Another AC anomaly?
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From: WM on 21 Dec 2009 12:02 On 21 Dez., 14:17, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > In article <2fac8bb1-4c90-4421-b559-1ea7f0301...(a)e27g2000yqd.googlegroups..com> WM <mueck...(a)rz.fh-augsburg.de> writes: > > On 18 Dez., 15:12, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > ... > > > Why need I to think about a last one (which there isn't) to be able to > > > think about a set that contains all natural numbers? Apparently you > > > have some knowledge about how my mind works that I do not have. > > > > Yes. A very convincing and often required proof of completenes of a > > linear set is to know the last element. > > Oh, is it often required? Except in matheology it is always required. > > > T talk about all in case there > > is no last is silly. > > And I think it is silly to require there being a last to be able to talk > about all. That's why you love matheology. > > > > > > > > Right, but there is no finite initial segment that contains them > > > > > > > all. > ... > > > > > Sorry, I have no knowledge of the bible. But live without that axiom > > > > > when you can't stomach it. And do not attack mathematicians who > > > > > live with that axiom. > > > > > > > > To live with that axiom does not create uncountability. See the proof > > > > here: > > > >http://groups.google.com/group/sci.logic/browse_frm/thread/46fa18c8bb= > > ... > > > > > > Where is the proof there? I see only you writing a bit of nonsense and > > > two rebuttals. > > > > One of the rebuttals has meanwhile been changed. Peter Webb > > recognized: It is true that you cannot show pi as a finite decimal, > > but you can't show 1/3 as a finite decimal either. > > So what? That is not contested and it does not show in *any* way that the > axiom of infinity does not create uncountability. So no proof at all. It may create what you like. Either 1/3 can be identified at a finite digit or 1/3 cannot be identified at a finite digit. Even a matheologian should understand that: If there is no digit at a finite place up to that the sequence 0.333... identifies the number 1/3, then there is no digit at a finite place up to that the number 1/3 can be identified. > > > Just what I said. > > And just wat I said: see the quote above: > > > > > > > Right, but there is no finite initial segment that contains them > > > > > > > all. > > which you contested. I did not contest it. I said, if there is a sequence that identifies 1/3, then the identifying digits must be at finite places. But we know that for every finite place d_n, there is a sequence d_1, ..., d_n that is not 1/3 but is identical to the sequence of 1/3. Therefore we can conclude that there is no sequence identifying the number 1/3 by means of digits at finite places only. > > > > > > The infinite paths because you stated a priori that your tree did > > > > > not contain infinite paths. So it is impossible to construct in > > > > > your tree infinite paths by the axiom of infinity. > > > > > > > > The axiom of infinity establishes the set N from finite numbers. > > > > > > It establishes the *existence* of a set N of finite numbers. > > > > What else should be established? > > Does not matter. The axiom of infinity does *not* construct infinite paths > in your tree, beacuse you stated that your tree did not contain infinite > paths a priori. The union of finite ínitial segments cannot ield an infinite initial segment? Does the sequence of 1/3 not consist of a union of all finite initial segments? Regards, WM
From: WM on 21 Dec 2009 12:15 On 21 Dez., 14:23, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > In article <9ad63bf1-549b-4822-bf86-839dd7c64...(a)j4g2000yqe.googlegroups.com> WM <mueck...(a)rz.fh-augsburg.de> writes: > > On 18 Dez., 15:19, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > ... > > > > > No, it is a matter of convention. In mathematics > > > > > a, b, c, ..., z > > > > > means a, b, c, continue this way until you reach z. But starting > > > > > {1}, {1, 2}, {1, 2, 3} > > > > > and going on you never reach > > > > > {1, 2, 3, ...} > > > > > > > > That is true. Therefore it does not exist. > > > > > > That you can not get there step by step does not mean that it does not > > > exist. > > > > That you cannot get step by step to 1/0 does not mean that it does not > > exist? > > Indeed. On the projective line (that precedes Cantor by quite some time as > far as I know) it does exist. But does the projective line exist? > > > > > However, see Cantor, > > > > collected works, p 445: > > > > 0, 1, 2, 3, ... w_0, w_0 + 1, ..., gamma, ..., > > > > He seems to reach far more. > > > > > > Right, he uses a convention that is no longer used. > > > > Wrong, it is used presently, for instance by myself. > > But you are not a mathematician. Do you think so? I studied mathematics and a university council appointed me to teach mathematical lessons. Have you better insights than they? Or have you only a different definition of mathematics? Do you mean that I am not a matheologian? Then you are right! > > > > > No. But the union contains two paths. > > > > > > Wrong. If we look at the paths as sets, they are sets of nodes. Their > > > union is a set of nodes, not a set of paths. And as a set of nodes we > > > can form from them seven different paths. > > > > Wrong. The nodes of two paths give exactly two paths. > > Darn, the paths 0.000 and 0.100 contain the following nodes: > 0.000 = {0., 0.0, 0.00, 0.000} and 0.100 = {0., 0.1, 0.10, 0.100} > where a node is named by the path leading to it. Their union contains > the following paths: > 0., 0.0, 0.00, 0.000, 0.1, 0.10, 0.100 > and I count seven. The nodes of two maximal paths give two maximal paths. You count initial segments. > > > > > Let every finite path of every infinite path be mapped on the elements > > > > of omega. That was simple. > > > > > > By your statements infinite paths do not exist. But pray give such a > > > mapping. Until now you have only asserted that such a mapping exists > > > without showing that. > > > > Do you accept the mapping from omega on SUM{k = 1 to n} 3*10^-k yields > > the infinite decimal expansion of 1/3? > > *What* mapping? Do you mean from n in omega -> SUM...? In that case the > infinite decimal expansion of 1/3 is unmapped. The sum of all finite segments is not the infinite path. Interesting. Regards, WM
