Cantor Confusion
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From: mueckenh on 25 Nov 2006 04:13 Virgil schrieb: > > > By induction one can even prove that always a finite number of words > > has been mapped on the natural numbers. > > By induction one can also map an infinite set of words onto the natural > numbers. Then you think the natural numbers are covered by induction? You read my heart. I knew they were not actually infinite. > > > > Now, after you have "completed" this finite list of words, take it and > > construct a diagonal number. > > I prefer to take the infinite set of words. I know, but you don't get it by induction. > > > This is always a finite number, as the list is finite. > > Maybe yours is, mine isn't. > > > Fine. > > So we have proved that there is a finite word which was missing in the > > "complete" list. > > You haven't proved anything, If the set 1,2,3,...,n is finite, then the set 1,2,3,..., n+1 is finite too. Therefore, by induction we (that is: those who can think logically) prove that the generated set is always finite. Regards, WM
From: mueckenh on 25 Nov 2006 04:16 Dik T. Winter schrieb: > In article <1164126272.884271.43260(a)m7g2000cwm.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > ... > > > > I + I = II is very fine and reliable mathematics. Absolutely true. And > > > > this approach can be put forward --- very far. > > > > > > What do you mean with those symbols? How do you define "+" and "="? > > > What is the meaning of "I" and "II"? > > > > What do you mean with "There" or "exists" or "a"? > > How do you define "set" and "which" and "has"? > > What is the meaning of "no" and "elements"? > > > > I think these symbols and expressions deserve closer examination than > > "+" and "II". > > Oh, perhaps. But now you are entering the field of philosphy end exiting > the field of mathematics. Philosophy of mathematics belongs to mathematics. It is its foundation. The most solid one is MatheRealism. Regards, WM
From: mueckenh on 25 Nov 2006 04:18 Dik T. Winter schrieb: > 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 do not require that one digit represents the number 1/3 in Cantor's list. You do not require that any digit there represents an infinite sequence. But you require that one digit (node) represents 1/3 in the tree? But you require require that one digit (node) in my tree represents an infinite sequence? Are you trolling, Dik? There is no node in the tree which represents 0.000... and no node which represents 0.1 But these real numbers are represented in the tree by paths. Regards, WM
From: mueckenh on 25 Nov 2006 04:20 Dik T. Winter schrieb: > 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? No. > 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. That is one of the reasons why set theory is wrong. Regards, WM
From: mueckenh on 25 Nov 2006 04:22
Dik T. Winter schrieb: > 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) For *finite, linearly ordered* sets, the quantifiers can be exchanged. Every line is a finite, linearly ordered set. Therefore the quantifiers can be exchanged. Regards, WM |