Cantor Confusion
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From: Virgil on 26 Oct 2006 15:01 In article <1161860460.122685.294590(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > "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. Given a sequence of sets, S_n, what is your definition of "lim [n --> oo] S_n" ? To the best of my knowledge there is no such expression defined in standard mathematics, so it requires defining before it has meaning. So far, " lim[n-->oo] {1,2,3,...,n}" is totally without meaning.
From: David Marcus on 26 Oct 2006 15:02 mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > I already mentioned several books that you could use. > > It is not me who always faisl to understand. I don't see that you've understood anything said by all the people who have pointed out your errors. However, if you still assert that you have something mathematical to say, please do say it, rather than just talking about saying it. > But here is not he place to discuss physical theories. Might as well discuss physics since you don't seem to be capable of discussing mathematics. -- David Marcus
From: David Marcus on 26 Oct 2006 15:13 mueckenh(a)rz.fh-augsburg.de wrote: > 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? Better? No. The chance of guessing correctly is rather slim since what you claim is provably false. However, I'll take a stab at it. Construct a binary tree by starting with one node and connecting each node to two nodes at the next level. There is a level for each natural number. The connection between a node and a node at the next level is an edge. So, the number of edges is countably infinite. A path is a sequence of edges that starts at the root node, goes from node to node, and only goes forward (to higher numbered levels). The number of paths is uncountable. At this point you claim that you can create a bijection between the edges and paths, but your bijection isn't really a bijection because you use pieces of an edge. This isn't too surprising since we can prove that there is no bijection between the edges and paths. -- David Marcus
From: William Hughes on 26 Oct 2006 15:13 Tony Orlow wrote: > mueckenh(a)rz.fh-augsburg.de wrote: > > Tony Orlow schrieb: > > > > > >>> How would you construct an actually infinite set? Pair, power, union? > >>> They all stay in the finite domain if you start with existence of the > >>> empty (or any other finite) set. Comprehension or replacement cannot go > >>> further. So, how would you like to achieve it? > >>> > >>> Regards, WM > >>> > >> Inductive subdivision of the unit continuum? We certainly seem to be > >> able to specify, or approximate arbitrarily closely, some values with > >> infinite strings of digits. It seems obvious that any finite interval in > >> the continuum has more than any finite number of points within it. So, > >> isn't that an actually infinite set, albeit with linear finite measure > >> and bounds? > > > > > > Sorry Tony, you are in error. We cannot approximate sqrt(2) arbitrarily > > close. We can visualize it by the diagonal of a square and we can name > > it. But we cannot approximate it better than to an epsilon of > > 1/10^10^100. It woud be nice if we could, but assuming we can manage > > it, only because otherwise mahematics becomes too difficult, is a bit > > too simple. > > > > Regards, WM > > > > Can you justify that limit of accuracy in mathematical terms, without > resorting to discussions of the size of the universe? I see no > theoretical, mathematical reason for it. And, whether we can specify > every point or not, especially if we can't, there are more than any > finite number of reals in any unit interval. Or, do you claim there are > a finite number? It is interesting to note that when WM feels theatened by something he will simple abandon a subthread (sometimes announcing this, sometimes not). Turn about is fair play. - William Hughes > > TO
From: Virgil on 26 Oct 2006 15:16
In article <1161862972.647270.89100(a)i3g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > Lester Zick wrote: > > > On Mon, 23 Oct 2006 22:01:25 -0400, David Marcus > > > <DavidMarcus(a)alumdotmit.edu> wrote: > > > > > > >Han de Bruijn wrote: > > > > > > [. . .] > > > > > > >> > > > >> There are many readers here who DO understand Mueckenheim's binary > > > >> tree. > > > >> And no, binary trees will not be found in Halmos' "Naive Set Theory". > > > >> Because it's too naive, I suppose .. > > > > > > > >You mean all the cranks think they understand it. > > > > > > So this set of all cranks. Would that be those who disagree with you? > > > > That's not the definition of "crank". Wikipedia has a definition of > > crank. > > > > You can readily detect cranks because they never clearly state what > > their words mean, they keep repeating the same things rather than > > understanding and replying to what others say to them, > > but always asking for an explanation in the only anguage they believe > to understand. And never acknowledging those explanations, not providing any of their own in intelligible form. > > And if delivering an answer, then they don't know how to interpret it, > neither in their nor in another language. > > > Since Mueckenheim claims his binary tree shows an inconsistency in > > standard mathematics (at least he sometimes claims that--at other times > > he claims to not be discussing standard mathematics), > > It shows that there is a countable set which has not less elements than > the set of real numbers. Not according to any standard mathematical constructions, it doesn't. No unambiguous injection from the set of paths to the set of edges has been shown. merely some idiotic and irrelevant handwaving about splitting edges. > > > if anyone really > > understood what Mueckenheim was saying, they could translate it into > > standard mathematics and demonstrate the inconsistency. > > > > Some people claim they understand it, but fail to give their > > translation. > > I guess I can see now the reason of your problems: You don't > understand because you are looking for a tanslation always. There is no > translation required. There is if it is to be comprehensible in standard mathematics. > The tree is that mathematics which deserves this > name. It is outside of your model, independent of ZFC, but generally > valid and, therefore; covering ZFC too. An infinite binary tree can be modeled within ZFC (or NBG), and since all infinite binary trees are easily seen to be tree-isomorphic, whatever is true of any of them is true of one in ZFC. For example: let T be the set of all functions from N to {L,R}, with each function, f, being a path and each f(n) being the appropriate edge, a left (L) or right(R) branch as appropriate. hen T is an infinite binary tree in ZFC. |