Cantor Confusion
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From: MoeBlee on 18 Oct 2006 14:29 mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > > > 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? > > Meanwhile there are many who understand the binary tree. Perhaps you > will follow the discussion, then you may understand it too. I just asked a questions. More basically, what is your context? Is your argument about binary trees supposed to be within set theory or not? MoeBlee
From: MoeBlee on 18 Oct 2006 14:43 mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > MoeBlee schrieb: > > > > > > You have an interpretation of a thought experiment that differs from > > > > the interpretation of other people. That doesn't make set theory > > > > inconsistent. It just makes set theory not suitable for your intuitions > > > > regarding the thought experiment. > > > > > > 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. > > > > > > This is the contradiction. No matter what the result (3) may be. > > > > In other words, just as I said, you have an interpretation of a thought > > experiment that differs from the interpretation of other people, > > And according to your intuition, the axioms of ZFC and the rules of > logics are more likely to believe and do not contradict the fact that > accumulating numbers will not yield an empty set? The balls in a vase problem is not within set theory. The balls in a vase problem is an infinitary thought experiment - premised on conditions that cannot (or at least, I don't know how they could) occur in any physical experiment, that, while interesting in certain informal ways, is not one that I know to be crucial to any of the mathematics of set theory. As to "accumulating numbers", if you provide me with anything IN SET THEORY (not just in thought experiments that are NOT in the language of set theory) that is a contradiction, then I will be interested. And my thoughts about set theory axioms are not reducible to whether I "believe them" but rather as to concerns for whether they axiomatize mathematics, are consistent, and, as a bonus, correspond to informal concepts in the understanding of abstractions such as sets and operations with sets. MoeBlee
From: William Hughes on 18 Oct 2006 14:55 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: > > > > > > > > > > > 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. > > Correct. > > > 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. > > Correct. It is not possible to have the value "all natural numbers". Piffle. It is perfectly possible for a function to have the value "all natural numbers". However, this function does not have this value. > > > 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. > > Correct >. Therefore the assertion that all natural numbers were outside > the vase at noon is unjustified like any quantitative assertion about > all natural numbers. > Piffle. Just because we cannot use the limit, does not mean we cannot use something else. - William Hughes
From: Virgil on 18 Oct 2006 15:00 In article <1161158151.840234.251500(a)f16g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1161027429.194776.277830(a)m73g2000cwd.googlegroups.com>, > > 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? > > > > Not even that. > > How then do you know that all the balls have been removed? > > Regards, WM The problem states that for a given set of balls and an initially empty vase there is a time at which each ball is inserted into the vase and a subsequent time at which it is removed and then the problem asks whether a time later than any of these times whether all balls have been removed. Since each ball must have been removed by that later time, all balls must have been removed by that time.
From: Virgil on 18 Oct 2006 15:06
In article <1161158289.063574.300240(a)b28g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > Unless someone can dispute it, how can anyone claim that there are any > > balls in the vase at noon? > > It is obviously nonsense to see only the one side as set theorists do. > Infinity has two sides. The other side says the opposite. > > Further the result o the gedankenexperiment must not depend on > switching numbers. Removing balls 1, 11, 21, ... does not change the > quantities in fact, but according to set theory it does. Therefore set > theory has been contradicted. If under one rule ball 2 is specifically removed and under another it is specifically not removed, then the two rules do not both match the same experiment. According to the original statement of the gedankenexperiment, for every ball, n, there is a time of insertion before noon, ti_n and a time of removal before noon, tr_n, with the time of removal never before the time of insertion and always before noon. I.e., ti_n <= tr_n < noon. In addition ti_n <= ti_{n+1} for all n and tr_n < tr_{n+1} for all n. Does anyone dispute that all this is valid for the original gedankenexperiment? Unless someone can successfully dispute it, how can anyone claim that there are any balls in the vase at noon? |