Cantor Confusion
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From: mueckenh on 26 Oct 2006 04:26 David Marcus schrieb: > > > Yes, just as I said, the discussion is about the philosophy of > > > mathematics and set theory (and, I should add, about informal concerns > > > and motivations), but there is not, WITHIN the set theory discussed > > > there, a definition of 'actually infinite' and 'potentially infinite'. > > > > Does the study of formal languages really make incapable of > > understanding plain text? What is written above means: "INFINITY" IN > > SET THEORY IS ALWAYS "ACTUAL INFINITY". > > Exactly. There is only one kind of infinity in modern set theory (and > modern mathematics). We no longer distinguish between "potential > infinity" and "actual infinity. These distinct notions were important > historically until the concept of infinity became better understood. LOL. By such students as you, i.e., by people who refuse to understand anything but compete with computers for the most formal formalism? Regards, WM
From: mueckenh on 26 Oct 2006 04:35 David Marcus schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > David Marcus schrieb: > > > Binary Tree > > > Unfortunately, it was described in a way that I can't understand it. A > > > wild guess on my part is that you mean to set up a correspondence > > > between edges and sets of paths. > > > > I am sorry, but if you need a wild guess to understand this text, then > > we should better finish discussion. Observe just how the discussion > > runs with all those who understood it, like Han, William, jpale. > > Han doesn't understand it (although he probably thinks he does). William > and jpale simply pick the mathematically meaningful statement that is > closest to what you write and go from there. The model is that simple that any student in the first semester could understand it. Every paths which branches into two paths necessarily needs two additional edges for this sake. It is only your formalistic attitude that blocks your understanding. But you must not think that anybody is blocked like you. > I could do that too, but I > suspect that what you are actually thinking is rather far from their > guesses. That means, you have a better guess? It seems you think you are the best guesser of mankind, or at least here? > I'd rather wait until you say something coherent before I > comment on it. You mean, until you are able to *recognize* something coherent? Regards, WM
From: Han de Bruijn on 26 Oct 2006 05:02 mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > >>Han de Bruijn wrote: >> >>>David Marcus wrote: >>> >>>>I don't think so. Bohmian Mechanics is 100% deterministic. All of the >>>>uncertainty in the results of an experiment is due to uncertainty in >>>>setting up the initial conditions of the experiment. >>> >>>Sure. Back to the dark ages of Laplacian determinism. >> >>Don't you mean Newton? > > I am sure, Han, did not mean Newton. Newton was not a determinist at > all. He accused Leibniz, who was a bit more deterministic, of being an > atheist. Newtonian mechanics at least raises the _suggestion_ of being completely deterministic. But I guess it has been Laplace who formulated that idea explicitly. Anyway, there is _no_ modern physicist who still believes in in determinism these days. Due to Quantum Mechanics, the idea has become entirely outdated. If David Marcus adheres to "Bohmian Mechanics" which "is 100% deterministic", then he adheres to outdated ideas. And there is no doubt that he will be stigmatized by the physics community with the same name as he calls us: a crank. Han de Bruijn
From: mueckenh on 26 Oct 2006 05:59 Dik T. Winter schrieb: > Within the *real* numbers the limit does exist. And a decimal number is > nothing more nor less than a representative of an equivalence classes. So we are agian at this point: The real numbers do exist. For the real numbers we have LIM 10^(-n) = 0. Therefore, i ther limit n--> oo tere is no application of Cantor's argument. > I do not understand why you want to tie that list in with the definition > of the reals? > Because the real numbers are used in that list. > > > But you should try to understand. In Cantor's list, there are > > limits of sequences, not equivealence classes. > > The list is a list of equivalence classes. In the decimal case > representatives of those equivalence classes are used. And that are limits. > > > Where in the construction above did I use the limit omega? In using any infinite sequence already > > > > You will need it in order to construct a real number and its decimal > > representation for a Cantor list. > > No. By the theory, each decimal number is a representative of an > equivalence class of sequences of rational numbers. By the construction > we get another decimal number that is also a representative of an > equivalence class of sequences of rational numbers. No omega is needed. Why then do you think omega is needed at all in mathematics? Real numbers without the axiom of infinity would be a nice construction. Omega is introduced only by the axiom of infinity. Regards, WM
From: mueckenh on 26 Oct 2006 07:01
Dik T. Winter schrieb: > > > > lim [n-->oo] {-1,0,1,2,3,...,n} = N > > > > > > > > is obviuously wrong too. > > > > > > Depends on how you define the limit. > > You did not state why that one was obviously wrong. You think it must be proved that -1 is not a natural number? As it must be proved that an always increasing positive function does not converge to zero? > > > > > Therefore lim [n-->oo] {1,2,3,...,n} = N. > > > > > > Depends on how you define the limit. > > > > That *is* the definition of the this limit. It is a wrong definition > > only in case N does not actually exist. > > You state "therefore", in general definitions are not conclusions from > earlier arguments. But N obviously exist; it is simply a letter. "N" is simply a letter. N, or better |N, as we understand it by convention in mathematics is a set most instructively expressed by lim [n-->oo] {1,2,3,...,n} = N. > > > But you never give a definition of the limit of sets. You only state that > > > definitions are *wrong* (although I fail to see why a definition can be > > > wrong). But you never state a proper definition. > > > > Look here: lim [n-->oo] {1,2,3,...,n} := N. > > Finally. A definition. So you define the notation lim for a particular > sequence of sets. But can you, in that case, use that definition in your > arguments? This is not the only sequence of sets for which you did use > the limit notation. Not the set is here the important thing with my definition. The definition of the operator "lim [n-->oo]" covers every case where n grows without upper bound such that either it comes never to a end or such that a non-natural number, namely omega, is reached. In the first case, we have 1/n < epsilon for every positive epsilon and we may *define or put* lim [n-->oo] 1/n = 0. In the second case we have without further ado lim [n-->oo] 1/n = 0. That is the difference between potential and actual infinity. > > > > > Because the contents of the vase increases on and on. Such a process > > > > cannot lead to emptiness in any consistent system - independent of any > > > > "intuition". > > > > > > That requires proof. > > > > LOL. Idle discussion. > > Perhaps, but you use it as argument, and I want to know whether it is a > valid argument, and so I want to see a proof. How do you know that the infinite set omega = N, which contains {n} or n+1 if it contains n, does have a positive number of elements? How do you know that it has more than one element? If you can prove that, then the positive contents of the vase at noon, i.e., for t --> oo, is proved too. Regards, WM |