Cantor Confusion
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From: mueckenh on 19 Oct 2006 12:49 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. > > > Why then? > > Depending on the person who writes about the problem. One possibility is > defining a set of functions dependent on time t that tells whether at time > t (not t now goes from -1 to 0) ball n is in the vase or not. Addition of > those functions gives the number of balls present in the vase at time t. > But all depends on the exact mathematical formulation of the problem. There > is none (in my opinion, but I have already had discussions with others > about this point, so I will not repeat them). > > 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. If the vase is not empty at noon, than one can never have all natural numbers together. Then one cannot enumerate a Cantor list to show that at least one real number is missing etc. Then set theory is meaningless. There is no complete infinite set. 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. Then there are no irrational numbers and several other parts of mathematics. Therefore set theory cannot be a solid foundation of mathematics Regards, WM
From: MoeBlee on 19 Oct 2006 12:50 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. MoeBlee.
From: mueckenh on 19 Oct 2006 13:02 David Marcus schrieb: > > Therefore, in the limit {2,4,6,...} there are infinitely many finite > > natural numbers m > |{2,4,6,...}| > > Once again you use words and phrases that are not part of standard > mathematics, i.e., "in the limit {2,4,6,...} there are infinitely many > finite natural numbers m > |{2,4,6,...}|". What does "in the limit > {2,4,6,...}" mean? > You said (repeatedly) that standard mathematics contains a > contradiction, so please state the contradiction in standard > mathematics. If you use new terms, please define them. Sorry, I do not know what you state of knowledge is. {2,4,6,...} means obviously "all natural numbers". That is the usual notation in mathematics. 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. Perhaps you will step by step understand it. > In standard mathematics, a finite set of natural numbers has a largest > element. Please prove that a set of elements consisting of 100 bits has a largest element. > So, what are you actually saying? Are you saying that you don't > like standard mathematics? I am saying that standard mathematics is false. As I have shown there are finite sets in mathematics which have no largest elements. > I've never seen "potential infinity" or "actual infinity" in any > textbook I've used. So you read not the right books or too few. > Bohmian Mechanics is a deterministic theory that avoids the measurement > problem, satisfies Bell's Inequality (as do all theories of quantum > mechanics), agrees with all experiments, and doesn't produce negative > probabilities. So, it seems to be a better theory than the one you > constructed. > And, Bohmian Mechanics is non-relativistic. Therefore I did not adopt it. My theory is even a local-hidden-variables theory, which satisfies Bell's inequlities, a goal never matched by any other theory. Regards, WM
From: mueckenh on 19 Oct 2006 13:10 Han de Bruijn schrieb: > There are many readers here who DO understand Mueckenheim's binary tree. Thanks, Han. I also made this experience. Many mathematicians understood it on the first glance (and felt its dangereous implications - may be that some biased mathematicians do not want to understand it). I think it is not more difficult than Cator's proofs. Alas it contains new ideas. By the way, I am constructing a website on Mathe-Realism with links to many interesting sites likes yours. I hope you agree? Here are some first quotes: "When the objects of discussion are linguistic entities [...] then that collection of entities may vary as a result of discussion about them. A consequence of this is that the "natural numbers" of today are not the same as the "natural numbers" of yesterday." (David Isles). "But numbers are symbolic constructions; a construction does not exist until it is made; when something new is made, it is something new and not a selection from a preexisting collection." (Edward Nelson) Regards, WM
From: Mike Kelly on 19 Oct 2006 13:30
mueckenh(a)rz.fh-augsburg.de wrote: > 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. > > > > > Why then? > > > > Depending on the person who writes about the problem. One possibility is > > defining a set of functions dependent on time t that tells whether at time > > t (not t now goes from -1 to 0) ball n is in the vase or not. Addition of > > those functions gives the number of balls present in the vase at time t. > > But all depends on the exact mathematical formulation of the problem. There > > is none (in my opinion, but I have already had discussions with others > > about this point, so I will not repeat them). > > > > > 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. > > If the vase is not empty at noon, than one can never have all natural > numbers together. Then one cannot enumerate a Cantor list to show that > at least one real number is missing etc. Then set theory is > meaningless. There is no complete infinite set. > > 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. Then there are no irrational numbers and several other parts > of mathematics. Well, no. The value of a function at a point need not be the limit of that function approaching that point from the left. Not the same thing. -- mike. |