Cantor Confusion
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From: Virgil on 29 Nov 2006 18:28 In article <1164834278.356038.117340(a)80g2000cwy.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > In your terminology every digit 3 in 0.333... represents a number. The > first 3 represents the number 0.3, the second 3 represents the number > 0.33 In my number system it represents 0.03. > and the 10th 3 represents the number 0.3333333333. In my number system it represents 0.0000000003. > What does this > interpretation have to do with the existence of the decimal > representations of the number 1/3, namely 0.333... in a list of real > numbers? The way you have stated it, nothing. > As there is no digit "3" at an infinite distance from the "0." in > 0.333..., there is no infinite decimal expansion. There is that same confusion surfacing again. An endless sequence of single decimal digits creates an "infinitely long" expansion with no digit more that finitely far from the decimal point. Until WM learns this, his attempts to work with set theory will continue to fail. > EIT: > > > Eh? N is the set of natural numbers. The quantifiers can be exchanged > > when N is finite. Not when something completely different is finite. > > We consider only finite lines. Within every line the quantifiers can be > exchanged. But when what is being quantified are the lines themselves, that is not "within" a line, and cannot in general be 'exchanged". AxEy f(x,y) and EyAx f(x,y) are not in general equivalent. E.g. For every woman x there is (or at least once was) a woman y such that y is x's mother But is it the case that there is a woman y such that for every woman x, y is x's mother? Only in EB's family!
From: Dik T. Winter on 29 Nov 2006 21:19 In article <1164816790.379338.139370(a)h54g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > MoeBlee schrieb: > > > I think if one swells to explosion about his knowledge of set theory, > > > he should at least know the very foundation. But I know, that you do > > > not even understand the simple texts of Fraenkel et al. > > > > No, YOU radically MISunderstand what Fraenkel, Bar-Hillel, and Levy > > wrote. > > How can you judge about that without the slightest idea of what they > wrote? Only for the lurkers: Fraenkel et al. write: "Platonistic point > of view is to look at the universe of all sets not as a fixed entity > but as an entity capable of "growing", i.e., we are able to "produce" > bigger and bigger sets." So a set (like the set of all sets) is not a > fixed entity. There is nothing in that that shows that a set is not a fixed entity. You are able to produce bigger and bigger sets, but they are all different. When the universe has grown it allows bigger sets than where originally allowed, but the sets originally allowed are still sets and still the same and did not grow. How you conclude from the above statement that sets themselves are growing escapes me. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Virgil on 29 Nov 2006 21:29 In article <1164834775.086587.74220(a)l39g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > In article <1164637800.628277.302340(a)14g2000cws.googlegroups.com> > > mueckenh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: > > ... > > You wrote: > > > > > > > Cantor wrote: "It is remarkable that removing a countable set > > > > > > > leaves the plaine *being connected*." But it is not remarkable > > > > > > > that removing an uncountable set leaves the plaine being > > > > > > > connected? What a foolish assertion. > > ... > > > > > Yes, but Cantor considered only point sets which are dense. > > > > > > > > You did not state that above. > > > > > > I wrote about algebraic and transcendental numbers. So being dense is > > > implied and need not be mentioned. > > > > Perhaps. But that was not clear from what you wrote. And you even > > let the examples where the inner portion of a circle were removed go, > > while the apparently do *not* satisfy what Cantor considers! > > You cannot expect that I carefully read and answer every sentence of > Mr. Bader. It is only by accident that I saw and answered his posting > (because he seems to be very interested in my papers and to study them > in great detail). I usually ignore him. > > > > > > But under *those* conditions there there are still uncountable sets > > > > that > > > > leave the plane connected and also sets that leave the plane > > > > disconnected. > > > > > > Of course, but Cantor did not recognize it. And as far as I know nobody > > > recognized it before I did. At least nobody mentioned it. > > > > Of course nobody mentions it, because it is highly unremarkable! > > Well, after I showed it. Cantor's proof cannot be used to find an > uncountable set which leaves the plane connected. As Cantor's proof does not touch on connectivity in the plane, that is irrelevant. Besides which, it is quite easy to find uncountable subsets of the plane whose excision leaves the remainder not only connected but simply connected.
