Cantor Confusion
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From: mueckenh on 9 Nov 2006 05:15 MoeBlee schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > MoeBlee schrieb: > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > MoeBlee schrieb: > > > > > > > > > > > > > For ordinals, > > > > > > > > > > x<y <-> xey > > > > > > > > > > where 'e' is the epsilon membership symbol. > > > > > > > > That is identical with my definition. Take for instance Zermelo's > > > > definition of the naturals or Cantor's own (Collected works, p. > > > > 289-290), then you can see it. Your definition has been simply > > > > translated from Cantor's. Please don't conclude from your own ignorance > > > > on mine. > > > > > > You wrote: > > > > > > "My question is : Do you maintain omega > n for all n e N? I know that > > > modern set theory says so. If something can be larger than a number, > > > then it must be a number." > > > > > > Given that "n < omega" stands for "n e omega", and given that N is > > > omega, what point is there in asking anyone whether they maintain that > > > n < omega for every n e N? You're asking someone whether he or she > > > maintains that n e omega for every n e omega. What is the point of > > > asking that? > > > > > > If omega > n then we cannot have a diagonal with omega digits in a > > matrix the lines of which have only n (= less than omega) digits, > > because the digits of the diagonal are digits of the lines. > > That's just another reenactment of your dogmatic pronouncements. I gave > you a proof in set theory, from axioms using only first order logic. > You give an argument that does not use any recognizable logic and from > no stated axioms. The diagonal cannot have more positions than every line. In order to see this you need not study set theory or logic or whatever. It is simply the fact which everybody accepts who has not deliberately been blinded by assuming the existence of a whole number larger than any natural number but counting them. This number does not exist. The diagonal has not omega positions. Regards, WM
From: mueckenh on 9 Nov 2006 05:20 Virgil schrieb: > > > > The length of the diagonal is less than omega. > > > > > > The number of 1's in the diagonal is aleph_0. So, what you wrote is > > > false. In fact, I have no clue what you could possibly be thinking that > > > would lead you to make such an obviously incorrect statement. The length > > > of the diagonal is clearly the same as the length of the first column, > > > and you just wrote above that the length of each column is omega. > > > > The length of the diagonal is clearly not more than the length of any > > line. > > If every line is finite but at least of length equal to is line number, > then even more clearly, the diagonal is longer that every line. That is outright nonsense. Its acceptance by set theorists makes set theory suspicious as a theory without any value. > So that WM could hardly be more wrong if he tried. >From the side of a set theorist I consider that as a compliment. Regards, WM
From: mueckenh on 9 Nov 2006 05:28 MoeBlee schrieb: > > Wrong.. You have not understood the meaning. We have: The set of all > > sets does not exist. But if all mathematical entities including all > > sets do exist in a Platonist universe, how can it be that the set of > > all sets does not exist? > > That is NOT Fraenkel, Bar-Hillel, and Levy's point at all! You admitted that you do not understand their philosophical remarks. So let it be. > > So you have not made ANY point regarding your own views by quoting > Fraenkel, Bar-Hillel, and Levy. > > > > That does not suggest that > > > Fraenkel, Bar-Hillel, and Levy consider that a set itself can grow or > > > have different members at different times or anything like that. > > > > They do not consider it for themselves but they report that some > > mathematicians could adhere to that point of view. As far as I > > remember, they do *not* say hat this point of view is wrong or > > illogical or silly. > > And they don't advocate it at all. They mention this point of view as a possible one, and, as far as I remember, as the only possibility for a platonist. > If you want a theory in which there > are growing sets, then, by all mean, go ahead and develop one. Not necessary. It has already been there. It is the well known mathematics without set theory. > > > > For > > > such axiomatic set theories, membership in a set is definite and sets > > > are determined strictly by membership. And Fraenkel, Bar-Hillel, and > > > Levy do not dispute that. > > > > If sets are strictly determined by membership, and if all sets do > > exist, why then doesn't just that set exist the members of which are > > sets with no further specification. > > Because no one who works with Z set theory claimed that for any > possible description there is a set that has the properties described. It is not important whether one claims that or not. > And what do you mean "all sets exist"? Do you mean all sets that exist > do exist? (Which is of course true.) No, it is wrong, because the set of all those sets does not exist. > Or do you mean that for any > specification of properties, there exists a set having those > properties? Not even all sets with a well-defined specific property do exists. Already the definition of the natural numbers n > 0 by all sets with cardinality n fails, because the set of all sets equinumerous to n does not exist. Regards, WM
From: Virgil on 9 Nov 2006 05:33 In article <1163067090.943471.189230(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > David Marcus schrieb: > > > > > > > > Here is my translation: It is allowed to understand the new number > > > > > omega as limit to which the (natural) numbers n grow, if by that we > > > > > understand nothing else than: omega shall be the first whole number > > > > > which follows upon all numbers n, i.e., which is to be called larger > > > > > than each of the numbers n. > > > > > > > > That's not a definition. It is just a remark. > > > > > > It is a definition. You should not try to judge about topics which you > > > must reject because you do not understand them. Please learn: A > > > definition is an explanation in other words. Nothing more can be done. > > > > No, YOU are IGNORANT of the subject of mathematical defintions. A > > mathematical definition is NOT just any explaination in words. > > What is it then, according to your understanding? > > > > > 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"? > > > > No, it's just that you haven't given MATHEMATICAL definitions of them. > > I quoted Cantor. You cannot expect that I give a mathematical > definition of a notion which, in my eyes and considered objectively, is > nonsense. The expressions "number", "limit", and "grow", however, > should be known to the audience of this thread. There are mathematical definitions of "number" and "limit" though they are quite contest sensitive, but EB does not seem to be using either word in any mathematically acceptable sense. I do not know of any standard scientific meaning for "grow" outside of biology.
From: mueckenh on 9 Nov 2006 05:35
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). Regards, WM |