Another AC anomaly?
in [Logic]
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From: Virgil on 7 Dec 2009 16:07 In article <cccaffe3-72fe-4a1f-9514-a018e9379309(a)j24g2000yqa.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 7 Dez., 16:37, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > In article <yvCdnW28VrXBqIXWnZ2dnUVZ_s-dn...(a)giganews.com> "K_h" > > <KHol...(a)SX729.com> writes: > > > > ... > > �> > > > When you mean with your statement about N: > > �> > > > � � �N = union{n is natural} {n} > > �> > > > then that is not a limit. �Check the definitions about > > �> > > > it. > > �> > > > > �> > > It is a limit. That is independent from any definition. > > �> > > > �> > It is not a limit. �Nowhere in the definition of that > > �> > union a limit is used > > �> > or mentioned. > > �> > > �> Question. �Isn't this simply a question of language? > > > > Not at all. �When you define N as an infinite union there is no limit > > involved, there is even no sequence involved. �N follows immediately > > from the axioms. > > Nevertheless it is a limit ordinal. It is an ordinal which is neither the first ordinal nor a successor ordinal, which does not in any way require it to be formed by any limiting process. The only thing that requires an axiom of infinity at all in ZF is the fact that unions in ZF are only defined for unioning of members of a set, and not for unioning sets which are not know to be members of set.
From: Virgil on 7 Dec 2009 18:40 In article <87my1ubkbf.fsf(a)dialatheia.truth.invalid>, Aatu Koskensilta <aatu.koskensilta(a)uta.fi> wrote: > Virgil <Virgil(a)home.esc> writes: > > > In article <KuAGqH.FrI(a)cwi.nl>, "Dik T. Winter" <Dik.Winter(a)cwi.nl> > > wrote: > > > >> (And if I remember right, a limit ordinal is an ordinal that has no > >> predecessor, see, again no limit involved.) > > > > Other than the first ordinal, though that restriction is not really > > relevant here. > > This is a matter of taste. For example, in Jech's _Set Theory_ (Third > Millennium edition), p. 20, we learn: > > If alpha is not a successor ordinal, then alpha = sup {beta | beta < > alpha} = union of alpha; alpha is called a /limit ordinal/. We also > consider 0 a limit ordinal and define sup {} = 0. Okay.
From: Dik T. Winter on 8 Dec 2009 09:22 In article <247baff4-5209-49dc-a203-7ccfe492b5b0(a)d21g2000yqn.googlegroups.com> WM <mueckenh(a)rz.fh-augsburg.de> writes: > On 7 Dez., 16:32, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > > 1) N is a set that follows (as omega, but that is not important) from > > > the axiom of infinity. You can take it "from the shelf". > > > > Actually: N is the smallest inductive set that starts with 1. The axiom > > of infinity states that that set does exist, but with that definition it > > is *not* defined as a limit. > > 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. > > > 2) N is the limit of the sequence a_n = ({1, 2, 3, ...,n}) > > > > You have first to define the limit of a sequence of sets before you can > > state such. > > No. Oh, you like to use things for which you have not given a definition? Would you agree that N is also the foo of the sequence a_n? > > N is (in set theory) defined before even the concept of the > > limit of a sequence of sets is defined (if that concept is defined at > > all in the particular treatise). > > 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. > > > 3) N is the limit, i.,e. the infinite union of singletons {1} U {2} > > > U ... > > > > No, the infinite union is *not* a limit. It is defined without even the > > presence of the concept of a limit of a sequence of sets. > > 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. > 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. I have no idea of a definition of the limit of a sequence of sets where they are always necessarily equal. > > > But if (3) is correct, then N must also be the limit of the process > > > described in my > > >http://www.hs-augsburg.de/~mueckenh/GU/GU12.PPT#394,22,Folie22 > > > without and *with* the intermediate cylinder. > > > > I see no process there, only a picture of a cylinder with the digits 1, > > 2, 3, 4 and 5, and an open cube. > > It is a good tactics to play possum or play stupid. As I have no access to powerpoint and have to look at it using openoffice, that is indeed all I see, without any explanation. Apparently there is more in it, but I do not see it. -- 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 8 Dec 2009 10:07 In article <76816a30-dac0-404d-91c1-92b851f212d8(a)b2g2000yqi.googlegroups.com> WM <mueckenh(a)rz.fh-augsburg.de> writes: > On 7 Dez., 16:41, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: .... > > > > > > No. I asked you for a mathematical definition of "actual > > > > > > infinity" and you told me that it was "completed infinity". > > > > > > Next I asked you for a mathematical definition of "completed > > > > > > infinity" but you have not given an answer. So I still do > > > > > > not know what either "actual infinity" or "completed > > > > > > infinity" are. > > > > > > > > > > Both are nonsense. But both are asumed to make sense in set > > > > > theory. > > > > > > > > No, set theory does not contain a definition of either of them. > > > > So in what way do they make sense in set theory? As I have not seen a > > definition of them at all, I can not see what that means. > > 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? 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. > > > > > The axiom of infinity is a definition of actual infinity. > > > > > "There *exists* a set such that ..." > > > > > Without that axiom there is only potential infinity, namely > > > > > Peano arithmetic. > > > > > > > > I see neither a definition of the words "actual infinity" neither > > > > a definition of "potential infinity". Or do you mean that > > > > "potential infinity" is Peano arithmetic (your words seem to imply > > > > that)? > > > > > > > > So we can say that in "potential infinity" consists of a set of > > > > axioms? > > > > > > Here is, to my knowledge, the simplest possible explanation. > > > > Darn, I ask for a definition, not for an explanation. > > 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. > 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. > > > Paint all paths of the form > > > 0.111... > > > 0.0111... > > > 0.00111... > > > 0.000111... > > > and so on. > > > Potential infinity then says that every node and every edge on the > > > outmost left part of the tree gets painted. > > > Actual infinity says that there is a path 0.000... parts of which > > > remain unpainted. And that is wrong. > > > > So "potential infinity" and "actual infinity" are theories? But I do not > > know of any theory that states that there is any part of the path 0.000... > > that is unpainted. > > That is the inconsistency of set theory. What is the inconsistency. Apparently you think that "actual infinity", whatever that may be, apparently at this point a theory, states that parts of 0.000... remain unpainted. As I do not know of any theory that states that I wonder whether you could give me an outline of such a theory. Set theory does not state that. > 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. -- 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 8 Dec 2009 10:13
In article <cccaffe3-72fe-4a1f-9514-a018e9379309(a)j24g2000yqa.googlegroups.com> WM <mueckenh(a)rz.fh-augsburg.de> writes: > On 7 Dez., 16:37, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > In article <yvCdnW28VrXBqIXWnZ2dnUVZ_s-dn...(a)giganews.com> "K_h" <KHol...= > @SX729.com> writes: > > ... > > > > > > When you mean with your statement about N: > > > > > > N = union{n is natural} {n} > > > > > > then that is not a limit. Check the definitions about > > > > > > it. > > > > > > > > > > It is a limit. That is independent from any definition. > > > > > > > > It is not a limit. Nowhere in the definition of that > > > > union a limit is used > > > > or mentioned. > > > > > > Question. Isn't this simply a question of language? > > > > Not at all. When you define N as an infinite union there is no limit > > involved, there is even no sequence involved. N follows immediately > > from the axioms. > > 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. > > > Now w is a limit ordinal so the ordered set N is, in the > > > ordinal sense, a limit. Of course w is not a member of N > > > becasuse then N would be a member of itself (not allowed by > > > foundation). > > > > Note here that N (the set of natural numbers) is *not* defined using a > > limit at all. That w is called a limit ordinal is a definition of the > > term "limit ordinal". It does not mean that the definition you use to > > define it actually uses a limit. (And if I remember right, a limit > > ordinal is an ordinal that has no predecessor, see, again no limit > > involved.) > > Karel Hrbacek and Thomas Jech: "Introduction to Set Theory" > Marcel Dekker Inc., New York, 1984, 2nd edition, p. 134 > > 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. > 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. -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |