Another AC anomaly?
in [Logic]
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From: WM on 10 Dec 2009 09:56 On 10 Dez., 15:40, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > Have a look at <http://en.wikipedia.org/wiki/Lim_inf> in the section > titled "Special case: dicrete metric". An example is given with the > sequence {0}, {1}, {0}, {1}, ... > where lim sup is {0, 1} and lim inf is {}. > > Moreover, in what way can a definition be invalid? It can be nonsense like the definition: Let N be the set of all natural numbers. Regards, WM
From: Dik T. Winter on 10 Dec 2009 10:29 In article <69368271-d841-4c3e-9f73-57259312f585(a)g12g2000yqa.googlegroups.com> WM <mueckenh(a)rz.fh-augsburg.de> writes: > 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. The reason is that omega is a limit > without axiom of infinity, and omega is older than that axiom. Without the axiom of infinity omega would not be immediately existing. So apparently there is a definition of omega without the axiom of infinity. Can you state that definition? > > > 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. There are no concepts of mathematics without definitions. > > > 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. It *is* according to one of the axioms of ZF, and as such it is not a limit. > > > 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. Where? Why do you think taking a limit and taking cardinality should commute? Should also the limit of te sequence of integral of functions be equal to the integral of the limit of a sequence of fuctions? -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 10 Dec 2009 10:39 In article <9f4850a5-268c-4b3d-a84b-34713cfaedd7(a)c34g2000yqn.googlegroups.com> WM <mueckenh(a)rz.fh-augsburg.de> writes: > 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. That is not more than opinion. > > > As a starting point, we use the fact hat each natural number is > > > identified with the set of all smaller natural numbers: n =3D {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. Yes, but the set of natural numbers (N) can *not* 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). Sorry, there are many definitions of pi, some use a limit, others do not use a limit. At least, that is the case in mathematics. But I know you prefer to use things without any definition at all. -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 10 Dec 2009 10:35 In article <fd82ac52-8f71-4930-8f8e-415187ae832b(a)g26g2000yqe.googlegroups.com> WM <mueckenh(a)rz.fh-augsburg.de> writes: > 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. Yes, and since than quite a lot of newer mathematics has been made available. Moreover, before 1908 mathematicians did use concepts without actually defining them, which is not so very good in my opinion. > > > 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. I see no definition, so what *is* the definition? > > > 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? Ok, so N is not a set. What is it? > > > 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! Eh? So there are no infinite paths in that tree? > 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? If there are no infinite paths in that tree, 1/3 is not in that tree. Otherwise 1/3 would be a rational with a denominator that is a power of 2 (each finite path defines such a number). > 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)? By the axiom of infinity there *are* infinite paths in that tree. So your statement that there are none is a direct contradiction of the axiom of infinity. -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: WM on 10 Dec 2009 12:12
On 10 Dez., 16:29, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > Without the axiom of infinity omega would not be immediately existing. > So apparently there is a definition of omega without the axiom of infinity. > Can you state that definition? Look into Cantor's papers. Look into my book. > > There are no concepts of mathematics without definitions. So? What is a set? > > > An infinite union *is* not at all. But if it were, it was a limit. > > It *is* according to one of the axioms of ZF, and as such it is not a limit. It *was* according to Cantor, without any axioms. > Where? Why do you think taking a limit and taking cardinality should > commute? Should also the limit of te sequence of integral of functions > be equal to the integral of the limit of a sequence of fuctions? If an infinite set exists as a limit, then it has gotten from the finite to the infinite one by one element. During this process there is no chance for any divergence between this set-function and its cardinality. Regards, WM |