Cantor Confusion
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From: Han de Bruijn on 11 Oct 2006 09:07 David R Tribble wrote: > 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 >> > David R Tribble schreef: > >>>I notice that there is no Y point at the rightmost X at "noon". >> > Han.deBruijn wrote: > >>True. That symbolizes the fact that there is no noon. >>It's also a fact that you cannot do something else with infinity >>than clipping it against the window, graphically speaking. > > I get it now. We can never "get to" noon, so it does not exist. > > Likewise, the infinite sum > s = 1/2 + 1/4 + 1/8 + 1/16 + ... > can never be "reached", so it is not actually equal to 1. Not quite likewise. The infinite sum (s) does NOT have to be clipped against a graphics window. Han de Bruijn
From: Han de Bruijn on 11 Oct 2006 09:10 Dik T. Winter wrote: > In article <1160496764.343276.115430(a)k70g2000cwa.googlegroups.com> > Han.deBruijn(a)DTO.TUDelft.NL writes: > > David R Tribble schreef: > > > 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". > > > > True. That symbolizes the fact that there is no noon. > > Ah, a reincarnation of Zeno. No. A limitation of the cosmic window. Han de Bruijn
From: Dik T. Winter on 11 Oct 2006 09:06 In article <1160551520.221069.224390(a)m73g2000cwd.googlegroups.com> "Albrecht" <albstorz(a)gmx.de> writes: > David Marcus schrieb: .... > > I don't follow. How do you know that the procedure that you gave > > actually "defines/constructs" a natural number d? It seems that you keep > > adding more and more digits to the number that you are constructing. > > What is the difference to the diagonal argument by Cantor? That a (to the right after a decimal point) infinite string of decimal digits defines a real number, but that a (to the left) infinite string of decimal digits does not define a natural number. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 11 Oct 2006 09:19 In article <1160562822.815626.82270(a)c28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: .... > > What *might* be a sensible definition of a limit for a sequence of sets of > > naturals is, that (given each A_n is a set of naturals), the limit > > lim{n = 1 ... oo} A_n = A > > Yes, in that manner the definition runs. Cantor does not write n = > 1...oo but puts only the n (he uses nue) under the limit. But the > meaning is clearly this one. > > > exists if and only if for every p in n, there is an n0, such that either > > (1) p in A_n for n > n0 > > or > > (2) p !in A_n for n > n0. > > In the first case p is in A, in the second case p !in A. > > > > With that definition, indeed, > > lim{n = 1 ... oo} A_n = N, > > but also > > lim{n = 1 ... oo} {n + 1, ..., 10n} = 0. > > > > I do not think you are meaning that definition. So what *is* your > > definition? > > I do *not* believe that omega exists or is a useful notion. Therefore I > do not give a definition, but, if necessary during the discussion, I > use the only posdsible one as given by Cantor (see above). So the definition I gave for a limit of a sequence of sets you agree with? Or not? I am seriously confused. With the definition I gave, lim{n = 1 .. oo} {n + 1, ..., 10n} = {}. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: mueckenh on 11 Oct 2006 10:31
imaginatorium(a)despammed.com schrieb: > The cranks universally proceed from some intuitions about the real > world that - essentially - there are no discontinuous functions in > physics. If there are no discontinuous functions, then it follows that > you can "swap limits" - and in particular that if balls(t) is a > function representing the number of balls in state IN at time t, then > lim t->0 (balls(t)) = balls(0). If there _are_ discontinous functions, > this clearly does not follow. > > I do find it hilarious that one of the cranks proudly displays a graph > of y = 1/x on his website, and seems to believe this tells us "the > value of 1/0". Well, a little education would be no idleness in > something or other, I don't doubt. > If discontinuous functions were easily allowed everywhere, why then do you think that lim{n-->oo} n < 10 or lim{n-->oo} 1/n > 10 would be wrong? Regards, WM |