Another AC anomaly?
in [Logic]
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From: Virgil on 7 Dec 2009 15:21 In article <KuAGqH.FrI(a)cwi.nl>, "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote: > In article <yvCdnW28VrXBqIXWnZ2dnUVZ_s-dnZ2d(a)giganews.com> "K_h" > <KHolmes(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. > > > My > > book on set theory defines omega, w, as follows: > > > > Define w to be the set N of natural numbers with its > > usual order > > < (given by membership in ZF). > > > > 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.) Other than the first ordinal, though that restriction is not really relevant here.
From: Virgil on 7 Dec 2009 15:23 In article <KuAGIy.FE1(a)cwi.nl>, "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote: > In article > <7772c857-57b0-4422-b688-9a4c8b923467(a)h10g2000vbm.googlegroups.com> WM > <mueckenh(a)rz.fh-augsburg.de> writes: > > On 3 Dez., 16:27, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > ... > > > > > The limit set is not necessarily "assumed" by the sequence of sets > > > > > (if by that you mean that there is a set in the sequence that is > > > > > equal to the limit set, if you mean something else I do not > > > > > understand it at all). > > > > > 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. > > > > 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. For some people, it starts with 0. > > > 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. 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). > > > 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. > > Rid yourself of all those ideas. > > > This is fact. > > It is not. > > > 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,Folie 22 > > 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.
From: Aatu Koskensilta on 7 Dec 2009 15:28 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. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Virgil on 7 Dec 2009 15:39 In article <247baff4-5209-49dc-a203-7ccfe492b5b0(a)d21g2000yqn.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > 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. That particular shelf does not exist until AFTER N has been created. > It is not defined as a limit (if you like so) although amazingly > omega is called a limit ordinal. For a different meaning of "limit". The word "limit" has more than one possible meaning, and its meaning depends on context. > > > > �> 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. Yes. Undefined processes are meaningless in mathematics. > > >�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. We think so, but you apparently disagree. > > > > �> 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. No definition of union requires any limit process. The union of any set S is defined to be the set T of all elements of elements of S. > Why did you argue that limits of > cardinality and sets are different We didn't do that exactly. What we argued, and still argue, is that the claim that they are ALWAYS the same is false. And, IIRC, Dik provided an example in which they are not the same. > > > �> 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. Which is why WM employes those ploys so often. But such tactics do not constitute proofs, which are the only winning tactics at math. And WM has show himself, time and again, incapable of proofs at any level above the trivial.
From: Virgil on 7 Dec 2009 15:54
In article <76816a30-dac0-404d-91c1-92b851f212d8(a)b2g2000yqi.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 7 Dez., 16:41, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > In article > > <5f8c7e7f-ec83-45ba-a584-b61475969...(a)j4g2000yqe.googlegroups.com> WM > > <mueck...(a)rz.fh-augsburg.de> writes: > > �> On 3 Dez., 16:27, "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. As yet, no one has shown that the axiom of infinity causes any problems with mathematics, at least in ZF or NBG, though it does cause some in those who are trying to be mathematicians. > > > > �> > �> 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 should know it or know where to find it. > (You can look it up in my book.) > > The definition of a potentially infinite set is given by > 1 in N > n in N then n+1 in N. If N is to be a set, then it must be actually infinite by your own definition. And if it is not to be a set, then it is improper to even speak of "all naturals". > > > > �> � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �Consider > > �> the infinite binary tree: > > �> > > �> � � 0 > > �> � � /\ > > �> � 0 �1 > > �> �/\ � �/\ > > �> 0 1 0 1 > > �> ... > > �> > > �> 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. Since, as far as I am aware, no set theory says that, the only 'inconsistency' is in WM's understanding. Which is why his math in Wolkenmuekenheim is such a mess. > > 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. WM allows that he set of nodes can be so formed even in WM's world , but then in WM's world, he says the set of paths cannot be formed. > > Nevertheless set theory says that there is something missing in a tree > thus constructed. What do you think is missing? The set of paths. Note that, as a set of nodes, each path must be an infinite set such that for each pair of nodes one is an initial string of the other. So that, among other things, in WM's world the set of nodes is not allowed to have a power set. > (If nothing is > missing, there are only countably many paths.) According to WM's rules there can be no set of paths to count, so it can't be countable. |