these elements, if not by counting them? If you count them you are adding properties back again. You need not to count to compare set sizes. > > > > What is the cardinality of the set of all finite natural numbers? > > > > > > It has none. > > > > Why not? Cardinality has been defined so it applies to *each* set. > > It is self-contradictory. Compare the binary tree. It is not self-contradictory. Compare the binary tree. -- 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 4 Sep 2006 10:38 In article <1157367467.816725.158560(a)i3g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > > > Then such numbers like 1/3 do not exist (in that representation). > > > > Indeed, in your tree with terminating edges, such numbers do not exist. > > > > > But > > > the same holds for Cantor's list. > > > > I do not see how you can jump to that conclusion. > > Cantor's list has as many lines as my tree. The diagonal has as many > digits as each path of my tree. The only difference is that the paths > split while the diagonal does not. And the difference is that given the list we can immediately state what the n-th digit of the diagonal is for arbitrary n. You can not state what the n-th path is for arbitrary n. > > > More to the topic: Do you know how Cantor's diagonal is constructed > > > from a list of reals? And how this list is constructed? > > > > Again, the second proof was *not* about the reals. > > Please spare your nonsense. Cantor did not consider anything else than > the reals. If today the proof concerns some wider range then this is > not due to Cantor. Please spare your nonsense. The second part of that article is *definitely* not about the reals. It uses the reals only "beispielsweise". > > But I know how the > > diagonal is constructed. I have no idea how the list is constructed. > > But again (it appears that I have to repeat myself a lot in this > > discussion), there is *no* construction of the list given. The proof > > is along the lines of: "given *any* list, I can show such". So the > > proof is valid without regards to the actual construction of the list. > > > The same is true for my tree. What is the tenth path in your tree? > > > > > There is no process of construction. If you have difficulties to > > > > > comprehend that: it is the same as with Cantor's list. The tree is > > > > > defined once and for all. That's it. > > > > > > > > Wrong. But I am not going to explain that again. > > > > > > What is the difference between my tree and Cantor's list. There was no > > > explanation and there is none, because there is no difference. > > > > You give a construction process for your tree. In the case of the list > > there is no construction process involved. The proof is about "given > > *any* tree". > > Forget the construction process of the tree. Take any tree which > contains all reals. As you say: My proof is about "given *any* > infinite tree". What is the tenth path of that tree? > > The list is not a list that is constructed, but a list that is given, or > > assumed. No construction involved. Whether the list contains finite > > binary expansions or not is completely irrelevant. The process with Cantors > > proof is more like the following: > > give me a list of infinite sequences of 0's and 1's, and I show that there > > is a sequence that is not in the list. > > Do you have problems with proofs by contradiction? > > Give my a tree of infinite paths consisting of 0's and 1's, and I show > that there are not less edges than paths. Indeed. If all edges terminate, also all paths terminate, and both are countable, and 1/3 is not in the tree. If edges do *not* terminale, also paths do not terminate, and both are not countable, and 1/3 is in the tree. > > > Please give a reference, or better: copy and paste your explanation. > > > > I have repeated my argument above. But I think you will not read it. > > 1/3 is not in your tree because all edges (and hence paths) are > > terminating. The diagonal is not in the list because it is defined > > in such a way that it can not be in the list. There is no relation > > between the two. > > The list, if existing, contains a diagonal (before anything is > exchanged). Doesn't it? Yup. > > If you want to do the same with 1/5 > > the case is similar. The problem appears when you want to count *all* > > edges. At each level, the number of edges is 2^n - 1 or somesuch. But > > also at each level we still have paths that need to be distinguished. > > You know that sets of order 2^omega are countable? No. Can you prove it? It is precisely Cantor's diagonal proof that shows that it is not countable. -- 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 4 Sep 2006 10:49
In article <1157367591.092721.79760(a)i42g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: .... > > > Because I have the German text available and because I do not want to > > > be blamed of mistranslating. > > > > So you can blame other persons of mistranslating? > > Those who cannot yet read German but are interested in the origins of > set theory should learn to do it. Yes, everybody who is not able to read German is not allowed to speak about set theory and the origins... > > > > Cantor: So while a changing quantity x that successively takes the > > > > various values of finite numbers 1, 2, 3, ..., v, ... , is a > > > > potential infinite, on the other hand, a through the axioms > > > > completely determined set (N) of all integral finite number is > > > > an example of an actually finite quantity. > > > > > > Not through the axioms, but through "a law" (ein Gesetz). > > > > What is the difference? > > A law is derived from the natural properties of arithmetics. Oh. What law derived from the natural properties of arithmetics is he talking about when he gets the completely determined set of all integral finite numbers? > > He uses his (I think, but that is from memory) > > second completion law. In current terminology, that is an axiom. > > Do you mean first and second Erzeugungsprinzip? Cantor disliked axioms. > He called them Hypothesen. Perhaps. But his Erzugungsprinzip is (I think) in current terminology an axiom. > > > Cantor's changing quantity is a variable. A set is never potentially > > > infinite according to Cantor. > > > > You missed my "in current set theory"? > > In current set theory there is no potential infinity at all. Ask > Virgil. I disagree with Virgil. That is possible, yes? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |