Cantor Confusion
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From: Virgil on 13 Nov 2006 16:01 In article <1163430459.318473.317960(a)i42g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > > If you add only one element to each column, you get the order type > omega + 1 for the length of he matrix. The "length", being a cardinality rather than an ordinality, is unaffected, since Card(omega) = Card(omega+1).
From: Virgil on 13 Nov 2006 16:09 In article <1163431235.417059.113430(a)e3g2000cwe.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > > > > Well, my only advise is, read it. > > > > > > If he says so, then it wil not be a good idea to waste my time with it. > > > > Do you really think the node 1/3 is finitely far from the root in the tree? > > Dik, are you joking? There is no node yielding any number like 1/3. Precisely the point!!! > > The first two levels of the tree contain the following initial > segments: > 0.00... > 0.01... > 0.10... > 0.11... > which are to be continued in the third and following levels. I use the > tree because it makes notation easy (and reduces the digits necessary > to write down for infinitely many numbers because the first digit for > all is either 0 or 1) and it assures that here is no number left out. > (There is no diagonal argument possible.) On the contrary, this construction follow Cantor's original argument based on two-valued strings of characters. Theorem: Any list of endless two valued strings must be incomplete. Proof: A string having in its nth place the opposite of the nth character of the nth string differs from every string the list.
From: Virgil on 13 Nov 2006 16:16 In article <1163431763.731213.16430(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > No set of the form {1,2,3..,n} can be counted by an unnatural > > number like omega. The set {1,2,3,...} is not a set of the form > > {1,2,3,...,n} > > If only natural numbers are within, that it is of that form. Although > we may not know the last one, we can be sure that it is a natural. We cannot even b e sure that there is a last one, as there is no way to stop n -> n+1. > > > > > This leads to the problem |{1,2,3,...}| = > > > omega and |{1,2,3,...,omega}| = omega + 1. So we have |{1,2,3,...,a}| = > > > a or a + 1, corresponding to the kind of a. If |s| represents the cardinality of set s then |omega| = |omega + 1| > > > > Yes. This is true. It is not however a problem. > > It shows that by setting omega we loose a number in between. Only for those who deliberately conflate ordinality with cardinality. > It should > disturb a mathematician. What WM does does. > > > > [Sometimes this is used to argue that starting at 0 is more elegent. > > That is nonsense. Von Neumann was not noted for nonsense. > The first natural number is that one which is called > the "first" and not the zeroth. 0 is a perfectly nice number, with nothing unnatural about it.
From: Virgil on 13 Nov 2006 16:18 In article <1163432644.765437.254240(a)h54g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > > If you would at least say clearly just what your argument about trees > > is supposed to be about - to what theory it is supposed to apply - then > > that would help. > > This argument simply shows that there are not more real numbers than > natural numbers, where real and natural numbers are those defined in > modern mathematics. Since the "argument" has not been made clearly enough to be understood, it does not yet show anything. > BTW: You are correct in assuming the the finiteness of the univers does > not play a role in this argument. > > Regards, WM
From: Virgil on 13 Nov 2006 16:21
In article <1163432735.818253.258930(a)f16g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > > Franziska Neugebauer schrieb: > > > > > >> mueckenh(a)rz.fh-augsburg.de wrote: > > >> > > >> > Franziska Neugebauer schrieb: > > >> [...] > > >> >> Are there really three vertices in WM's "triangle"? > > >> > > > >> > If finished infinities [...] > > >> > > >> Verbiage. > > > > > > Yes. But, sorry to see, it is the fundament of modern mathematics. > > > > "Finished infinities" is your wording. > > Precisely describing the fundament of modern mathematics. It may describe WM's fundament, but need not describe anyone elses'. |