An uncountable countable set
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From: mueckenh on 4 Sep 2006 06:48 Virgil schrieb:> > "Mueckenh" 's claim implies that any infinite set must surject onto its > power set. But that has been shown to be false. Learn the difference between a surjection and an intercession. Regards, WM
From: mueckenh on 4 Sep 2006 06:51 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. > > > > Note, you may critique Cantor's set theory, but that was set theory in > > > its infancy. It still did contain inconsistencies and errors. Since > > > that time quite a bit has been developed and corrected. > > > > In particular by intermingling the different meanings of infinity. > > In particular by just separating the meanings of potential and actual > infinity. Modern set theorists refuse to accept these notions. > Regards, WM
From: mueckenh on 4 Sep 2006 06:53 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. > > > > 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? Counting is the most primitive version of addition. > > > > 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. Regards, WM
From: mueckenh on 4 Sep 2006 06:57 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. > > > > Oh, well. As in mathematics the reals require a construction process, so > > > also your tree requires a construction process. In mathematics the reals > > > are constructed from the rationals (and I know at least four methods to > > > do that, that can be shown to give equivalent results). And the rationals > > > are constructed from the integers. I asked you before, but you never did > > > reply. Do you know how the rationals are constructed from the integers in > > > mathematics? More basic, do you know how arithmetic on naturals is > > > defined, based on the Peano axioms? > > > > 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. > 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. > > > > 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". > > 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. > > > > You seem to misunderstand the tree. If the diagonal is in Cantor's > > > > list, then 1/3 is in my tree. Can you give a reason why it should not > > > > (other than that then set theory is inconsistent)? > > > > > > I have explained already many times, but you are not willing to listen. > > > > 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? > > > The number 1/3 has only finite digit positions at finite levels of my > > tree. Where should any uncountable edge appear? > > Depends on how you count. If you count only the edges leading to 1/3 you > can do that (but never reach 1/3). That is self-evident for all counting processes. Why do you stress it? Because you try to argue the counting of all edges would fail for that reason? That is wrong. > 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? > > > > Either you agree that all the edges of my tree are countable or you > > > > agree that Cantor's diagonal is uncountable. Or you state at least how > > > > many infinite path in my tree you would consider to have completely > > > > countable edges. > > > > > > This question makes not sense to me. > > > > The problem is the following: You assert that the digit positions of > > 1/3 are countable as well as the levels of my tree (all of them), but > > you deny that the edges at these levels are not all countable. > > As I have stated again and again, if your edges *do* terminate, the number > of edges is countable, but 1/3 is not in your tree. There is no terminating > edge that leads to 1/3. "Countable" means that you can count on and on ad that for any element a natural number is reserved which is mapped on that element. "Countable" does ot mean that we reach an end. Give up your arguing. All set theorists know that the edges of my tree are countable. Even Virgil knows it. Regards, WM
From: mueckenh on 4 Sep 2006 06:59
Dik T. Winter schrieb: > > > Again translated (why do you post so much German in an English speaking > > > newsgroup while you should know that most readers are not able to read > > > German?): > > > > 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. > > > > 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. > 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. Von Hypothesen ist in meinen arithmetischen Untersuchungen über das Endliche und Transfinite überall gar keine Rede, sondern nur von der Begründung des Realen in der Natur Vorhandenen. Sie hingegen glauben nach Art der Metageometer Riemann, Helmholtz und Genossen auch in der Arithmetik Hypothesen aufstellen zu können, was ganz unmöglich ist.... So wenig sich in der Arithmetik der endlichen Anzahlen andere Grundgesetze aufstellen lassen, als die seit Alters her an den Zahlen 1,2,3,... erkannten, ebensowenig ist eine Abweichung von den arithmetischen Grundwahrheiten im Gebiete des Transfiniten möglich. "Hypothesen" welche gegen diese Grundwahrheiten verstoßen, sind ebenso falsch und widersprechend, wie etwa der Satz 2 + 2 = 5 oder ein viereckiger Kreis. Es genügt für mich, derartige Hypothesen an die Spitze irgend einer Untersuchung gestellt zu sehen, um von vorn herein zu wissen, daß diese Untersuchung falsch sein muss. (To Veronese, 17.11.1890) > > > > Nice that you found the quote I have alluded to, and that you did deny > > > of existing, but that I could not find back. > > > > > > What Cantor is stating here (and I did already indicate that in an > > > earlier response), is, translated to current set theory: > > > The set N is potentially infinite, > > > > No. The changing quantity is here a variable. > > > > >the size of N is actually infinite. > > > (In current terminology a set is not a quantity.) > > > > 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. Regards, WM |