Another AC anomaly?
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From: Virgil on 3 Dec 2009 15:21 In article <27aee2d6-5966-4b62-a026-fea13e0bad6c(a)h2g2000vbd.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 3 Dez., 13:52, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > In article > > <0c635f53-f5c4-4320-8825-de05f021a...(a)m3g2000yqf.googlegroups.com> WM > > <mueck...(a)rz.fh-augsburg.de> writes: > > �> On 2 Dez., 16:27, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > �> > Limit cardinality is confusing because it can either mean the > > cardinality > > �> > of the limit set or the limit of the cardinalities. > > �> > > �> If the limit set is "assumed" by the sequence of sets, if for instance > > �> N is given by unioning all naturla numbers, then tghe limit set has to > > �> have the limit cardinality. > > > > 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. Every mathemtical "limit" notion has to be defined, as there is no ur-limit notion which is universal in mathematics. Of course, what goes on in Wolkenmuekenheim is largely irrelevant to mathematics. > > So below is your argument: > > > > �> If you want to prove that the determinant of a matrix M with not > > �> ecclusively linear independent rows is det(M) = 0, you go two ways: > > �> You multiply a row of the matrix M by 0, and you empty a row of M by > > �> elementary operations which do not change the determinant. Then you > > �> have proved that the original matrix M has the determinant 0. > > > > So you argue that each matrix has determinant 0? �If not what do you > > mean with that? > > > > �> Same holds for cardinals of sets. You form a set by unioning its > > �> elements, resulting in the complete set. > > > > You can not unite the elements if the elements are not sets in themselves > > and when the elements are sets, the union of the elements is different > > from the original set. �Consider: > > � �A = {{a, b}, {b, c}, {c, d}} > > uniting the elements gives: > > � �B = {a, b, c, d} > > quite different. > > Not at all different with respect to being a limit process. Quite different in mathematics, at least outside of Wolkenmuekenheim. > > > > �> � � � � � � � � � � � � � � � � � � � � �This cannot change the > > �> connection between cardinal number and set during the whole process of > > �> formation. For every step the cardinal number and the number of > > �> elements of the set are equal. > > > > What do you mean with "every step"? �Uniting sets is a single step > > operation. > > > > No. Uniting two sets or singlets or elements is a single-step > operation. > Uniting infinitely many sets or singlets or elements is a limit > process. Not when using the Peano axioms. > > > �> Same holds for the cylinder. Its contents is > > �> � {1}, {1}, {1}, {1} , ... > > �> and that is not different from > > �> � {1}, {2}, {3}, {4} , ... > > �> as you can see by renaming the elements. > > > > I do not understand this at all. > > I see. You seem to share that fate with most so-called matematicians. > > > > �> I think that this answers all the questions contained in the > > �> following. Therefore I extinguish it. > > > > 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. Then don't base your arguments on them. And neither is needed in any version of set theory outside of Wolkenmuekenheim. > The axiom of infinity is adefinition of actual infinity. > "There *exists* a set such that ..." > Without that axiom there is only potential infinity, namely Peano > arithmetic. Without that axiom, or something like it, there is no Peano arithmetic.
