Cantor Confusion
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From: Franziska Neugebauer on 7 Feb 2007 10:48 mueckenh(a)rz.fh-augsburg.de wrote: > On 6 Feb., 13:47, Franziska Neugebauer <Franziska- > Neugeba...(a)neugeb.dnsalias.net> wrote: >> mueck...(a)rz.fh-augsburg.de wrote: >> > It can be proven. {p(0), p(1), q(1), p(2), q(2), r(2),s(2) ... } >> > contains {p(0), p(1), p(2), ... } >> >> What does contain mean? > > The tree is a set of nodes. This is not true. A tree is "more" than a set of nodes. > It contains paths as subsets. 0. In the usual definition of tree T = { v, e } this is obviously not true since neither of the non-trivial subsets {v}, {e}, {v, e} is a path at all. 1. Only in Virgil's view of tree T it is true that a tree is a set of paths { p1, ... } . 2. A set s is a subset of a superset S iff A x (x e s -> x e S) 3. I interpret your conjecture "It contains paths as subsets." as "there is (at least) a path which is a subset of the tree", formally E p (p c T) (*) which means E p (A x (x e p -> x e T)) Assume now that there is some p = p' for which (*) holds. We have then A x(x e p' -> x e T) (**) Now the definition of path comes into play. As explained below the elements of p' are ordered pairs, i.e. have structure (number, node) or (number, edge). As one easily sees T only has paths as elements, i.e. has sequence structure { ..., (number, node), ... }. Therefore there is no (non-trivial) p' which is a subset of a tree. Hence your conjecture "there are paths which are subsets of a tree" is wrong. > The special meaning of "contain" is irrelevant as long as we refer to > a unique meaning, i.e., > either: something which is contained in the tree is really in the > tree, > or: something which is contained in the tree does belong to the tree > as a supremum. It is absolutely necessary to carefully distinguish between membership (element-relation "e") and the subset-relation ("c"). >> > Correct. And a set of nodes can be a path. >> >> No. A paths *is* >> >> a) a sequence of nodes, or >> b) a sequence of edges. >> > Yes to a and b. You should force yourself to take a decision. Nonetheless we notice that the *elements* of a path are ordered pairs, i.e. have structure (number, node) or (number, edge). > But not every set of nodes or edges is a path. No non-trivial set of nodes is a path at all since those sets do not posess the necessary sequence-structure. >> You have simply stated the M�ckenheim axiom >> >> X is not finite -> there must be an x in X which is infinite >> >> which is alas not part of ZFC or any other modern set theory. > > > I did not start off with this assumption. You do persistently refer to that axiom but you don't grasp that you do. F. N. -- xyz
From: William Hughes on 7 Feb 2007 10:48 On Feb 7, 10:12 am, Han de Bruijn <Han.deBru...(a)DTO.TUDelft.NL> wrote: > William Hughes wrote: > > And since the limit is not a finite number the fact that there > > is a contradiction for all finite numbers does not mean that > > there is a contradiction for the limit. > > In my calculus class, the fact that the "limit is not a finite number" > just always meant: the limit _does not exist_, i.e. there IS NO limit. .. If the limit doesn't exist there is no contradiction. If you progress beyond introductory calculus you will find that it is possible to define a limit that is not a finite number. But it does not matter. Whether or not you definite such a limit there is no contradiction. - William Hughes
From: Franziska Neugebauer on 7 Feb 2007 10:59 mueckenh(a)rz.fh-augsburg.de wrote: > On 6 Feb., 23:07, Virgil <vir...(a)comcast.net> wrote: > >> > The basic way to establish IV c V is to use the numbers in their >> > basic >> > form IIII c IIIII. (Numbers *are* their representations.) >> >> Numerals are no more numbers than names are people. > > Wrong! People can exist and do exist without names. Numbers cannot. Who has told you that? Whatever, M�ckenheim axiom 2: "Numbers cannot exist without name" Let's call that numbers "named numbers". If the length of names "must" be finite one can easily proof, that any set of named numbers has cardinality <= card(omega). F. N. -- xyz
From: Franziska Neugebauer on 7 Feb 2007 11:20 mueckenh(a)rz.fh-augsburg.de wrote: > On 6 Feb., 23:07, Virgil <vir...(a)comcast.net> wrote: > >> > The basic way to establish IV c V is to use the numbers in their >> > basic >> > form IIII c IIIII. (Numbers *are* their representations.) >> >> Numerals are no more numbers than names are people. > > Wrong! People can exist and do exist without names. Numbers cannot. Who has told you that? Whatever, M�ckenheim axiom 2: "Numbers cannot exist without name" Let's call that numbers "named numbers". If the length of names "must" be finite one can easily prove, that any set of named numbers has cardinality <= card(omega). F. N. -- xyz
From: Virgil on 7 Feb 2007 13:31
In article <1170851625.932474.129740(a)q2g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 6 Feb., 13:21, Franziska Neugebauer <Franziska- > Neugeba...(a)neugeb.dnsalias.net> wrote: > > mueck...(a)rz.fh-augsburg.de wrote: > > > On 5 Feb., 05:10, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > > [...] > > > > > Of course the requirement to decide whether n is in N is more circular > > > than the statement that 1 is in N. > > > > Contemporary set theories do not have these kind of problems. > > > > [...] > > > > >> it is impossible to talk about > > >> it, at least mathematically. > > > > > Present mathematics has nothing in common with existence. > > > > In present mathematics "existence" does not mean *physical* existence. > > In present mathematics "existence" is meaningless. In present mathematics WM is meaningless. > > > > [...] > > > > >> Existence is a mathematical thing when you can establish it by axioms > > >> or through theorems based on axioms. Anything else is merely > > >> phylosophy. > > > > > There is something called reality and another thing called matheology. > > > > The latter is taught in Augsburg. > > No. That is a wrong statement. Certainly if what WM "teaches" corresponds to what WM claims here, WM does not teach anything that can be seriously called mathematics. |