Cantor Confusion
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From: Virgil on 18 Oct 2006 18:07 In article <1161183932.920021.201110(a)b28g2000cwb.googlegroups.com>, 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? > > Regards, WM According to the axioms of ZFC, when every natural numbered ball that has been inserted into the vase has been removed as well, the vase does not contain any of those balls.
From: Virgil on 18 Oct 2006 18:12 In article <1161200547.493424.95130(a)k70g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Neglecting the powers 10^(-n) converts an infinite sequence which can > possibly yield a meaningful result into an impossible sequence, which > cannot be treated at all. But I believe your intuition will hinder you > accept that. "Mueckenh" seems to think that when two real numbers can only be shown to differ by 10^(-n), for some natural number n, they need not be regarded as different at all.
From: Virgil on 18 Oct 2006 18:17 In article <1161200729.156817.65510(a)m7g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1161079685.233073.120000(a)k70g2000cwa.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > MoeBlee schrieb: > > > > > > There is no "equal weight" in the proof. > > > > > > > > > > You haven't yet noticed it? Each digit of the infinitely many digits of > > > the diagonal number has the same weight or importance for the proof. > > > > It is only necessary that each digit have non-zero weight in order for > > the difference between the diagonal and one of the list to be non-zero. > > > > > > > In > > > mathematics, the weight of the digits of reals is 10^(-n). > > At least in decimal notation, but as that makes all the weights > > > non-zero, that is sufficient to distinguish the diagonal from each of > > the listed numbers. > > Only for a finite diagonal. In the infinite case we have for example 1 > = 0.999... But if the diagonal is not allowed to contain any 1's, 0's or 9's, as in standard constructions of "diagonals", such cases cannot occur. > > > > >Infinite > > > sequences of digits with equal weight are undefined and devoid of > > > meaning. > > > > As no such sequences are involved, the comment is irrelevant. > > The diagonal is not an infinite sequence? Its digits do not have equal weight, as "Mueckenh" well knows, but each digit has a non-zero weight, and non-zero is enough.
From: Virgil on 18 Oct 2006 18:19 In article <1161200818.318369.114170(a)k70g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1161103769.338793.195850(a)i42g2000cwa.googlegroups.com>, > > 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.) > > > > Yes. All of them are. > > So you act and think biased. That is not science and not mathematics. It is mathematical induction, which is mathematics at least: Ball one is removed before noon, and for each ball n removed before noon, so is ball n+1. Therefore ALL balls are removed before noon.
From: Virgil on 18 Oct 2006 18:24
In article <1161201490.714487.312730(a)f16g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > jpalecek(a)web.de schrieb: > > > > The set of constructible numbers is countable. Any diagonal number is a > > > constructed and hence constructible number. > > > > No. Definition (from MathWorld): Constructible number: A number which > > can be represented by a finite number of additions, subtractions, > > multiplications, divisions, and finite square root extractions of > > integers. > > > > How do you represent the diagonal number (which is sort of a limit of a > > series) > > via FINITE number of +,-,*,/,sqrt( ) ? > > Which digit should not be constructible by a finite number of > operations? > > > > > Every list of reals can be shown incomplete in exactly the same way as > > > every list of contructible reals can be shown incomplete. > > > > No. > > A constructible number is a number which can be constructed. Definition > obtained from 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." > > Definition (by me): A number which can be constructed like pi, sqrt(2) > or the diagonal of a list is that what I call constructible. If you > dislike that name, you may call these numbers oomflyties. Anyhow that > set is countable. And that set cannt be listed. Therefore the diagonal > proof shows that a set of countable numbers is uncountable. That presumes, contrary to fact, that a list can only list constructible numbers if a diagonal is to be constructed from it. In fact it only requires that one be able to determine the nth decimal digit of the nth number in the list, which allows these numbers to be unconstructible. |