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From: mueckenh on 18 Oct 2006 11:05 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
From: Randy Poe on 18 Oct 2006 11:20 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 Crucial phrase missing: The cardinality f(t) of the set of balls in the vase at time t STRICTLY LESS THAN ZERO... > are also > natural numbers. f(t) = 9, 18, 27, ... which grow without end. How can > such a function take on the value zero? Because the set of balls at t=0 is not one of these sets. t=0 is not a time strictly less than zero. - Randy
From: imaginatorium on 18 Oct 2006 12:43 mueckenh(a)rz.fh-augsburg.de wrote: > imaginatorium(a)despammed.com schrieb: > > > > But the function of balls/numbers removed from the vase is a > > > continuously (stepwise) increasing one, containing all natural numbers > > > at noon? > > > > Uh, yes, unless I mysteriously misunderstand you... If takenout() is a > > function from time to the power set of the integers (i.e. it maps to a > > set of integers) then each natural number m is included in the set that > > takenout() maps to from time = -1/m. So by time zero, all natural > > numbers are included. > > > > Was there a question with that? > > And this result would change, if the numbers of the balls were > exchanged, for instance multiplied by 10 after having been inserted? Can you clarify what "exchanged" means? Obviously, if you do exactly the same thing with exactly the same balls, it makes no difference what is written on them, but perhaps that is not what you mean. Brian Chandler http://imaginatorium.org
From: MoeBlee on 18 Oct 2006 13:17 mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > > Han.deBruijn(a)DTO.TUDelft.NL wrote: > > > David Marcus schreef: > > > > > > > Han de Bruijn wrote: > > > > > > > > > > How can we know, heh? Can things in the real world be true AND false > > > > > (: definition of inconsistency) at the same time? > > > > > > > > That is not the definition of "inconsistency" in Mathematics. On the > > > > other hand, I don't know of any statements in Mathematics that are both > > > > true and false. If you have one, please state it. > > > > > > What then is the precise definition of "inconsistency" in Mathematics? > > > > How many times does it have to be posted? > > > > G is inconsistent <-> G is a set of formulas such that there exists a > > formula P such that P and its negation are both members of G. > > Like: The vase is empty a noon and the vase is not empty at noon. How many times does it have to be said that "The vase is empty at noon" is not a sentence in the language of set theory? You don't prove the inconsistency of a theory just by providing an interpretation of a thought experiment that contradicts other people's interpretation of a thought experiment. Set theory is inconsistent if and only there is a sentence IN THE LANGUAGE OF SET THEORY such that that sentence and its negation are both theorems of set theory. MoeBlee
From: MoeBlee on 18 Oct 2006 13:26
mueckenh(a)rz.fh-augsburg.de wrote: > That is correct. That is precisely the reason why the natural numbers > appear to be infinitely many. There are at most 10^100 different > numbers but there are far larger values. The set is finite, but has not > a largest element. Since you provide no theory, no axioms, no primitives, no definitions from primitives, such assertions as above are but counterintuitive doubletalk. MoeBlee |