Cantor Confusion
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From: mueckenh on 16 Oct 2006 15:57 William Hughes schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > William Hughes schrieb: > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > William Hughes schrieb: > > > > > > > > > > > > COLLECTED ANSWERS > > > > > > > > > Once you have chosen your set of 100 elements it will have > > > > > a largest size. Questions as to the method of choice have > > > > > no bearing on this. > > > > > > > > > Correct. Once you have chosen how to use all the bits of the universe, > > > > then you will have a largest number. > > > > > > And crucial to this discussion, once you have chosen > > > that you are going to use > > > all the bits in the universe you know that you will have > > > a largest number. > > > > No. I will not have a largest number > > My statement was not > "you will have a largest number" but "you know that you > will have a largerst number". The fact that you know > that you will choose your largest number sometime in > the future, and that you have not yet chosen the way > you will choose this number does not change > the fact that you know you will have a largest number. If you mean that there is a largest number ever defined, then you are right. But we cannot know which this number will be. Independent of this there is, as far as I see, no principle limitation to expressing a number as large as desired. > > > > That is correct. It depends on the ingenuity how to define a large base > > with few bits. There are no limits to ingenuity, therefore there is no > > largest number. > > Piffle. Ingenuity cannot change the fact that there is only a finite > time to describe the method of representation. That is correct. But why must the invention of a very powerful method of representation consume much time? > > > We could > > > do the usual formailization in terms of Turing > > > machines, but this would do little except free us from > > > such paradoxes as "let K be the largest integer that can be > > > described plus 1".(note the usual interpretation is that K has > > > not been described, you, however, claim that K cannot exist.)) > > > > > > And while the largest integer that will be described is not > > > yet known (even in theory) > > > > I cannot see that. > > > > You cannot know what the largest integer mentioned in the > New York times in 2007 will be (even in theory). Oh, I misplaced my comment. It should appear later > > > > the largest integer that > > > can be described is known (at least in theory). namely here: I cannot see that. > Ingenuity may or may not be limited, but there is a limited amount > of time to communicate the results of ingenuity. Yes. It can be communicated, however, in highly compressed form. Also the compression depends on ingenuity. > > > > > > > Let N be the set of finite integers. Either N has an upper bound > > > or it does not. If you claim that N has an upper bound we will > > > have to go our merry ways. Otherwise let K be subset of N that > > > does not have an upper bound. There, I have produced an > > > infinite set without using the three "...". > > > > You have produced a finite set without an upper bound. Remember the 100 > > bits. > > What you described is not a set. It is a large collection of sets. > None > of these sets is arbitrary (arbirtray describes a way of choosing > not what is chosen) Each of these sets has an upper bound. I disagree. But if you are correct, then the set of natural numbers has an upper bound. > > > However, you wish to do more. You want to show > > > that claiming "N does not have an upper bound and > > > N exists as a complete set" leads to a contradiction.] > > > > > That is true too. And it is easy to see: If we define Lim [n-->oo] > > {1,2,3,...,n} = N, then we can see it easily: > > > > For all n e N we have {2,4,6,...,2n} contains larger natural numbers > > than |{2,4,6,...,2n}| = n. > > So we have something that is true for finite sets. It is true for finite numbers and sets of finite numbers. Should it not be true for all sequences of finite even numbers, then there must be some of even finite numbers, X, in {2,4,6,...,2n, X} which care to push the cardinality without increasing the sizes. That is obviously impossible. > > > There is no larger natural number than aleph_0 = |{2,4,6,...}|. > > Contradiction, because there are only natural numbers in {2,4,6,...}. > Wiithout any justification whatsoever you state something > about infinite sets. I state something about finite numbers. > > Something that is true about finite sets does not have to > be true about infinite sets. That is your standard excuse. It seems that nothing can be true for infinite sets. > > > > This is a (slightly) obfuscated version of "{1,2,3,...,n} is bounded > > > for all n in N. > > > Therefore N is bounded". > > > > This conclusion is nearly true, in principle, though it should be > > stated more carefully in order to avoid misunderstanding: N is > > potentially infinite, i.e., always finite but without an upper bound. > > > > If N is potentially infinite it does not have an upper bound. > How can you say that the conclusion "Therefore N is bounded" > is nearly true? > It was some kind of politeness, because as an exception you were stating something which could be interpreted (as I did) as not completely wrong. > The brain is contstrained by physical laws. The concepts produced by > the brain are not contrained by physical laws. The puppet hangs on the string. The feet of the puppet hang on the puppet, but not on the string? They do not fall down when the string is cut? Regards, WM
From: Han.deBruijn on 16 Oct 2006 15:58 Virgil schreef: > In article <290c1$45333e14$82a1e228$8972(a)news2.tudelft.nl>, > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > > Dik T. Winter wrote: > > > > > In article <1160857746.680029.319340(a)m7g2000cwm.googlegroups.com> > > > Han.deBruijn(a)DTO.TUDelft.NL writes: > > > > Virgil schreef: > > > ... > > > > > I do not object to the constraints of the mathematics of physics when > > > > > doing physics, but why should I be so constrained when not doing > > > > > physics? > > > > > > > > Because (empirical) physics is an absolute guarantee for consistency? > > > > > > Can you prove that? > > > > Is it possible to live in a (physical) world that is inconsistent? > > The consistency of the physical world did not guarantee the consistency > of the Phlogiston theory of combustion. Being a physicist is not a > guarantee of being right, or of being consistent. Every physical theory > must be, at least in theory, falsifiable, so that none of them can be > held to be infallibly consistent. I'm not talking about a theory. I'm talking about the world as it IS. > HdB is conflating the consistency of the world with the infallibility of > its observers. The former is granted, the latter denied. Hey! THE FORMER IS GRANTED. We agree! Han de Bruijn
From: Virgil on 16 Oct 2006 16:00 In article <1161007554.513186.56640(a)i42g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Randy Poe schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > Randy Poe schrieb: > > > > > > > Han de Bruijn wrote: > > > > > > I merely note that there is no requirement in the problem that > > > > > > the limit be the value at noon. > > > > > > > > > > The limit at noon - iff it existed - would be the value at noon. > > > > > > > > Wrong. That is a flat out incorrect statement showing a > > > > fundamental misunderstanding about what limits mean. > > > > > > > > A CONTINUOUS function at x0 has the property that the > > > > limit of f(x) as x->x0 is f(x0). But not all functions are > > > > continuous. > > > > > > And you are in charge of determining which functions are continuous and > > > which are not? > > > > Where do you get this stuff from? > > > > How do you translate a statement that some functions are not > > continuous into "I am in charge of determining if some functions > > are continuous"? > > Because it seems a bit mysterious, how you know or can define that the > function of balls in the vase was not continuous. Or, may be, because > you will accept that the function is continuous if the balls are taken > out in the sequence 1, 11, 21, .... How is the putting of balls into the vase continuous? if that is not continuous, then why should anything else be? > > > > No, I am not in charge. Non-continuous functions are non-continuous > > now and forever. They were non-continuous before I existed, they will > > remain non-continuous after I'm gone. > > > > The number of balls in the vase is such a function. > > How do you acquire that knowledge? From the definition of continuity of a function at a point: A real function, f, is continuous at a real number r in R if and only if (1) f is defined at r in R, i.e., f(r) exists, (2) (lim_{x --> r} f(x) exists, and (3) (lim_{x --> r} f(x) = f(r) > > > > > The function f(t) = 9t is continuous, because the function 1/9t is > > > continuous. Wrong! The function 1/(9t) is not continuous at t = 0, but the function f(t) = 9t is. > > > For the t-th transaction 9t is the number of balls in the vase. But for 9t to be continuous, t must be time, not number of transactions. The number of transactions cannot be not continuous so that 9 times it cannot be either.
From: mueckenh on 16 Oct 2006 16:00 David Marcus schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > William Hughes schrieb: > > > However, you wish to do more. You want to show > > > that claiming "N does not have an upper bound and > > > N exists as a complete set" leads to a contradiction.] > > > > That is true too. And it is easy to see: If we define Lim [n-->oo] > > {1,2,3,...,n} = N, then we can see it easily: > > > > For all n e N we have {2,4,6,...,2n} contains larger natural numbers > > than |{2,4,6,...,2n}| = n. > > There is no larger natural number than aleph_0 = |{2,4,6,...}|. > > Contradiction, because there are only natural numbers in {2,4,6,...}. > > You appear to have written the following: > > Let N be the set of natural numbers. For all n in N, > > 2n > |{2,4,6,...,2n}| = n, No. I have written the following: For all n e N we have {2,4,6,...,2n} contains larger natural numbers than |{2,4,6,...,2n}| = n. I did not explicitly mention 2n. > > n < |{2,4,6,...}| = alpheh_0, > > {2,4,6,...,2n} is a subset of N. > > I follow this. But, you have the word "contradiction" in your last > sentence. Are you saying there is a contradiction in standard > Mathematics? If so, what is it? I don't see it. By induction we find the larger the set the larger the number of numbers contained in the set and surpassing its cardinality. The assumption that the infinite set would not contain such numbers neglects the question of what kind of numbers can increase the cardinality without increasing the number sizes. The problem is the same, in principle, as the vase and its balls. Regards, WM
From: David Marcus on 16 Oct 2006 16:00
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". Then you say the "edge is related to a set of path". 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. It seems that first the relation is between paths and edges, then edges and sets of paths, then pieces of an edge and sets of paths. Please restate the entire theorem and proof and include the precise definition of your "relation". -- David Marcus |