Cantor Confusion
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From: Virgil on 4 Feb 2007 14:42 In article <1170607157.002214.299710(a)p10g2000cwp.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 4 Feb., 14:34, Franziska Neugebauer <Franziska- > Neugeba...(a)neugeb.dnsalias.net> wrote: > > The path-length of the infinite path is the supremum of the set of > > paths-lengths of all finite paths. The latter is the supremum of the > > set of natural numbers. This supremum exists. > > No. What does it consist of? A supremum exists in many places but not > here. The simplest reason is that > omega - n = omega > for all n in N. WM does not seem to have much of a grasp of what a supremum is. Among ordinals, a supremum of a set of ordinals is an ordinal which is at least as large as any member of the set and is at least as small as any other ordinal having this property. For every n in omega, n < omega, and for any ordinal, k, not in omega, omega <= k. Thus omega is, according to the definition of supremum, the supremum of the set of ordinals < omega. > > > > > The total pathlength cannot be omega without omega being a pathlengt > > > > Omega is the path-length of the infinite path. > > The pathlengths is measured *and marked* by nodes. Only for finite paths, and not merely by nodes, by edges connected in a chain. > This is the > advantage of the tree. While you may insist that there is an infinite > umber of nodes we know that there is no node number omega. But the "number of nodes" is more thatn every finite natural or ordinal number. > > The pathlengths is measured *and marked* by nodes. Only for finite paths, and not merely by nodes, by edges connected in a chain. > > > > > "Infinite pathlength" means only that every finite pathlength is > > > surpassed by another finite pathlength. > > > > This is the wrong meaning in the framwork of contemporary set theory. > > What do you think is the correct meaning with respect to node numbers? That the "length" of the ultimate path is greater than every node number.
From: Virgil on 4 Feb 2007 14:47 In article <1170612113.718870.67100(a)h3g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 4 Feb., 15:31, "William Hughes" <wpihug...(a)hotmail.com> wrote: > > > > Similarly one can prove by induction: Every set of even positive > > > numbers contains numbers which are larger than the cardinal number of > > > the set. > > > > No. You make your usual mistake. > > No, you make your usual mistake by thinking that a potentially > infinite set can have an infinite cardinal number. A set can be only finite or not finite. There is no such thing as potential infiniteness distinct from non-finiteness. There is no third alternative. > > > Let E be the potentailly infinite set of even numbers. > > Two statements > > > > i: For every element e of E, the set > > F(e) = {2,4,6,...,e} contains cardinal numbers which are > > larger > > than the cardinal number of F(e) > > > > ii E contains numbers which are larger than the cardinal > > number of E > > > > Statement i is true and can be shown by induction. Statement > > ii is false. > > Of course, namely because there is no cardinal number for potentially > infinite sets. There is a cardinal for N everywhere except in WM-topia.
From: Virgil on 4 Feb 2007 14:59 In article <1170612592.876538.194120(a)k78g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 4 Feb., 17:33, Carsten Schultz <cars...(a)codimi.de> wrote: > > > >> > On Feb 2, 5:12 pm, Carsten Schultz <cars...(a)codimi.de> wrote: > > >> ... > > >> > > From that it follows that there are exactly four sets with four > > >> > > elements, since these are the elements of 4. > > > > > Is this intended to be a serious statement? > > > > Of course I am serious in pointing out that this is a consequence of > > what you have written, and had you not snipped your statement from which > > this is derived, evryone could see this. > > I did not snip anything here. (I did not answer you.) But what ever > you may have thought, it must be mad. > > > > >> > > It also follows that there > > >> > > is only one set with four elements, namely four. > > > > > With respect to the property "set having elements" one can in fact > > > take this position. There is only one set with four elements. > > > > So for example {1,2,3,4} = {7,42,666,1984}? WM declares absolutely that "There is only one set with four elements." But that would require {1,2,3,4} = {7,42,666,1984}, which is false.
From: David Marcus on 4 Feb 2007 15:34 Virgil wrote: > What is WM's definition of a set being finite anyway. He has often been > asked for it, but I do not recall his ever giving it. WM doesn't seem to understand the word "definition". I wonder if he has a name for what we call "definition". -- David Marcus
From: David Marcus on 4 Feb 2007 15:35
Fuckwit wrote: > On 4 Feb 2007 10:39:08 -0800, mueckenh(a)rz.fh-augsburg.de wrote: > > > That is set theory, not mathematics. > > Well, the last time I've checked it set theory was mathematics. I guess you haven't read WM's book. -- David Marcus |