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From: WM on 8 Dec 2009 11:10 On 8 Dez., 16:13, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > Nevertheless it is a limit ordinal. > > Yes, that does not mean that necessarily a limit is involved. It is a limit > in the sense that you do not get there by continuously getting at the > successor, in that case it is a limiting process. But when you define > N as an infinite union you do *not* go there by continuously getting at > a successor. The union of a collection (finite, countably infinite or > some other infinity) is defined whithout resorting to successor operations. > Moreover, they would even not make sens if the collection is infinite but > not countably infinite. That does not make sense in either respect, so or so. > > > As a starting point, we use the fact hat each natural number is > > identified with the set of all smaller natural numbers: n = {m in N : > > m < n}. > > Note that here N is apparently already defined, without using a limit. Natural numbers can be defined without using a set. > > > Thus we let w, the least transfinite number, to be the set N > > of all natural numbers: w = N = {0, 1, 2, 3, ...}. > > It is easy to continue the process after this 'limit' step is made: > > The operation of successor can be used to produce numbers following w > > in the same way we used it to produce numbers following 0. > > Yes. So what? That you can define things using a limit does *not* > imply that it is necessarily defined as a limit. O I see. That's like cardinality. The limit cardinality is not the cardinality of the limit (because the limit is not a limit). Regards, WM
From: A on 8 Dec 2009 13:36 On Dec 8, 11:10 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > On 8 Dez., 16:13, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > > Nevertheless it is a limit ordinal. > > > Yes, that does not mean that necessarily a limit is involved. It is a limit > > in the sense that you do not get there by continuously getting at the > > successor, in that case it is a limiting process. But when you define > > N as an infinite union you do *not* go there by continuously getting at > > a successor. The union of a collection (finite, countably infinite or > > some other infinity) is defined whithout resorting to successor operations. > > Moreover, they would even not make sens if the collection is infinite but > > not countably infinite. > > That does not make sense in either respect, so or so. > > > > > > As a starting point, we use the fact hat each natural number is > > > identified with the set of all smaller natural numbers: n = {m in N : > > > m < n}. > > > Note that here N is apparently already defined, without using a limit. > > Natural numbers can be defined without using a set. > > > > > > Thus we let w, the least transfinite number, to be the set N > > > of all natural numbers: w = N = {0, 1, 2, 3, ...}. > > > It is easy to continue the process after this 'limit' step is made: > > > The operation of successor can be used to produce numbers following w > > > in the same way we used it to produce numbers following 0. > > > Yes. So what? That you can define things using a limit does *not* > > imply that it is necessarily defined as a limit. > > O I see. That's like cardinality. The limit cardinality is not the > cardinality of the limit (because the limit is not a limit). > > Regards, WM A function f is said to be continuous at a point x in its domain if the limit of f(a), as a approaches x, is equal to f(x); in others words, the limit of the values of f is equal to the value of f at the limit, speaking loosely. Of course, not every function is continuous at every point in its domain, and some functions are not even continuous at any point in their domains at all. The situation for sets and cardinality is no more mysterious than that. The cardinality of a limit of subsets of the integers is not guaranteed to be the limit of the cardinalities of those subsets. You don't expect an arbitrary function to always be continuous, so perhaps it's unreasonable to expect the cardinality "function," defined on subsets of the integers, to be continuous.
From: Virgil on 8 Dec 2009 15:57 In article <69368271-d841-4c3e-9f73-57259312f585(a)g12g2000yqa.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 8 Dez., 15:22, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > �> I said you can take it from the shelf. It is not defined as a limit > > �> (if you like so) although amazingly omega is called a limit ordinal. > > > Yes, it is called a limit ordinal because by definition each ordinal that > > has no predecessor is called a limit ordinal (that is the definition of the > > term "limit ordinal"). �It has in itself nothing to do with limits. > > No, that is not the reason. Yes it is!!! > The reason is that omega is a limit > without axiom of infinity By what definition of "limit"? > > > �> N is a concept of mathematics. That's enough. > > > > Yes, and it is a concept of mathematics because it is defined within > > mathematics, and it is not defined as a limit. > > It is a concept of mathematics without any being defined. Not outside of Wolkenmuekenheim. > > > �> The infinite union is a limit. > > > > I do not think you have looked at the definition of an infinite union, if > > you had done so you would find that (in your words) such a union is found > > on the shelf and does not involve limits. �Try to start doing mathematics > > and rid yourself of the idea that an infinite union is a limit. > > An infinite union *is* not at all. But if it were, it was a limit. In ZF, any union of the sets which are members of a set is "defined" by the axiom of union, and in ZF there is no other form of union at all. So that what WM is saying about unions is false in ZF. > > > > �> � � � � � � � � � � � � � � � �Why did you argue that limits of > > �> cardinality and sets are different, if there are no limits at all? > > > > I have explicitly defined the limit of a sequence of sets. �With that > > definition (and the common definition of limits of sequences of natural > > numbers) I found that the cardinality of the limit is not necessarily > > equal to the limit of the cardinalities. > > That means that you are wrong. It shows that Dik is right and that WM is wrong. As usual! > > Regards, WM