From: Dik T. Winter on 29 Nov 2006 21:29 In article <1164834775.086587.74220(a)l39g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: .... > > Of course nobody mentions it, because it is highly unremarkable! > > Well, after I showed it. Cantor's proof cannot be used to find an > uncountable set which leaves the plane connected. Well, of course not, it was not targeted at that goal. And the fact that it is possible is totally unremarkable. > > Remove > > all irrational points and you still have a connected space. Remove in > > addition a line and you have a disconnected space. What is remarkable > > about that? What *is* remarkable is that whatever countable dense set > > you remove, the result is still connected. And the property of being > > dense is not even needed for both results. > > Just this shows that Cantor, who insists on it, was not aware of the > fact that uncountable sets leave the plane connected. Because that fact is false. Removing dense uncountable subsets can leave the plane connected or disconnected. Where did Cantor insist that removal of an uncountable set leaves the plane disconnected (as you assert he did insist on)? -- 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 29 Nov 2006 21:54
In article <1164834278.356038.117340(a)80g2000cwy.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > In article <1164637973.064418.67510(a)45g2000cws.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: > > ... > > > > Nothing more than opinion. You think there are no such representations, > > > > so everything that states that there are such representations is wrong. > > > > > > I proved that there are no such representations. If they were (the > > > paths in the infinite tree, for instance), then we had a contradiction. > > > > You are, with your tree, wrong. > > Please try to understand it first! Up to now you didn't. I try to understand it, but you are not much help. > > In your tree (where the nodes represent > > bits), it can be shown that all nodes represent a number: the collection > > of bits found when going from the root to the current bit. So the nodes > > represent all rational numbers in [0,1) where the denominator is a power > > of two. You state that the paths represent numbers. Let us analyse that, > > and begin with the finite paths that terminate at some finite edge. > > In my tree there are no finite (terminating) paths! Sorry, as far as I see your tree has also finite paths. I see paths from the root to the first nodes below the roots. > >When > > you consider such paths, you can assign numbers to each final edge: the > > same number as the node were it comes from (nodes represent bits, so when > > you come at an edge you can only add the bit of the last node you passed, > > you can not look in the future). So your paths (whether finite or > > infinite) all represent a number that is also represented by a node. > > In your terminology every digit 3 in 0.333... represents a number. The > first 3 represents the number 0.3, the second 3 represents the number > 0.33 and the 10th 3 represents the number 0.3333333333. What does this > interpretation have to do with the existence of the decimal > representations of the number 1/3, namely 0.333... in a list of real > numbers? What is the tree? What are the nodes? You again switch to something different. But in a decimal tree, where each node is connected by edges to ten subnodes and where each of those subnodes is assigned a decimal digit, it can be shown that also each node represents a finite decimal number. And in that tree 1/3 is not represented by any node. But 0.333... is not a path in that tree, just as I wrote earlier. And, indeed, 0.333... does not represent a number until that notation has been defined. > > In your infinite tree, all edges come fome a node > > at finite distance, and so all edges together only represent the numbers > > with a finite binary expansion. > > So all decimal representations represent finite decimal expansions? Or > what is the difference? The difference is that infinite decimal representations are not defined as given, there is a concept behind it: limit. > > > Then 1/3 is not in Cantor's list. > > > It is. > > Can you imagine to represent the numbers in Cantor's list by paths? No, your paths represent finite expansions only. > > Again, see above. As there is no node at an infinite distance, there is > > also not an edge that comes from such a node, and so all edges represent > > a number with a finite binary expansion. > > As there is no digit "3" at an infinite distance from the "0." in > 0.333..., there is no infinite decimal expansion. > What is the difference to a representations by paths? Limits. > EIT: Eh? O, you are responding again to more than one article at once without giving proper references. > > Eh? N is the set of natural numbers. The quantifiers can be exchanged > > when N is finite. Not when something completely different is finite. > > We consider only finite lines. Within every line the quantifiers can be > exchanged. And due to your quoting all context has been lost. If I remember well it was about: forall{n in N} thereis(m in N} {etc.} and: thereis{m in N} forall{n in N} {etc.} The quantifiers are about N. And N is not finite, so you can not exchange them. What is stated in "etc." is irrelevant. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |