Cantor Confusion
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From: David Marcus on 16 Oct 2006 19:39 mueckenh(a)rz.fh-augsburg.de wrote: > > David Marcus schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > David Marcus schrieb: > > > > > > > > > > Consider the binary tree which has (no finite paths but only) infinite > > > > > > > paths representing the real numbers between 0 and 1. The edges (like a, > > > > > > > b, and c below) connect the nodes, i.e., the binary digits. The set of > > > > > > > edges is countable, because we can enumerate them > > > > > > > > > > > > > > 0. > > > > > > > /a \ > > > > > > > 0 1 > > > > > > > /b \c / \ > > > > > > > 0 1 0 1 > > > > > > > ..................... > > > > > > > > > > > > > > Now we set up a relation between paths and edges. Relate edge a to all > > > > > > > paths which begin with 0.0. Relate edge b to all paths which begin with > > > > > > > 0.00 and relate edge c to all paths which begin with 0.01. Half of edge > > > > > > > a is inherited by all paths which begin with 0.00, the other half of > > > > > > > edge a is inherited by all paths which begin with 0.01. Continuing in > > > > > > > this manner in infinity, we see that every single infinite path is > > > > > > > related to 1 + 1/2 + 1/ 4 + ... = 2 edges, which are not related to any > > > > > > > other path. > > > > > > > > > > > > Are you using "relation" in its mathematical sense? > > > > > > > > > > Of course. But instead of whole elements, I consider fractions. That is > > > > > new but neither undefined nor wrong. > > > > > > > > > > > > Please define your terms "half an edge" and "inherited". > > > > > > > > > > I can't believe that you are unable to understand what "half" or > > > > > "inherited" means. > > > > > I rather believe you don't want to understand it. Therefore an > > > > > explanation will not help much. > > > > > > > > In standard terminology, a "relation between paths and edges" means a > > > > set of ordered pairs where the first element of a pair is a path and the > > > > second is an edge. Is this what you meant? > > > > > > Yes, but this notion is developed to include fractions of edges. > > > > > > > > I am at a loss as to how "half an edge" can be "inherited" by a path. > > > > > > An edge is related to a set of path. If the paths, belonging to this > > > set, split in two different subsets, then the edge related to the > > > complete set is divided and half of that edge is related to each of the > > > two subsets. If it were important, which parts of the edges were > > > related, then we could denote this by "edge a splits into a_1 and a_2". > > > But because it is completely irrelevant which part of an edge is > > > related to which subset, we need not denote the fractions of the edges. > > > > > > > As > > > > far as I know, a path is a sequence of edges. Is this what you mean by > > > > "path"? > > > > > > Yes. A path is a sequence of nodes (bits, 0 or 1). The nodes are > > > connected by edges. > > > > > > > > Is edge a the line connecting 0 in the first row to 0 in the second row? > > > > > > Correct. This edge is related to all paths starting with 0.0. So half > > > of edge a is related to all paths starting with 0.00, and half of it is > > > related to all paths starting with 0.01 > > > > > > > Or, is it the line connecting 0 in the first row to 1 in the second row? > > > > Or, is it something else? > > > > > > I have tried above to make it a bit clearer. > > > > First you refer to a "relation between paths and edges". > > Correct. > > > Then you say > > the "edge is related to a set of path". > > Is there any contradiction? Using standard mathematical terminology, there most certainly is a contradiction. You have apparently given two different descriptions as to what the elements of your "relation" are. In one, the elements are paths, but in the other they are sets of paths. A path is not a set of paths. Please state the definition of your "relation" clearly. If you use a word in other than its standard mathematical meaning, then please give a mathematical definition. > > Then you say that "half of that > > edge is related to each of the two subsets". I'm sorry, but I can't > > follow this. > > I warned you that this point is new: The edges are split in shares of > 1/2, 1/4, and so on. But when fractions were introduced in mathematics, > most people may have had the same problems as you today. I am sure you > can understand it from the written text above. (Many others have > already understood it.) -- David Marcus
From: David R Tribble on 16 Oct 2006 19:40 mueckenh wrote: >> You see, your proof is rubbish. B will have a largest element. And the >> set of all numbers ever used in the universe in eternity also will have >> a largest element. But it has not yet. > How do you know? How do you know that the universe is of a finite age? William Hughes schrieb: >> Therefore it is unknown. However, it is not arbitrary. > mueckenh wrote: > The largest element possible with 100 bits can be very different, > according to my arbitrary choice of representation. Of course. I can choose a representation whereby each bit represents a power of 10^100, for example. Thus with 100 bits the largest number I can represent is (10^100)^100-1. However, there are only 2^100 unique combinations of values using 100 bits. So regardless of the largest _value_ I can represent, I can represent far fewer _total_ unique values.
From: David Marcus on 16 Oct 2006 19:43 cbrown(a)cbrownsystems.com wrote: > Han.deBruijn(a)DTO.TUDelft.NL wrote: > > I talked about the real world, physics as an empirical science, > > not about artifical theoretical constructs. In the real world, > > Schrodinger's cat is dead :-( > > I thought you kept up with physics? > > http://physicsweb.org/articles/news/4/7/2 > > The device is conducting electricity in a clockwise fashion; and the > device is not conducting electricity in a clockwise fashion. That interpretation of the experiment is probably dependent on theory. Try this: http://www.mathematik.uni-muenchen.de/~bohmmech/BohmHome/bmstartE.htm -- David Marcus
From: David Marcus on 16 Oct 2006 19:45 MoeBlee wrote: > Han.deBruijn(a)DTO.TUDelft.NL wrote: > > MoeBlee schreef: > > > > > Han de Bruijn wrote: > > > > Sigh! Start digging into my website. I've said more about mathematics > > > > than anybody else in 'sci.math'. > > > > > > That's hilarious! I didn't even have dig at all to find you proposing > > > an inconsistent set of axioms and blaming not yourself but set theory > > > for the inconsistency - on the very first page I saw at that web site! > > > > Dig deeper! > > "Dig deeper." It has the sound of a punchline to a joke. I thought it was a straight line, but I decided to refrain from giving the punch line. -- David Marcus
From: William Hughes on 16 Oct 2006 19:58
mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueck...(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > > > No. The fact that everything that is true about the infinite > > > > must be justified in the finite, does not mean that everything > > > > that can be justified in the finite must be true about the > > > > infinite. > > > > > > > > You prove that something is true in the finite case. You > > > > do not justify your transfer to the infinite case. > > > > > > Who has ever justified such a proof? In fact that is impossible because > > > there is no infinity. Therefore all such "proofs" are false. But if we > > > assume the existence of the infinite, then the sum of the geometric > > > series is the most reliable entity at all. (Niels Abel: With the > > > exception of the geometric series no series has ever been calculated > > > precisely.) > > > > If you wish to assme that infinity does not exist, knock yourself > > out. However, if you are trying to show that the assumption > > that infinity does exists leads to a contradiction you need > > to justify the proofs that you make using that assumption. > > > > > > > > > > > The axiom of infinity > > > > > applies to the paths. They are nothing but representations of real > > > > > numbers. These exist according to set theory, therefore the paths > > > > > exist too. > > > > > > > > > > > > > Yes, but the question is not "do the paths exist?". > > > > > > There are two questions: Do the infinite paths exist and does the > > > geometric series wit q = 1/2 have a limit? I don't need any further > > > infinities. > > > > > > > No, there is a third question: "What is the connection between the > > infinite paths and the limit of the series?" You have only shown a > > connection between finite paths and partial sums. > > Wrong. The connection between finite paths and partial sums of edges > leads to > (1-(1/2)^n+1)/(1 - 1/2) edges per path. > And this is the only connection you have ever shown. You then take a limit and get 2. But you have never shown that this limit is connected to anything. The fact that you have a connection in the fnite case is not enough to show that the same connection holds in the infinite case. - William Hughes |