Cantor Confusion
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From: David Marcus on 7 Dec 2006 19:20 Mark Nudelman wrote: > On 12/7/2006 3:15 AM, Eckard Blumschein wrote: > > Just try and refute: > > > > Reals according to DA2 are fictitious > > Reals according to DA2 are fictitious > > With fictitious I meant: They must not have a directly approachable > > numerical address. This was the basis for the 2nd DA by Cantor afer an > > idea by Emil du Bois-Raymond. > > > > Good luck > > Can you define what you mean by a "directly approachable numerical address"? He hasn't so far, despite being asked to. > Do you mean that SOME reals don't have such an address, or that NONE of > them have such an address? Is "sqrt(2)" a "directly approachable > numerical address"? > > If you're saying that some reals do not have a finite decimal > representation, that's true, but so what? Why does that make them any > more "fictitious" than the integers? Ah, but it sounds better to say they don't have a "directly approachable numerical address" than to say they don't have a "finite decimal representation". Much more mysterious-sounding. -- David Marcus
From: David Marcus on 7 Dec 2006 19:26 William Hughes wrote: > > mueckenh(a)rz.fh-augsburg.de wrote: > > <statements which make it clear that certain > things which were though to be settled are not settled> > > Terminology: If we say that X exists > then we can use X in a proof. > > On Dec 4 I wrote: > > You now agree that a potentially infinite set can have > a cardinal number and that this cardinal is not > a natural number. > > As your latest post points out, this is not (or > no longer) true. > > Stop me when I make a statement you disagree with Although, just because WM agrees today doesn't seem to imply that he will agree tomorrow. -- David Marcus
From: Dik T. Winter on 7 Dec 2006 19:28 In article <1165493179.371372.259190(a)16g2000cwy.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > In article <1165403079.475990.150660(a)f1g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: .... > > > Please do not mistake your misunderstandings for > > > errors of mine. For instance I never stated that nodes represent > > > numbers, as you erroneously believed. > > > > Pray reread what I wrote: "the nodes can be made to represent numbers in > > your tree". That is an easy exercise, I even did show it. The same for > > the edges, I did show that too. So actually the nodes and edges also > > represent numbers in some way. > > In the same way as the first few digits of a real number represents a > number. 3.1, 3.14, and so on represent numbers in some way. But that is > not at all important or interesting for the tree argument. So why did you state that I erronously believed that nodes represent numbers? > > > Not at all! I represent numbers by standard binary notations. > > > > It is using the limits where you are doing something non-standard. > > I do nothing. The tree cares that even in the limit the number of paths > cannot become uncountable. 2^n remains the cardinal number of a > countale set, even in the limit n --> oo. That's why I devised the > tree! And it is exactly that what is wrong. For each finite n 2^n is the cardinal number of a countable set (even of a finite set), that does not make something like that also true in the limit. It is easy enough to construct a bijection between the natural numbers and the edges, because the edges are countable. Contrary to what you write elsewhere, you have *not* constructed a surjection from the edges to the paths. If you think you did that present us with an edge that maps to 1/3, and show how the mapping is constructed. -- 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 7 Dec 2006 19:21 In article <1165531790.881743.134350(a)n67g2000cwd.googlegroups.com> cbrown(a)cbrownsystems.com writes: > If you claim to have constructed T, then you claim that you have > constructed a function T such that for any edge e, say the first edge > to the left, T(e) is a path. It is easy enough to construct a surjection from the edges to the binary rational numbers with terminating expansion. But indeed, WM does not show a surjection at all. -- 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 7 Dec 2006 19:29
In article <1165493310.671029.265800(a)16g2000cwy.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > In article <1165403177.893469.76590(a)j72g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: > > ... > > > > > To have a big mouth is not enough to create a world, not even a notion. > > > > > You would see that if you tried to say where the assumed object > > > > > existed. > > > > > > > > In my mind. I can reasonably think about the set of all natural numbers. > > > > Honest, I have no problem with it. > > > > > > You believe you could. That is a difference to "can". Other people > > > think they can reasonably think of having an immortal soul. > > > > How do you know I can not? How do you know what I can think? > > Perhaps you think so. But facts are contrary. Just as in case of > infinity. Your facts only. For me it is easy enough to think about an infinite set. When I think about the number line, I think about an infinite set of points. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |