Cantor Confusion
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From: Virgil on 9 Oct 2006 13:08 In article <1160403637.269409.243190(a)k70g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1160295281.279569.143920(a)m7g2000cwm.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > If a set of axioms yields the theorems A and nonA, then this set is > > > useless. The axioms of ZFC yield the theorems "the vase is empty at > > > noon" and "the vase is not empty at noon". > > > > Not so. One has to add "Mueckenh"'s, or at least other assumptions, to > > ZFC to get a nonempty vase at noon. > > > > The only ZFC analysis coincides with: > > > > Let A_n(t) be equal to > > 0 at all times, t, when the nth ball is out of the vase, > > 1 at all times, t, when the nth ball is in the vase, and > > undefined at all times, t, when the nth ball is in transition. > > > > Note that noon is not a time of transition for any ball, though it is a > > cluster point of such times. > > > > let B(t) = Sum_{n in N} A_n(t) represent the number of balls in the vase > > at any non-transition time t. > > > > B(t) is clearly defined and finite at every non-transition point, as > > being, essentially, a finite sum at every such non-transition point. > > > > Further, A_n(noon) = 0 for every n, so B(noon) = 0. > > Similarly when t > noon, every A_n(t) = 0, so B(t) = 0 > > > > There is no ZFC-compatible argument for having a non-empty vase at or > > after noon that does not require assumptions beyond those of ZFC. > > > > And it is with those additional assumptions that "Mueckenh" and others > > make, that the conclusions of ZFC conflict. > > The assumption is that "lim{n->oo} 9n = 0" is wrong. If this assumption > is not wrong in ZFC then ZFC is useless. > > Regards, WM No one but "Mueckenh" makes that limit assumption (B(t) is specifically NOT continuous at t = 0), so that ZFC is only useless to "Mueckenh".
From: Virgil on 9 Oct 2006 13:15 In article <1160404241.528341.215260(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1160308871.194701.44520(a)c28g2000cwb.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > 0) There is a bijection between the set of balls entering the vase and > > > |N. > > > > When? > > Independent of time, all the balls are enumerated. WRONG! At any time before noon, not all the balls have entered the vase. If "Mueckenh" means all the balls that will be entered, "entering" does not carry that meaning in English. > > > > > 1) There is a bijection between the set of escaped balls and |N. > > > > When? > Independent of time, the balls are enumerated. Similar comment. The set of balls which have escaped is very much time dependent. >But the time t can be > enumerated by the balls leaving the vase. Or vice versa. > > > > > 2) There is a bijection between (the cardinal numbers of the sets of > > > balls remaining in the vase after an escape)/9 and |N. > > > > This does not occur ever. > > The contents of the vase is A(t) = 9t. It will no be reset for t --> > oo. Lim {t-->oo} 9t = 0 is as false as Lim {k-->oo} Sum{1 to k} = 0. After an escape carries the sense of before the next escape. Only finitely many balls will have been moved at that point in time.
From: Virgil on 9 Oct 2006 13:22 In article <1160405917.541616.241630(a)b28g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > No. I have the estimation Lim{n-->}n > 0. If you call this "intuition", > then some more fundamental laws of mathematics are invalid than can be > recovered by set theory. I would call it, at best, a typo. But there is no necessity that the state at t >= noon be determined by the limit you may have meant.
From: Virgil on 9 Oct 2006 13:24 In article <1160407082.513322.311190(a)b28g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > My conclusion is: > > > Either > > > (S is covered up to every position <==> S is completely covered by at > > > least one element of the infinite set of finite unary numbers > > > > Straight quatifier dyslexia. The fact that "for every x there exists > > a y such that" does not imply "there exists a y such that for every x" > > A nonsense argument. Your assertion is wrong in a linear set. Give an > example where the linear set covers a number which is not covered by > one member of the linear set. > > Regards, WM An argument claimed to be nonsense replied to by an argument which actually is nonsense.
From: David R Tribble on 9 Oct 2006 17:37
Mueckenheim wrote: >> But you cannot derive that the vase is not empty at noon from the >> observation that its contents cannot decrease? > Han de Bruijn wrote: > A picture says more than a thousand words. [Doesn't] it? > > http://hdebruijn.soo.dto.tudelft.nl/jaar2006/ballen.jpg I notice that there is no Y point at the rightmost X at "noon". |