Another AC anomaly?
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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
From: Virgil on 4 Dec 2009 03:01 In article <5f8c7e7f-ec83-45ba-a584-b614759696f4(a)j4g2000yqe.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > 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. What WM falsely calls "actual infinity" does not say any such thing. What actual mathematics actually says is that the path 0.000... is not included in the set of paths {0.111..., 0.0111..., 0.00111..., ...} although there is a sense in which it is the limit of that sequence of paths, just as 0 is the limit of the sequence 1, 1/2, 1/4, 1/8,....
From: Virgil on 4 Dec 2009 03:20 In article <949e3b4b-04c5-4d6e-ba64-7f95d37db42f(a)r24g2000yqd.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > 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". From what shelf? If, as WM does, one rejects ZFC and NBG, there does not seem to be any such shelf available to take it from. > 2) N is the limit of the sequence a_n = ({1, 2, 3, ...,n}) According to what definition of "limit"? Without a specific definition of limit, your claim is at best dubious. > 3) N is the limit, i.,e. the infinite union of singletons {1} U {2} > U ... According to what definition of "limit"? Without a specific definition of limit, your claim is at best dubious. > >�Under the definitions given in this thread, you are > > simply wrong. > > The definition lead to the empty set. And *that result* is simply > wrong. Dik gave a definition of limit under which it is right. WM has given no definition of limit at all, so is 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. Does WM claim that the "limit" is 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 point is to determine, if possible, the number on that ball when the process is over and done with, but it cannot be done. > > >�The bottom line is that none of this proves > > any inconsistency in set theory. > > > Paint all paths of the form > 0.111... > 0.0111... > 0.00111... > 0.000111... > and so on. > Potential infinity then says There is no such thing as "potential infinity". There are sets which are finite sets and sets which are not finite, and no other sort of sets. > Actual infinity says that there is a path 0.000... parts of which > remain unpainted. And that is wrong. There is no "actual infinity" saying anything like that, at least none outside Wolkenmuekenheim. Actual mathematics would say that while every node and every edge is included in the sequence, there is no member of the sequence having all of them at once. > > 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. Such claimed implications require proofs, which WM is incapable of producing for two reasons: 1. If it were true, WM would not be competent to prove it. 2. It is not true.
From: WM on 4 Dec 2009 05:31
On 4 Dez., 09:20, Virgil <Vir...(a)home.esc> wrote: > In article > <949e3b4b-04c5-4d6e-ba64-7f95d37db...(a)r24g2000yqd.googlegroups.com>, > > > Actual mathematics would say that while every > node and every edge is included in the sequence, there is no member of > the sequence having all of them at once. Everey member covers the first node. The n-the member covers all nodes including 1 and n. So logic forces us (i.e. those who can) to conclude that if all nodes are covered then there is a path covering all nodes. But as there is no such path. Hence there is no "all nodes". Some matheologians claim that there are all nodes and that all nodes are covere by all paths but that there is no path covering all nodes. This would... But it is really useless to try to teach such ... Regards, WM |