From: Marshall on 21 Dec 2009 13:20 On Dec 21, 9:15 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > On 21 Dez., 14:23, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > > > > > > Right, he uses a convention that is no longer used. > > > > > > Wrong, it is used presently, for instance by myself. > > > But you are not a mathematician. > > Do you think so? I studied mathematics and a university council > appointed me to teach mathematical lessons. Have you better insights > than they? Obviously. Marshall
From: Jesse F. Hughes on 21 Dec 2009 14:17 WM <mueckenh(a)rz.fh-augsburg.de> writes: >> But you are not a mathematician. > > Do you think so? I studied mathematics and a university council > appointed me to teach mathematical lessons. Have you better insights > than they? Or have you only a different definition of mathematics? Do > you mean that I am not a matheologian? Then you are right! That you teach mathematics is a damned shame. I'm not sure why you draw attention to this sad mistake and I don't know why the administration hasn't the sense to fix their error. -- Jesse F. Hughes "[I]f gravel cannot make itself into an animal in a year, how could it do it in a million years? The animal would be dead before it got alive." --The Creation Evolution Encyclopedia
From: Virgil on 21 Dec 2009 16:34
In article <65fa0fa6-3723-4b81-ad46-1c3ab274fccc(a)t42g2000yqd.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 21 Dez., 14:17, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > In article > > <2fac8bb1-4c90-4421-b559-1ea7f0301...(a)e27g2000yqd.googlegroups.com> WM > > <mueck...(a)rz.fh-augsburg.de> writes: > > �> On 18 Dez., 15:12, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > ... > > �> > Why need I to think about a last one (which there isn't) to be able to > > �> > think about a set that contains all natural numbers? �Apparently you > > �> > have some knowledge about how my mind works that I do not have. > > �> > > �> Yes. A very convincing and often required proof of completenes of a > > �> linear set is to know the last element. > > > > Oh, is it often required? > > Except in matheology it is always required. What WM calls "Matheology" is merely a few of those parts of standard mathematics that WM is too inept to deal with > > > > �> � � � � � � � � � � � � � � � � � � � � T talk about all in case there > > �> is no last is silly. > > > > And I think it is silly to require there being a last to be able to talk > > about all. > > That's why you love matheology. (those parts of standard mathematics that WM is too inept to deal with) > > > > �> > �> > �> > Right, but there is no finite initial segment that contains > > them > > �> > �> > �> > all. > > ... > > �> > �> > Sorry, I have no knowledge of the bible. �But live without that > > axiom > > �> > �> > when you can't stomach it. �And do not attack mathematicians who > > �> > �> > live with that axiom. > > �> > �> > > �> > �> To live with that axiom does not create uncountability. See the > > proof > > �> > �> here: > > �> > > > �>http://groups.google.com/group/sci.logic/browse_frm/thread/46fa18c8bb= > > �> ... > > �> > > > �> > Where is the proof there? �I see only you writing a bit of nonsense > > and > > �> > two rebuttals. > > �> > > �> One of the rebuttals has meanwhile been changed. Peter Webb > > �> recognized: It is true that you cannot show pi as a finite decimal, > > �> but you can't show 1/3 as a finite decimal either. > > > > So what? �That is not contested and it does not show in *any* way that the > > axiom of infinity does not create uncountability. �So no proof at all. > > It may create what you like. Either 1/3 can be identified at a finite > digit or 1/3 cannot be identified at a finite digit. > > Even a matheologian should understand that. We understand it, but find it quite irrelevant to whether 1/3 can be identified with an infinite sequence of digits, which it can, in many ways. > > > > �> Just what I said. > > > > And just wat I said: see the quote above: > > �> > �> > �> > Right, but there is no finite initial segment that contains > > them > > �> > �> > �> > all. > > > > which you contested. > > I did not contest it. I said, if there is a sequence that identifies > 1/3, then the identifying digits must be at finite places. But we know > that for every finite place d_n, there is a sequence d_1, ..., d_n > that is not 1/3 but is identical to the sequence of 1/3. Therefore we > can conclude that there is no sequence identifying the number 1/3 by > means of digits at finite places only. > > > > �> > �> > The infinite paths because you stated a priori that your tree did > > �> > �> > not contain infinite paths. �So it is impossible to construct in > > �> > �> > your tree infinite paths by the axiom of infinity. > > �> > �> > > �> > �> The axiom of infinity establishes the set N from finite numbers. > > �> > > > �> > It establishes the *existence* of a set N of finite numbers. > > �> > > �> What else should be established? > > > > Does not matter. �The axiom of infinity does *not* construct infinite paths > > in your tree, beacuse you stated that your tree did not contain infinite > > paths a priori. � > > The union of finite �nitial segments cannot ield an infinite initial > segment? Does the sequence of 1/3 not consist of a union of all finite > initial segments? > > Regards, WM |