From: Virgil on 8 Dec 2009 16:15 In article <fd82ac52-8f71-4930-8f8e-415187ae832b(a)g26g2000yqe.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 8 Dez., 16:07, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > �> You have seen the axiom of infinity. It say that an infinite set > > �> exists and that implies that infinitely many elements of that set > > �> exist. That is actual infinity. > > > > Oh, so actual infinity means that a set with infinitely many elements > > exists? > > Yes. > > > In that case you should reject the axiom of infinity. �You are allowed to > > do > > that, and you will get different mathematics. �But you can not claim that > > mathematics with the axiom of infinity is nonsense just because you do not > > like it. �But go ahead without the axiom of infinity, I think you have to > > redo quite a bit of mathematics. > > Before 1908 there was quite a lot of mathematics possible. > There was quite a lot of possible mathematics. Most of which has been extended considerably since. Even if WM wants to restrict himself to only those parts of mathematics which precede 1908, he has not the right, or the power, to impose such restrictions on anyone else, except possibly his poor misled students. > > > �> The definition of an actually infinite set is given in set theory by > > �> the axiom of infinity. > > > > You are wrong, the axiom of infinity says nothing about "actually infinite > > set". �Actually the axiom of infinity does not define anything. �It just > > states that a particular set with a particular property does exist. > > That is just the definition of actual infinity. That may be WM's definition of "actual infinity", but his definitions carry weight only in Wolkenmuekenheim, and are of no importance anywhere else. > > �> The definition of a potentially infinite set is given by > > �> 1 in N > > �> n in N then n+1 in N. > > > > That does not make sense. �Without the axiom of infinity the set N does not > > necessarily exist, so stating 1 in N is wrong unless you can prove that N > > does exist or have some other means to have the existence of N, but that > > would be equivalent to the axiom of infinity. > > N need not exist as a set. If n is a natural number, then n + 1 is a > natural numbers too. Why should sets be needed? If WM chooses to work in a mathematics without sets, he is quite free to do so, but has no power to impose such limits on anyone else, except possibly his poor captive students. > > > > �> The complete infinite binary tree can be constructed using countably > > �> many finite paths (each one connecting a node to the root node), such > > �> that every node is there and no node is missing and every finite path > > �> is there and no finite path is missing. > > > > Right. > > > > �> Nevertheless set theory says that there is something missing in a tree > > �> thus constructed. What do you think is missing? (If nothing is > > �> missing, there are only countably many paths.) > > > > And here again you are wrong. �There are countably many finite paths. > > �There > > are not countably many infinite paths, and although you have tried many > > times you never did show that there were countably many infinite paths. > > There is not even one single infinite path! Then one does not have a COMPLETE infinite binary tree, but only a non-set of (finite) nodes. But there is every path > which you believe to be an infinite path!! Which one is missing in > your opinion? Do you see that 1/3 is there? In your model I do not see any infinite paths, since every infinite path requires an infinite set of nodes and your model disallows all infinite sets. > > What node of pi is missing in the tree constructed by a countable > number of finite paths (not even as a limit but by the axiom of > infinity)? Unless one is allowed to have a SET OF INFINITELY MANY nodes, one can not have even the fractional part of pi. Similarly for any proper fraction whose denominator is not a power of 2. A sequence which is not (actually) infinite (an image of the SET N), must have a last term, and thus be incapable of converging.
From: Virgil on 8 Dec 2009 16:26
In article <9f4850a5-268c-4b3d-a84b-34713cfaedd7(a)c34g2000yqn.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 8 Dez., 16:13, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > �> Nevertheless it is a limit ordinal. > > > > Yes, that does not mean that necessarily a limit is involved. �It is a limit > > in the sense that you do not get there by continuously getting at the > > successor, in that case it is a limiting process. �But when you define > > N as an infinite union you do *not* go there by continuously getting at > > a successor. �The union of a collection (finite, countably infinite or > > some other infinity) is defined whithout resorting to successor operations. > > Moreover, they would even not make sens if the collection is infinite but > > not countably infinite. > > That does not make sense in either respect, so or so. What does not seem to make sense to WM makes very good sense to those not constrained by the artificial rules governing Wolkenmuekenheim > > > > �> As a starting point, we use the fact hat each natural number is > > �> identified with the set of all smaller natural numbers: n = {m in N : > > �> m < n}. > > > > Note that here N is apparently already defined, without using a limit. > > Natural numbers can be defined without using a set. As soon as one wants to talk about any property of ALL NATURALS, one inevitably gets the set of them. > > > > �> � � � � Thus we let w, the least transfinite number, to be the set N > > �> of all natural numbers: w = N = {0, 1, 2, 3, ...}. > > �> � It is easy to continue the process after this 'limit' step is made: > > �> The operation of successor can be used to produce numbers following w > > �> in the same way we used it to produce numbers following 0. > > > > Yes. �So what? �That you can define things using a limit does *not* > > imply that it is necessarily defined as a limit. > > O I see. No, you do not see at all. > That's like cardinality. The limit cardinality is not the > cardinality of the limit (because the limit is not a limit). The limit of a sequence of cardinalities of sets need not be the cardinality of the limit of a sequence of sets when the definitions of those two limit processes differ, as they do. For the definition of a limit of a sequence of sets, as given by Dik, and any reasonable definition of the limit of a sequence of cardinalities, there are examples for which the two limits differ. That WM does not like that result, does not make it false anymore than WM liking something makes it true. |