Cantor Confusion
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From: William Hughes on 9 Nov 2006 09:32 mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > In article <1162820871.886446.129490(a)k70g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: > > > > In article <1162557178.206693.187650(a)e3g2000cwe.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > ... > > > > > I believe that a taken maximum and a not taken supremum are so > > > > > different (although the difference is very tiny) that set theory, > > > > > necessarily assuming this, is wrong. > > > > > > > > There is not a taken maximum. > > > > > > Oh no? The lengths of the columns of the list > > > > > > 0.1 > > > 0.11 > > > 0.111 > > > ... > > > > > > are not omega? I was under the impression that there are aleph_0 lines > > > with less than aleph_0 columns. > > > > There are alpeh_0 lines and aleph_0 columns. But neither is a maximum. > > Both are supprema. There is no aleph_0-th line as there is no > > aleph_0-th column. So, where is there a maximum involved? > > There is the following difference: > The first and (any other) column has actually aleph_0 1's. > No line has aleph_0 1's. > So the first column is the maximum which is not reached by the lines. > > > > > > > The difference between maximum und supremum is, by the way, the reason > > > > > why I insist that an actually infinite set of natural numbers must > > > > > contain a non-natural number. > > > > > > > > Yes, you keep insisting that, but as there is no maximum, that insistance > > > > is futile. > > > > > > The number 0.111... or the diagonal has alep_0 1's as has the n-th > > > column but there i sno line having aleph_0 1's. > > > > Yes. What does that prove? *Not* that the infinite set of natural numbers > > contains a non-natural number. Because that is obviously nonsense. > > And this obvious nonsense results from the assumption that the set N > does exist. > > > > > > But let's take the actually infinite set of natural numbers. > > > > Suppose it contains your non-natural number. Now remove that non-natural > > > > number from that set. Isn't the set still actually infinite? > > > > > > That shows only the self-contradiction of the notion actual infinity. > > > > No self-contradiction at all. What *is* the self contradiction? > > It is the set which is equinumerous with the 1's in the column, but not > with the 1's in any line. > > > > > > What you are confused about is that the cardinality (and ordinality) of > > > > the set of natural number is indeed actually infinite; but the *contents* > > > > are only potentially infinite. > > > > > > You see not difference between the infinite man 1's of 0.111... and the > > > finite many 1's in every line? > > > > I see a difference, but I have no problem with that difference. > > The problem is that the bijection n <--> n fails. There are n numbers > in the set {1,2,3,...,n} > > 1 1 > 2 1,2 > 3 1,2,3 > ... ... > n 1,2,3,...,n > ... ... > omega 1,2,3,... > omega + 1 1,2,3,...,omega > ... ... > omega + n 1,2,3,...,omega + n + 1 > > > > > > > > > > you remember? Without an infinite number in the list there is no > > > > > > > infinite diagonal defined. > > > > > > > > > > > > I state (you know that) that the diagonal: 0.111... > > > > > > > > > > Yes, again mixing up maximum and supremum. > > > > > > > > No, not mixing at all. Supremum only. > > > > > > Then 0.111... is not different from the finite sequences of 1's? > > > > How do you conclude that? Given finite initial segments of the triangle > > we get a length, a width and a length of the diagonal that are all equal. > > When we look at the complete triangle (which exists by the axiom of > > infinity) all three become the supremum of the finite quantities, and > > so still are equal. No maximum involved. > > > The diagonal and the left side of triangle are complete with infinitely > many ones, but no line has infinitely many 1's. > > > > > > But the length is realized as a whole number by a set of elements > > > > > (lines) while the width is not realized by a set of columns. > > > > > > > > You are confused between two concepts. The length is not realised in the > > > > sense you think it is. > > > > > > The length every column is infinite and larger than any n while no line > > > is. > > > > Yes. What is the problem with that? The length is realised as omega as > > the supremum of all finite lengths, so is the width realised as omega as > > the supremum of all finite widths. Where is the difference? > > The length is not only a supremum but exists as a maximum (if all > natural numbers exist). > No. Consider the two sets. A={1,2,3,...} B = {1,2,3,...,omega} Both sets have omega as supremum. Set A does not contain its supremum, set B does. In the first case the supremem is not a maximum, in the second case it is. So it is not true that the supremum must also be a maximum. - William Hughes
