Cantor Confusion
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From: MoeBlee on 17 Oct 2006 12:44 mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > A good, if no the best source to learn about the different meanings of > > > infinity would be Cantor's collected works. > > > > Set theory has advanced since Cantor. The best source to learn about > > current set theoretic definitions of 'infinite' is not Cantor. > > There is no definition what infinity is and there is no definition what > a set is. There is a definitition in Z set theory of 'is infinite'. And we may also have a definition in Z set theory of 'is a set' (though that particular definition may be considered trivial). > Current set theory has forgotten about theoretic definitions > of the infinite at all and uses the notion "infinity" just as seems > necessary to avoid too obvious contradictions. You just don't know what you're talking about. Why don't you just read a textbook in set theory? MoeBlee
From: mueckenh on 17 Oct 2006 12:49 William Hughes schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > William Hughes schrieb: > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > William Hughes schrieb: > > > > > > > > > > But the end time of the problem (noon) does not correspond to > > > > > an integer (neither in standard mathematics, nor in your > > > > > system, whether or not you interpret the problem as dealing > > > > > with infinite integers as well as finite integers). So the function > > > > > 9n does not have a value at noon. There is no way > > > > > it can be continuous at noon. And since there is no > > > > > value of n that corresponds to noon, 9n cannot be used > > > > > to determine the number of balls in the vase at noon. > > > > > > > > But the function n can be used to determine the number of balls removed > > > > from the vase at noon? > > > > > > > > > > Nope. [There are no balls removed from the vase at noon] > > > > Arbitrary misunderstanding? > > > > > The function 9n has nothing to do with the number of > > > balls in the vase at noon. > > > > But the function n can be used to determine the number of balls having > > been removed > > from the vase at noon? > > > No. There are no balls removed from the vase at noon. > > Note, that there is no time "just before noon". At any time > before noon there remain an infinite number of steps. > > So no value of n is close to the end. > > The balls are removed during an infinite number of > steps. Please read carefully: But the function n can be used to determine the number of balls *having been* removed from the vase at noon? (That means up to noon.) Regards, WM
From: mueckenh on 17 Oct 2006 12:52 David Marcus schrieb: > Are you now saying that you have a proof of > > lim_{t->infty} 9t = 0 > > in standard mathematics? If so, please give the proof (and just the > proof). I prefer to hear what your result is, concerning the set of balls in the vase at noon. If you give an answer, I will show you the contradiction. > I gave my translation of the ball and vase problem. I don't see any > contradictions that follow from it. If you say it implies two > contradictory conclusions, please state them and your proof. What was the result of your translation? Are there balls in the vase at noon or not? Regards, WM
From: MoeBlee on 17 Oct 2006 13:22 mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > MoeBlee schrieb: > > > > > > > Han de Bruijn wrote: > > > > > Set Theory is simply not very useful. The main problem being that finite > > > > > sets in your axiom system are STATIC. They can not grow. > > > > > > > > Set theory provides for capturing the notion of mathematical growth. > > > > Sets don't grow, but growth is expressible in set theory. If there is a > > > > mathematical notion that set theory cannot express, then please say > > > > what it is. > > > > > > Obviously the notion of "rational relation" as used in the binary tree > > > cannot be expressed by mathematical notion: > > > 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. The set of paths is uncountable, but as we have seen, it > > > contains less elements than the set of edges. Cantor's diagonal > > > argument does not apply in this case, because the tree contains all > > > representations of real numbers of [0, 1], some of them even twice, > > > like 1.000... and 0.111... . Therefore we have a contradiction: > > > > > > Card(R) >> Card(N) > > > || || > > > Card(paths) =< Card(edges) > > > > What I see above is a lot of mathematical terms that are used in set > > theory with precise definitions, but for which I do not know your own > > personal definitions. > > There are no personal definitions. There is only one extension of > current state, which, however, is not in contradiction with any axioms, > namely that edges can be subdivided and the shares can be counted. > > You may not agree with the axioms of set theory, > > Which part of my proof is not in agreement with current axioms and > definitions of set theory? First you say the notion of 'rational relation' (whatever that means) "cannot be expressed by mathematical notion". Then you challenge me to say what part of your proof is in conflict with set theory. What is the notion of 'rational relation' that "cannot be expressed by mathematical notion"? Are defining a certain relation in set theory or are you definining a relation you claim not to exist in set theory? MoeBlee
From: William Hughes on 17 Oct 2006 13:27
mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > William Hughes schrieb: > > > > > > > > > > > > But the end time of the problem (noon) does not correspond to > > > > > > an integer (neither in standard mathematics, nor in your > > > > > > system, whether or not you interpret the problem as dealing > > > > > > with infinite integers as well as finite integers). So the function > > > > > > 9n does not have a value at noon. There is no way > > > > > > it can be continuous at noon. And since there is no > > > > > > value of n that corresponds to noon, 9n cannot be used > > > > > > to determine the number of balls in the vase at noon. > > > > > > > > > > But the function n can be used to determine the number of balls removed > > > > > from the vase at noon? > > > > > > > > > > > > > Nope. [There are no balls removed from the vase at noon] > > > > > > Arbitrary misunderstanding? > > > > > > > The function 9n has nothing to do with the number of > > > > balls in the vase at noon. > > > > > > But the function n can be used to determine the number of balls having > > > been removed > > > from the vase at noon? > > > > > > No. There are no balls removed from the vase at noon. > > > > Note, that there is no time "just before noon". At any time > > before noon there remain an infinite number of steps. > > > > So no value of n is close to the end. > > > > The balls are removed during an infinite number of > > steps. > > Please read carefully: But the function n can be used to determine the > number of balls *having been* removed from the vase at noon? (That > means up to noon.) > No. The function can be used to determine the number of balls having been removed from the vase at any time before noon. The function cannot be use to determine the number of balls having been removed from the vase at noon, because the function does not have a value at noon. We can take the limit of the function as time approaches noon, but we cannot say that this limit is the number of balls having been removed from the vase at noon without further analysis. - William Hughes |