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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
From: WM on 10 Dec 2009 12:18 On 10 Dez., 16:35, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > Before 1908 there was quite a lot of mathematics possible. > > Yes, and since than quite a lot of newer mathematics has been made available. Most of it being rubbish. > Moreover, before 1908 mathematicians did use concepts without actually > defining them, which is not so very good in my opinion. Cantor gave a definition of set. What is the present definition? > > > 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? N is a sequence of natural numbers. > > There is not even one single infinite path! > > Eh? So there are no infinite paths in that tree? In fact no, but every path that you believe in is also in the tree, i.e., you will not be able to miss a path in the 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. 1/3 does not exist as a path. But everything you can ask for will be found in the tree. Everything of that kind is in the 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. Try to find something that exists in your opinion but that does not exist in the tree that I constructed. Regards, WM
From: Virgil on 10 Dec 2009 15:19
In article <6d236085-34a7-4c43-bbe6-341bf8bb6faf(a)l13g2000yqb.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 10 Dez., 08:19, Virgil <Vir...(a)home.esc> wrote: > > In article > > <b963b9aa-c345-43cf-bf78-e9e27401f...(a)c34g2000yqn.googlegroups.com>, > > > > > > > > > > > > �WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > On 10 Dez., 01:58, "K_h" <KHol...(a)SX729.com> wrote: > > > > > > > When using an intermediate reservoir, as shown in my > > > > > lesson > > > > >http://www.hs-augsburg.de/~mueckenh/GU/GU12.PPT#394,22,Folie > > > > > 22 > > > > > it becomes clear that N cannot be generated by adding > > > > > number after > > > > > number. > > > > > > Why not? �Say we have an infinitely large sheet of paper and > > > > we print each natural number, n, on the paper at time > > > > t=1-1/(n+1). �Certainly at time t=1 we have all the naturals > > > > printed on the page. > > > > > It seems so. But it is wrong. You see it if you consider the > > > alternative process using an intermediate reservoir as "realized" in > > > my lesson above. > > > > > naturals - reservoir - paper > > > N � � � � � - �{ } � � � � � �- { } > > > N/{1} � � �- �{1} � � � � �- { } > > > N/{1,2} � �- {2} � � � � � - {1} > > > N/{1,2,3} - {3} � � � � �- {1,2} > > > ... > > > �N/{1,2,3, ...,n} - {n} - {1,2,3, ...,n-1} > > > ... > > > > > The set in the middle contains a number at every time after t = 0. > > > Hence this number cannot yet have been printed on the paper (because > > > it will be printed only after its follower will have entered the > > > reservoir). > > > > And when all the numbers have passed through your "reservoir", both into > > and out ofd it, as will have happened by t = 1, which numbers does WM > > claim will still be unprinted. > > Then a last one would have had to leave the intermediate reservoir. The whole point of infinite series and sequences is that there is no such thing as a "last one". Though when has gone through them all there is an end of the process of going through them. > > > > When going through the terms of an infinite sequence, as in the above, > > EITHER the process hangs up on a particular term of that sequence OR it > > goes through every term of the sequence, TERTIUM NON DATUR.- > > Or it is proved that the assumption of finished infinity is nonsense. But that is exactly what, among many other things, WM has not proved. > Secundum non datur. Then rimum non datur as well, at least in Wolkenmuekenheim. |