From: WM on 3 Dec 2009 16:11 On 3 Dez., 16:27, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > In article <27aee2d6-5966-4b62-a026-fea13e0ba...(a)h2g2000vbd.googlegroups.com> WM <mueck...(a)rz.fh-augsburg.de> writes: > > On 3 Dez., 13:52, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > ... > > > > If the limit set is "assumed" by the sequence of sets, if for instance > > > > N is given by unioning all naturla numbers, then tghe limit set has to > > > > have the limit cardinality. > > > > > > 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". 2) N is the limit of the sequence a_n = ({1, 2, 3, ...,n}) 3) N is the limit, i.,e. the infinite union of singletons {1} U {2} U ... This is fact. 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. Then the cylinder must be empty in the limit, and cardinality of the limit set in the cylinder must be 0. That, however, is wrong, simply because the cylinder is never empty. Regards, WM
From: Virgil on 3 Dec 2009 16:52 In article <7772c857-57b0-4422-b688-9a4c8b923467(a)h10g2000vbm.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 3 Dez., 16:27, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > In article > > <27aee2d6-5966-4b62-a026-fea13e0ba...(a)h2g2000vbd.googlegroups.com> WM > > <mueck...(a)rz.fh-augsburg.de> writes: > > �> On 3 Dez., 13:52, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > ... > > �> > �> If the limit set is "assumed" by the sequence of sets, if for > > instance > > �> > �> N is given by unioning all naturla numbers, then tghe limit set has > > to > > �> > �> have the limit cardinality. > > �> > > > �> > 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". > 2) N is the limit of the sequence a_n = ({1, 2, 3, ...,n}) According to what definition of "limit" ? > 3) N is the limit, i.,e. the infinite union of singletons {1} U {2} > U ... If your definition of "limit" of a sequence of sets is the union of all the sets in the sequence then you are using a different definition of limit than Dik is using, and what you claim about your limits is irrelevant to what he says about his limits. You do not get to redefine "limit" when it is already defined, and Dik has already defined it. > > This is fact. Indeed it is!
From: K_h on 3 Dec 2009 17:02 "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote in message news:Ku31LL.K3x(a)cwi.nl... > In article > <27aee2d6-5966-4b62-a026-fea13e0bad6c(a)h2g2000vbd.googlegroups.com> > WM <mueckenh(a)rz.fh-augsburg.de> writes: > > On 3 Dez., 13:52, "Dik T. Winter" <Dik.Win...(a)cwi.nl> > > wrote: > ... > > > > If the limit set is "assumed" by the sequence of > > > sets, if for instance > > > > N is given by unioning all naturla numbers, then > > > tghe limit set has to > > > > have the limit cardinality. > > > > > > 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. Question. Isn't this simply a question of language? 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). k
From: K_h on 4 Dec 2009 00:30
"WM" <mueckenh(a)rz.fh-augsburg.de> wrote in message news:0c635f53-f5c4-4320-8825-de05f021a428(a)m3g2000yqf.googlegroups.com... On 2 Dez., 16:27, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > Same holds for the cylinder. Its contents is > {1}, {1}, {1}, {1} , ... > and that is not different from > {1}, {2}, {3}, {4} , ... > as you can see by renaming the elements. You need to provide a definition for your idea of a limit set. Under the definitions given in this thread, you are simply wrong. I only have a partial listing of this thread but from what postings I have it looks like you are trying to make an argument about a supertask. Informally, the idea behind a limit is that that state of a system tends to some fixed value. So the limit of a sequence of sets, .like {1}, {0}, {1}, {0}, {1}, {0},..., does not exist whereas the limit of a sequence of sets like {1}, {1}, {1}, {1}, {1}, {1},...does exist and is the set {1} [assuming a definition along the lines of convergence]. So in the case of Thompson's lamp, {1}, {0}, {1}, {0}, {1}, {0},..., its value at the limiting time does not exist and so there is not much that can be said about it. But to this supertask: > Same holds for the cylinder. Its contents is > {1}, {1}, {1}, {1} , ... > and that is not different from > {1}, {2}, {3}, {4} , ... > as you can see by renaming the elements. Renaming the elements? If you have the idea in mind that there is one ball in the vase and it is being repainted with different labels, naturals numbers, {1}, {2}, {3}, {4} , ... , (say at times t=1-1/n), then you can say that at time t=1 there is one ball in the vase. But its painted label can be anything you want it to be at time t=1. That is, if at time t=1-1/n you paint the ball with the label {n} and then at time t=1-1/(n+1) you paint it with the label {n+1}, etc, then you can paint any kind of label you want to on the ball at time t=1. The bottom line is that none of this proves any inconsistency in set theory. k |