Cantor Confusion
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From: William Hughes on 3 Nov 2006 08:13 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueck...(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > > > Each diagonal digit is formed by a line. > > > > > > and by a column! > > > > Yes, and the number of columns is equal to the number of lines. > > The fact that no single line contains all the columns is not > > important. > > > The fact that *no* set of lines contains all the line is not important > too. > > I think we have obtained agreement, that we disagree in this point and > that further discussion will not lead to a result acceptable for both > parties. The final state is: > > 1) The number of columns is equal to the number of lines. > > 2) Both are infinite, according to set theory this means omega or > aleph_0. > > But you believe: omega or aleph_0 is the maximum of the set of lines > and the supremum of the set of columns, which, however, is not taken by > any column. You believe the different meaning is not important. > No. I believe that the length of the diagonal is the supremum of the length of the lines. This supremum does not have to be the length of any line. Call it omega. Then omega is the *supremum* of the set of lines (equally the supremum of the set of columns). -William Hughes
From: mueckenh on 3 Nov 2006 08:16 David Marcus schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > Dik T. Winter schrieb: > > > In article <1162405520.008395.100850(a)e64g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > > Dik T. Winter schrieb: > > > > > > Where in the above quote is 1/oo defined? > > > > > > > > It is defined that omega (which Cantor used later instead of oo) is a > > > > number larger than any natural number n. Omega is the limit ordinal > > > > number. Therefore 1/omega must be a number smaller than every fraction > > > > 1/n. > > > > > > Why? As long as 1/omega is not defined you can not talk about it. You > > > simply assume that 1/omega is a number. But that is not the case, it > > > is not defined. > > > > If omega is a number > n, then 1/omega is a number < 1/n. For all n e > > N > > What do you mean by "number"? Normally, "omega" denotes a certain > ordinal. Division is not normally defined for ordinals. If you want it > to be defined, then you have to define it. So, using a plausible > interpretation of your words "number" and "omega", your statement is > false because 1/omega is not an ordinal (since it is not defined). By "number" I understand those mathematical objects which can be compared by size or magnitude, i.e., which observe trichotomy. I know that division is not normally defined for transfinite ordinals, but with omega > n we can define 1/omega < 1/n for n e N as some kind of abbreviation. Don't forget, most things come into being by intuition, not by formalization. It is by logic we prove, it is by intuition that we invent. (Henry Poincaré) When Cantor introduced omega, there was no formal definition. Das Zeichen oo, welches ich in Nr. 2 dieses Aufsatzes gebraucht habe, ersetze ich von nun an durch omega, weil das Zeichen oo schon vielfach zur Bezeichnung von unbestimmten [d. h. potentiellen] Unendlichkeiten verwandt wird. [G. Cantor, Gesammelte Anhandlungen, p. 195] Regards, WM
From: mueckenh on 3 Nov 2006 08:37 David Marcus schrieb: > > I have posed for several times a version which can be understood and > > has been understood by him and by other mathematicians. > > Please name these logicians and mathematicians. Read this thread. > > What do you mean by "understood"? You concede that your logician agrees > that the argument is not valid in ZFC. He does not know how to translate it. >So, what logical system does your > logician say it is valid in? Do you really believe that ZFC is the only and ultimate system of thinking? > Since ZFC is here now, if you wish to publish your proof in a math > journal, you will need to wait until ZFC is gone. We'll let you know > when that happens. I would not rely on your message, I am afraid you will not notice that ZFC is gone. > > > The proof of uncountability of R therefore shows ZFC is false. > > The statement "ZFC is false" is meaningless. All you are saying is that > your proof cannot be given in ZFC. The proof has not yet been given. That is a bit different from cannot be given. > > > > A supremum is a least upper bound of a set (like the set of columns of > > a list). If there is no maximum, then the supremum is not taken, i.e., > > it does not belong to the set. > > It is true that if there is no maximum, but there is a sup, then the sup > is not an element of the set. So what? That doesn't prove anything about > the diagonal. For the diagonal (d_kk) of the list 0 1 2 3 4 5 6 7 8 9 .... we have the following mappings: d_kk --> d_mk and d_kk --> d_km with k, m eps N. It is curious that the set of terms of (d_km) has omega as a maximum for every fixed m eps N while the set of terms of (d_mk) has omega not as a maximum for every fixed m eps N. Regards, WM
From: mueckenh on 3 Nov 2006 08:45 David Marcus schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > Virgil schrieb: > > > > Any "diagonal" for this function will have to be of length greater than > > every n in N. > > > > This shows the inconsistency of ZF and NBG. > > It is amazing that you keep repeating claims after you yourself admit > that you have not shown the claim. Is this intentional or don't you > realize that you are contradicting yourself? It is amazing that you don't understand the meaning of my proof. I have shown that the set of paths of the binary tree has a cardinality not less than R. I have shown that the set of edges has the cardinality of N. And finally I have shown that the cardinality of the set of paths is not larger than the cardinality of the set of edges. In ZFC Cantor's proof is valid, according to which the cardinality of R is larger than that of N. Therefore there is a false result obtained from ZFC. Regards, WM
From: mueckenh on 3 Nov 2006 08:53
David Marcus schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > > David Marcus schrieb: > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > The diagonal is an infinite sequence. So the diagonal is longer than > > > > any of the finite sequences. But the diagonal consists of elements of > > > > the finite sequences. So it cannot be longer than the maximum of the > > > > finite sequences. If this maximum does not exist, you cannot take the > > > > supremum omega for it, because the supremum is not a member of the > > > > sequences and does not supply elements of the diagonal. > > > > > > Let's try a simpler problem. Consider the following list. > > > > I would hesitate to call it a list, because injectivity is lacking. > > I've never seen injectivity required for a list. Usually, people use > "list" to mean sequence. If your lists require injectivity, you should > explicitly state that. People can't read your mind. "List" is no a technical term but taken from civil life. There it is used to list different items, not the same item twice or more. So, if we adopt this term, then we should adopt it wih a meaning one would expect. But as I said: > > > But that is not important. > > > > > > 1 > > > 1 > > > 1 > > > ... > > > > > > In other words, consider the sequence x where x(n) = 1 for n a natural > > > number. How long is this sequence? > > > > You expect the answer omega. > > I expect nothing. That expectation is obviously wrong. Regards, WM |