Another AC anomaly?
in [Logic]
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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
From: WM on 4 Dec 2009 01:31 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. > > > 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. > > 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. 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. Regards, WM
From: WM on 4 Dec 2009 01:42
On 4 Dez., 06:30, "K_h" <KHol...(a)SX729.com> wrote: > "WM" <mueck...(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. 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 ... > Under the definitions given in this thread, you are > simply wrong. The definition lead to the empty set. And *that result* is 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 we can say that the lamp itself does exist at every time and so in the limit too.This is the case with one ball in the vase. > 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 point is that there is one ball as the limit set and 1 as limit cardinality. > The bottom line is that none of this proves > any inconsistency in set theory. No. If the limit set is empty and the limit cardinality is 1, then the limit set is not the set corresponding to limit cardinality. Set theory tries to convince us that the infinite actually exists. In the present case that is obviously inconsistent. Another inconsistency is here: 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. But unless such paths remain unpainted we can paint the complete binary tree by countably many strokes. This implies the existence of only countably many real numbers. Regards, WM |