Cantor Confusion
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From: Dik T. Winter on 19 Oct 2006 22:11 In article <1161276322.150252.120060(a)i42g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > > Again replying to more than one article at once without giving proper > > references. That makes it very difficult to follow threads. > > I am sorry, I have only 15 shots per several hours, but far more opponents. Why do you think that is the case? > I repeat only that part of the discussion which I answer to. Yes, and meantime you are breaking threads so that it becomes difficult to see what you are actually replyint to. > > Yes, according to Cantor. And according to modern set theory that is > > wrong. To count the elements of a set with the ordinal omega you need > > only the finite ordinals. > > I do not understand you. We cannot count all naturals without saying > "omega" and being finished. I do not understand you. You cannot count all naturals and at some time saying that you are finished. (1) The set of natural numbers exist, by the axiom of infinity. (2) You will never terminate when counting through the natural numbers. > > > I think that, compared to Cantor, in modern set theory potential and actual > > infinity are split up again. The contents of the set N form only a > > potential infinity, on the other hand, the *size* is an actual infinity. > > But in modern set theory omega = {1,2,3,...} is the contents and > aleph_0 = omega is the size of the set. In one formulation of set theory. > Modern set theory has no use for potential infinity. Well, actually the terms 'potential infinity' and 'actual infinity' are meaningless in mathematics. Now, if you could provide a *mathematical* definition, perhaps more can be done. === (Yes, I am trying to separate the threads.) > > This one is in reply to <J7BuCw.LJr(a)cwi.nl>: > > > > Let me ask. You *did* state that "The function f(t) = 9t is > > > > continuous, because the function 1/9t is continuous", yes? Do > > > > you not see that that reasoning is wrong? That if 1/f(t) is > > > > continuous that does not mean that f(t) itself is continuous? > > > > You can not answer this question? > > See below. There is no answer to the question below. > > > We can talk about continuity in the finite domain. We can calculate the > > > limit in case of f(t) --> 0 and we can from that define a limit oo for > > > the case 1/f(t). > > > > I have my doubts about the latter, because oo is not a real number, and > > doing arithmetic with that one is impossible. > > > > > From these procedures we can obtain that the function > > > 1/f(t) will never yield the limit 0. That is the continuity requirement > > > I mentioned. > > > > You are doing arithmetic with oo as if it is a real number. > > "As if" it is larger than any real number. But, yes, according to > LAUGWITZ we can use it as a very large number. In any case I am sue > that oo > 1 is valid and can be used. Which Laugwitz? There are so many. But you (as a physicist) can possibly use it as a very large number, that is not a mathematical formulation of it as a number in mathematics. There go so many things wrong when you try to use it as a number... > > However, in > > standard mathematics you can state that lim{t -> oo} 1/f(t) = 0. That does > > *not* mean that 1/f(oo) equals 0, because that one is not defined, neither > > does it state that there is continuity, and also it does state nothing about > > the value of f(t) at t = oo, because that is also not defined. > > Then we cannot say anything about f(oo). That is in fact the truth, but > not in set theory. Set theory says about oo: omega is finished. Yes to the first, no to the second. f(oo) does not belong to set theory, it belongs to analysis. Also note that Cantor at some time refrained from using oo and started to use w, because there was too much confusion. The oo from pre-Cantorian mathematics is different from the omega of Cantorian mathematics. === > > > Any list gives a constructrible diagonal number. Any diagonal number is > > > constructed: > > > > Yes. You can construct a number from any list, but you can not construct > > a constructable number from a list that is itself not constructable. > > Every diagonal number is constructed and, therefore, is constructible. When constructed from a constructable list. > > For a constructible number the whole process to derive that number should > > be construction. The list is the first step in derriving that number, so > > for the diagonal to be constructable, that list must be constructable. > > No. You ignore that for a number to be constructible, every step in its construction is necessarily constructible. > > > > > Fraenkel, Abraham A., Levy, Azriel: "Abstract Set Theory" (1976), p. > > > 54: "Why, then, the restriction to the digits 1 and 2 in our proof? > > > Just to kill the prejudice, found in some treatments of the proof, as > > > if the method were purely existential, i.e. as if the proof, while > > > showing that there exist decimals belonging to C but not to C0, did not > > > allow to construct such decimals." > > > > What are C and C0 in this context? > > p. 52 explains it: "Lemma. Given any denumerable subset C0 of C, there > exist members of C which are not contained in C0; that is to say, C0 is > a proper subset of C." So C is the continuum R and C0 is the set of > list numbers Yes, any diagonal number is constructed from the list. That does *not* make any diagonal number constructible. Obviously if the list is not constructable, the diagonal number is constructed from the list, but is itself not constructable. You apparently understand 'constructable' as being able to be constructed from something, whatever that is. But that term has a specific mathematical meaning. And constructible in the context of numbers has a very specific meaning. > > > If you cannot understand that, then try this: The set of diagonal > > > numbers is countable. Therefore the diagonal proof proves the > > > uncountability of a countable set. > > > > Why is the set of diagonal numbers countable? How do you derive that? > > Because only a countable set of numbers can be constructed using a > language with a finite alphabet. Using strings with
From: Dik T. Winter on 19 Oct 2006 22:28 In article <1161276574.792436.186750(a)i3g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > In article <1161183237.727249.154740(a)b28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: > > > > > > > In article <1161079802.120515.175530(a)i3g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > > ... > > > > > The inconsistency is that > > > > > 1) For the balls inserted until noon, you can find the result: > > > > > It is the set N. > > > > > 2) For the balls removed until noon, you can find the result: > > > > > It is the set N. > > > > > 3) For the balls remaining at noon, the same arguments of > > > > > continuity which lead to (1) and (2) cannot apply. > > > > > > > > There are quite a few obvious reasons. > > > > (1) 1) is not because of continuity > > > > The reason that continuity plays no role is because the function of the > > number of balls in the vase when written as a function of t is > > discontinuous at infinitely many positions. > > But it is stepwise continuous. Yes, but it is infinitely many times not continuous in each neighbourhood of the limit point. > > You can *model* the problem in different ways in mathematics, and one of > > those models leads to 0 balls in the vase at time 0. And there are > > presumably other models that give different results. So, unless the > > problem is stated in a mathematically proper way, actually nothing can > > be said about it. > > Translate balls as numbers and vase as set variable. More is not > required. That is still not a mathematical formulation. More is required. You need to actually state what you mean with 'number of balls in the vase at noon' or 'natural numbers in the set at noon'. Also there is a time dependency that is not clearly stated. But let me try: Increasing from t = -1 to t = 0, and starting with S = {}, at step t = -1/(2^k) replace S by S u {10k - 9, ..., 10k) \ {k} What is set S at t = 0? To be honest, I have no idea. It depends on how you wish to model it. > If the vase is empty at noon, then can obtain from set theory that the > limit of a sequence in no way can be determined by the terms of he > sequence. The limit can in mathematics *always* be determined by the terms (if there is a limit). But the limit in no case defines the function value at the limit point. > Then there are no irrational numbers and several other parts > of mathematics. The irrational numbers are by definition the limits of particular sequencens. And that is not set theory. > Therefore set theory cannot be a solid foundation of mathematics Well, if you think so, start your own foundation. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Han de Bruijn on 20 Oct 2006 03:30 MoeBlee wrote: > Han de Bruijn wrote: > >>David Marcus wrote: >> >>>Dik T. Winter wrote: >>> >>>>I think that, compared to Cantor, in modern set theory potential and actual >>>>infinity are split up again. The contents of the set N form only a potential >>>>infinity, on the other hand, the *size* is an actual infinity. >>> >>>I've never seen "potential infinity" or "actual infinity" in any >>>textbook I've used. >> >>True. And you haven't seen any binary tree either. > > What are you talking about? Graphs, edges, paths, trees, binary trees, > et. al are all discussed in many textbooks and in many advanced > textbooks in which set theory and graph theory intersect. Yes. That's why I'm surprised that David Marcus hasn't met them before. Han de Bruijn
From: Han de Bruijn on 20 Oct 2006 03:33 mueckenh(a)rz.fh-augsburg.de wrote: > By the way, I am constructing a website on Mathe-Realism with links to > many interesting sites likes yours. I hope you agree? You're quite welcome, Wolfgang! Han de Bruijn
From: Han de Bruijn on 20 Oct 2006 03:36
Virgil wrote: > In article <d6483$45372f33$82a1e228$5556(a)news2.tudelft.nl>, > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > >>imaginatorium(a)despammed.com wrote: >> >>>I expect all the non-cranks would agree with my statement, and find it >>>astonishing. Cranks like yourself occasionally agree with all sorts of >>>things, more or less by accident - so what? >> >>Cranks, cranks .. You're clearly running out of _arguments_, aren't you? > > As there are no arguments that will penetrate a crankhood as profound as > HdB's, it would hardly matter if he did run out. Yeah, it's a long time ago that a mathematician (Socrates) could explain the proof of Pythagoras' Theorem to a slave, huh .. Who is to be blamed? Han de Bruijn |