Another AC anomaly?
in [Logic]
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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
From: Marshall on 4 Dec 2009 09:46 On Dec 3, 10:31 pm, WM <mueck...(a)rz.fh-augsburg.de> wrote: > On 3 Dez., 16:27, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > > 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 > ... http://www.flickr.com/photos/22984501(a)N06/2961175776/ Marshall
From: Virgil on 4 Dec 2009 17:18
In article <ce07a2f1-cf4c-4e53-b5d5-b1fcd2d80bb2(a)s20g2000yqd.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > 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. That WM's particular form of logic is rife with contradictions. For every 0 node there is a path in WM's set NOT containing it, and for every path in WM's set there is a node NOT contained in it, so why does WM insist that there is a path in your set containing every node? Not even God knows. > > But as there is no such path. Hence there is no "all nodes". Or WM is wrong again about the need for one. > > 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. Only WM claims that. Mathematicians claim that in ZFC there is a set of all nodes each represented by a finite. possibly empty, binary string of, say, zeroes and ones, and a set of all paths, each path being a function from N to {0,1}. Two such paths coincide on some initial string, the string of nodes preceding the first node at which they differ. Any two paths determine a > This would... But it is really useless to try to teach such ... It is not so much what WM does not know than hurts him, and his students, it's what he "knows" that ain't so, and then tries to teach to his poor captive students. |