Cantor Confusion
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From: Dik T. Winter on 23 Nov 2006 19:50 In article <1164126395.211430.7520(a)h54g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: .... > > But > > indeed, let's stop this non-discussion. Unless you come with a proof. > > You are not even able to comprehend potential infinity. > > Tell me which point is not accepted: > 1) Every line which contains an index of the diagonal, contains all > preceding indexes of the diagonal too. > 2) Every index of the diagonal is in a line. > 3) In order to show that there is no line containing all indexes of the > diagonal, there must be found at least one index, which is in the > diagonal but not in any line. This is impossible. > 4) There is no line with infinitely many indexes. > 5) Conclusion: There is no diagonal with infinitely many indexes. I do not accept (3). It effectively states: forall{n in N}thereis{m in N} (index m is not in line n) -> thereis{m in N}forall{n in N} (index m is not in line n) which is false. Quantifier dislexia again. -- 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 23 Nov 2006 19:54 In article <1164126570.219569.222590(a)k70g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > > > Well, the same in set theory. In one form it has AC as a fundamental > > > > truth, in another version it is not a fundamental truth. What is the > > > > essential difference between the cases? > > > > > > > That there is not an "underlying plane" which it is related to. > > > > So there should be an "underlying plane", whatever that may mean? > > For geometry, it is required, yes, for arithmetic it is not. And you do not think there is an underlying "something" to which set theory with AC and without AC are related? I still do not understand the essential difference. Moreover, AC has *nothing* to do with standard arithmetic. > > > You may call these entities numbers or not. My opposition stems from > > > their use in Cantor's list and the lacking trichotomy. > > > > So, now we may call them numbers (as you do). And, we are back to basics. > > Your misunderstanding of Cantor's argument and whatever. > > They have no representation which could be used for any form of > Cantor's list. Well, according to set theory they have such representations. -- 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 23 Nov 2006 20:01 In article <1164126713.968092.237570(a)k70g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > > > There are infinite paths > > > > in your tree, but they do not contain a node that represents (for > > > > instance) 1/3. So, if the nodes represent numbers (as you have said), > > > > > > Do you have a reference? > > > > Not needed. Just above you state that the nodes represent the bits 0 or > > 1. I have shown how you could concatenate the representation of a node > > with the representations of its parent nodes to get a number. > > A number consists of bits. Some numbers consist even of one bit. But > you must not mix up these terms. > A number like 1/2 consist of the bit sequence 0.1000.... That is a path > in my tree: As an infinite bit sequence, yes, but there is *no* node in your tree that represents that bit sequence. > > > > 1/3 is not in your tree. > > Of course it is, like 0.333... is in Cantor's list of decimals. No, there is *no* node in your tree that represents 1/3. Because there is *no* node in your tree that represents an infinite sequence. On the other hand, Cantor's diagonal proof is about infinite sequences. > > You are not clear about what the numbers in > > > > your tree are. Are they the nodes? Are they the paths? Sometimes > > > > you say one thing other times you say something different. So to get > > > > proper understanding. What are the things that represent numbers? > > > > > > Infinite paths. > > > > I do not understand. You stated the nodes represent bits. > > Yes. That's the stuff numbers are built from. This makes it still less clear. -- 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 23 Nov 2006 20:10 In article <34i6m255sfv5upv6nenqm6menle2gp8l8b(a)4ax.com> Lester Zick <dontbother(a)nowhere.net> writes: > On Tue, 21 Nov 2006 03:04:42 GMT, "Dik T. Winter" <Dik.Winter(a)cwi.nl> > wrote: > >In article <8m8pl2pj1icbeven2hq7rp4hq1rufqh1u2(a)4ax.com> Lester Zick <dontbother(a)nowhere.net> writes: > >... > > > Why are square circles unimaginable? > > > >They are not. With the Manhattan measure of the plane, each circle is > >a square. > > Then what is a square? Pray, tell me. I would say that a straight line in the Euclidean plane is a line of the form ax + by = c. With the standard formulas for angles it is easy enough to get rectangles. And I would say that a rectangle is a square when the sides have equal length (this is the point where the measure creeps in). So we have a rectangle enclosed by the lines: x + y = 1 x - y = 1 - x + y = 1 - x - y = 1 Now define the Manhattan measure: d((x1,y1), (x2,y2)) = ||x1 - x2| + |y1 - y2|| and we see easily enough that the figure enclosed by the lines above is a square with sides with length 2. A circle is a figure where each point has the same distance to a common centre, and it is also easy to show that the points on the boundary of that square have the same distance to the origin: 0. -- 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 23 Nov 2006 20:25
In article <4564875D.4040504(a)et.uni-magdeburg.de> Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> writes: > On 11/10/2006 3:03 AM, Dik T. Winter wrote: > > But in mathematics comparing numbers comes in quite late in > > the process of definition. > > With mathematics you perhaps mean the possibly questionable attempt to > formally justify what has proven successful. What has been shown successful in some cases. The formalisation allows us to find under precisely what conditions it will be successful, and why it is successful under those conditions. In addition it allows to provide (possibly) more general alternatives that work under less strict conditions. > > It starts when the ordering axioms are > > given. In a set a ardering relation may be can be defined. Let us use > > >= as the basis. > > Mathematicians like you are hopefully aware of the trifle that the > relation >= cannot be applied to the really real numbers. Oh. What are "really real numbers"? What you are missing is that mathematics provides an idealisation of arithmetic (amongst others), and from that background provides processes to do "real" calculations to get results in (amongst others) the physical world. The whole field of numerical mathematics could barely have been build without the idealised mathematical background. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |