Cantor Confusion
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From: William Hughes on 18 Oct 2006 16:44 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. Nope. By the definitions you use, that set is not countable. > And that set cannt be listed. And here is your problem. Uncountable means unlistable. - William Hughes
From: Virgil on 18 Oct 2006 16:48 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. > Which side is heads and which is tails? > Further the result o the gedankenexperiment must not depend on > switching numbers. Whyever not? Suppose we "switch numbers" no that none of the odd numbered balls ever get inserted, does that not change the experiment? >Removing balls 1, 11, 21, ... does not change the > quantities in fact, but according to set theory it does. Not at any time before noon. And it changes things at noon because changing numbers leaves balls in the vase that were not left in the vase in the original experiment. > Therefore set > theory has been contradicted. Only "Mueckenh"'s version of set theory have been contradicted, but as "Mueckenh" includes assumptions that standard set theory does not, standard set theory is unaffected by peculiarities of the"Mueckenh" version. > > Regards, WM
From: MoeBlee on 18 Oct 2006 16:52 mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > > MoeBlee wrote: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > 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. In > > > > mathematics, the weight of the digits of reals is 10^(-n). Infinite > > > > sequences of digits with equal weight are undefined and devoid of > > > > meaning. > > > > > > The proof doesn't contradict the fact that the members of the sequence > > > are divided by greater and greater powers of ten. That fact is > > > mentioned in the previous proof showing the correspondence between the > > > sequences and real numbers. We prove that every sequence corresponds to > > > a real number where the real number is the limit of the sum of the > > > sequence made by taking greater and greater powers of ten in the > > > denominators, and that every real number corresponds to such a > > > sequence. THEN we proceed to the diagonal argument. > > > > > P.S. Again, if you disagree with the proof, then please just say what > > axiom or rule of inference you reject. In the meantime, again, there is > > no rational basis whatsoever for disputing that the argument does > > indicate a proof from the axioms per the rules of inference. > > 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. You just went right past what I wrote. You just completely ignored what I wrote, which was in direct response to you. We do NOT neglect the exponentiation. MoeBlee
From: Virgil on 18 Oct 2006 16:56 In article <1161182462.511657.72280(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > > > It is impossible to show a "valid" mathematical proof against set > > > theory. > > > > Ah, so you agree that you can not prove an inconsistency using mathematical > > terms of proof. > > I agree that set theorists will ever accept any proof of inconsistency > as valid. At least until one has been presented which does not require assumptions alien to standard set theories. > > > > > We have discussed the vase and I would not have believed in > > > advance that anybody could maintain arguments here like Virgil and > > > William and others. > > > > No. You never would believe that anybody would use mathematical proofs > > against your intuition. > > It is not *intuition* to find a proof of lim{n-->oo} n = 0 is wrong. As that particular limit has never been claimed by set theoreticians, it is irrelevant > > > Nothing, as soon as we withdraw to call set theory mathematics. > > > Oh. In that case, please do not call it mathematics. > > The copyright of this name is protected for a science that is one of > the eldest on earth. Wrong! mathematics is not a science now, and has never been one, though it has often been applied by scientists. > It has been occupied illegally by a gang of > gamblers but will be reinstalled within the next years. Not in "Mueckenh"'s lifetime will any such crime be committed as "Mueckenh" so looks forward to.
From: Virgil on 18 Oct 2006 17:00
In article <1161182545.370808.223780(a)k70g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Randy Poe schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > Randy Poe schrieb: > > > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Virgil schrieb: > > > > > > > > > > > > According to the ZFC system: The vase is empty at noon, because > > > > > > > all > > > > > > > natural numbers left it before noon. > > > > > > > By means of the ZFC system we can formulate sequences and their > > > > > > > limits > > > > > > > in mathematical language. From this it follows that lim {n-->oo} > > > > > > > n > 1. > > > > > > > And from this it follows that the vase is not empty at noon. > > > > > > > > > > > > By what axiom do you conclude that the limit as t increases towards > > > > > > noon > > > > > > of any function and the value of that function at noon must be the > > > > > > same? > > > > > > > > > > By that or those axiom(s) which lead(s) to the result lim {t-->oo} > > > > > 1/t > > > > > = 0. > > > > > > > > Here is your theorem: Let f(x) be any function f:R->R. Then > > > > lim(x->0-) f(x) = f(0). That is, the limit of f(x) as x approaches > > > > 0 from the left is f(0). > > > > > > > > Can you show me how the axiom(s) you describe prove > > > > that theorem? > > > > > > > > Can you then show me how the theorem applies to this > > > > function? f(x) = 1 if x<0, f(x) = -1 if x>=0. > > > > > > If there is no stepwise continuity in f(t) = n, can you show me why the > > > set of balls/numbers removed from the vase is containing all natural > > > numbers at noon after the number of transactions t --> oo? > > > > Simple. Because n being a natural number => there is a removal time > > t n < noon. Therefore every natural is a member of the set of > > balls removed before noon. > > The cardinal numbers of the sets of balls residing in the vase are also > natural numbers. f(t) = 9, 18, 27, ... which grow without end. How can > such a function take on the value zero? The "number of balls in the vase" as a function of time is highly discontinuous, particularly around noon. > > > > What I don't understand is how anyone can think there are balls > > with removal times which are still in the vase. > > > What you don't understand is that it s impossible to catch the whole > set N. if you think you have done so, then there are many other > infinite sets of natural numbers not yet caught. It is simply Hilberts > hotel. It s simply an inconsistency of the idea that N could be > completed consistently. Except that the statement of the problem /presumes/ that N can be "completed consistently". One cannot even state the problem, much less solve it, unless that assumption is granted. |