From: mueckenh on 9 Nov 2006 12:49 David Marcus schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > > David Marcus schrieb: > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > David Marcus schrieb: > > > > > > > > Modern mathematics does not use the term "finished infinity". If you > > > > > wish to use it, please define it. > > > > > > > > Modern mathematics assumes that there are sets which are larger than > > > > infinite sets. That is the same as to say after one finished infinity > > > > we consider the next infinity. > > > > > > I asked you for a definition of "finished infinity". Is what you wrote > > > supposed to be the definition? Normally, a definition should start out > > > something like "Finished infinity is ... ". > > > > It is tedious to explain things to you. > > How would you know, since you never explain anything you say? I do explain. But a certain basic vocabulary is necessary to understand. > > > Please consult a book. Hint: > > "Some mathematicians object to the Axiom of Infinity on the grounds > > that a collection of objects produced by an infinite process (such as > > N) should not be treated as a finished entity." (Karel Hrbacek and > > Thomas Jech: "Introduction to set theory" Marcel Dekker Inc., New York, > > 1984, 2nd edition.) > > > > Now substitute "entity" by the most significant property of the set N. > > So, "finished infinity" isn't even in the book you reference. Kind of > silly for you to refer us to a book to look up what a phrase means when > the book doesn't even use it. Do you always repeat only full sentences you read in a book? Did you never try to learn substituting words in sentences? If 2 + 3 = x you can be sure that x + 5 = 10, even if that is not written down in a book. Now apply this to "finished entity" and "a collection of objects produced by an infinite process" Regards, WM
From: mueckenh on 9 Nov 2006 12:51 Virgil schrieb: > Every part of the discussion is crystal clear except for WM's > maunderings which seem more and more to be deliberately obfuscating. > > If every line is finite, say, for example, the nth line is of length > 2*n, and the diagonal is not finite , then the diagonal MUST be longer > that every line. By definition the diagonal of a matrix cannot be longer than every line of the matrix. Considering your reaction to the vase, I guess you would even accept a circle the diameter of which is longer than its circumference. Please let us stop here. I see no point in explaining to you any logical truth. It is useless to continue this discussion. Regards, WM
From: mueckenh on 9 Nov 2006 12:55 Virgil schrieb: > > > That's a wrong definition of the word "definition", at least in > > > Mathematics. > > > > As you believe? > > As mathematicians believe. A definition, at least in the mathematical > sense, is merely a form of abbreviation. It allows one to say more > briefly what could be said without it but at greater length. Correct. An expression, a long one or a short one, is explained in other words, more briefly or more detailled, respectively. > > > > > > Would you understand: "The new number omega is the limit to which the > > > > (natural) numbers n grow"? Or is even hat too incomprehensible to you, > > > > because you don't know the meanings of "number", "limit", and "grow"? > > Which natural number(s) does WM suggest grows? I do not suggest anything here. This is a sentence by Cantor. Regards, WM
From: mueckenh on 9 Nov 2006 12:58
Virgil schrieb: > What I claim is that there is a line for every MEMBER of N (or omega). > and if each line numbered n has at least n entries, there is a column > for each member of N ( or omega). > I do not say that the "number" of either is "equal" to omega, as I do > not know what that means unless it means no more than what I said above > that there is a member for each member of omega. But you think that there are *all* members of omega. And you think that their number is omega. > > > It contradicts the square matrix which is required for Cantor's > > diagonal argument. > > I do not recall anything in the Cantor proof about square matrices. > > The only requirement in the Cantor "diagonal" proof is that the nth > number listed have a determinable nth decimal digit. > > And that does not require any sort of "square matrix". This condition describes a square matrix (if actual infinity is a number). For every number there is a corresponding digit and for every digit there is a corresponding number. And there is a numbers which counts all the numbers and digits. Regards, WM |