Cantor Confusion
in [Math]
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From: William Hughes on 9 May 2007 09:36 On May 9, 9:21 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > On 9 Mai, 00:56, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > > On May 8, 5:26 pm, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > On 7 Mai, 21:32, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > > On May 7, 3:19 pm, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > > > That is not the point here. The point is that there are subsets of > > > > > countable sets which do not allow for a bijection with N. > > > > > No. > > > > Why "no"??? > > > Because you made the statement " there are subsets of > > countable sets which do not allow for a bijection with N." > > > The "No" was meant to indicate that this statement is false. > > What is true is that there are subsets of a finitely definable > > set which are not finitely definable. The difference is... > > > WAHH! WAHH! WAHH! I'm not listening. > > > > > There are subsets of a finitely definable set which are not > > > > finitely definitable. > > > > And they do allow for a bijection with N? > > > Yes, but not a finitely definable one. The difference is... > > > WAHH! WAHH! WAHH! I'm not listening. Gosh, I am sure that this line read M: WAHH! WAHH! WAHH! I'm not listening. when I wrote it. I must be wrong. Only a total slimeball would edit a line and then present it attributed. > > If you can recover for some time, please take notice: > Either: In mathematics there is nothing existing, unless it is > finitely definable. > Or: There exists a bijection between paths and nodes but it is not > finitely definable. > There is a bijection is between paths and sets of nodes. Look! Over there! A pink elephant! There is a bijection is between paths and nodes. - William Hughes
From: WM on 9 May 2007 09:39 On 9 Mai, 01:10, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > > But why do you write down these paths: > > > > > Let p'(1) be the path 0111... > > > > Let p'(2) be the path 00111... > > > > Let p'(3) be the path 000111... > > > > > ? > > > > Any set p(1) up to p(n) can be substitued by a single path. > > > In fact? Why then did you write down so many useless paths? > > > > Look! Over there! A pink elephant! > > > > The set P' can be substitued by a single path. > > > Oh, does the set P' consist of any paths which are not paths p(n) with > > a finite argument n? > > i: Any set of paths {p(1) ... p(n)) can be replaced by a single > path. Why then did you write the unnecessary paths? > > ii: The set P' is the union of sets {p(1) ... p(n)} and as such a set of nodes. In order to avoid the clumsy expression of path bunches, I call a path bunch now a finite path. A finite path ends at some node. The infinite path {0.000...} is the union of all finite paths with only 0's, namely {0.} U { 0.0} U {0.00} U ... The union is one infinite path and, therefore, has not more paths than were unified. All nodes of the tree are last nodes of finite paths. The union over all finite paths is the set of all infinite paths. This set has not more elements than were unified. > > Look! Over there! A pink elephant! > > iii: The set P' can be replaced by a single path. You need a pink elephant to see that? Regards, WM
From: Phil Carmody on 9 May 2007 09:40 William Hughes <wpihughes(a)hotmail.com> writes: > On May 9, 9:21 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > WAHH! WAHH! WAHH! I'm not listening. > > Gosh, I am sure that this line read > > M: WAHH! WAHH! WAHH! I'm not listening. > > when I wrote it. I must be wrong. Only a total slimeball > would edit a line and then present it attributed. Yes, he's a slimeball as well as being utterly ignorant about the foundations of modern mathematics. Killfile him, and help keep the signal-to-noise ratio slightly further away from zero. Phil -- "Home taping is killing big business profits. We left this side blank so you can help." -- Dead Kennedys, written upon the B-side of tapes of /In God We Trust, Inc./.
From: WM on 9 May 2007 09:52 On 9 Mai, 01:21, William Hughes <wpihug...(a)hotmail.com> wrote: > On May 8, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > time does not play a role in set theory > > and > > To branch off is a temporal act. > Therefore all paths which can "branch off" have done so. Or are there some exceptions for pink Billy? Regards, WM
From: WM on 9 May 2007 09:59
On 9 Mai, 13:05, Carsten Schultz <cars...(a)codimi.de> wrote: > WM schrieb: > > > > > > > On 8 Mai, 20:32, Virgil <vir...(a)comcast.net> wrote: > >> In article <1178636838.282834.114...(a)w5g2000hsg.googlegroups.com>, > > >> WM <mueck...(a)rz.fh-augsburg.de> wrote: > >>> On 7 Mai, 23:50, Virgil <vir...(a)comcast.net> wrote: > >>>>>http://books.google.com/books?id=Er1r0n7VoSEC&pg=PA23&ots=1afi1e6mts&... > >>>>> k++Jech+set+theory+function&sig=Zx5hPqZZ2icNy3mkguHi9kyrFVA#PPA24,M1 > >>>>> /Introduction to Set Theory/, by Karel Hrbacek, Thomas J. Jech > >>>>> (1999) [pages 23-4] > >>>>> : > >>>>> : * 3. Functions * > >>>>> : > >>>>> : Function, as understood in mathematics, is a procedure, a > >>>>> : rule, assigning to any object /a/ from the domain of the > >>>>> : function a unique object /b/, the value of the function > >>>>> : at /a/. A function, therefore, represents a special type > >>>>> : of relation, a relation where every object /a/ from the > >>>>> : domain is related to precisely one object in the range, > >>>>> : namely, to the value of the function at /a/. > >>>>> : > >>>>> : * 3.1 Definition * A binary relation /F/ is called a > >>>>> : /function/ (or /mapping/, /correspondence/) if /aFb_1/ > >>>>> : and /aFb_2/ imply /b_1 = b_2/ for any /a/, /b_1/, and > >>>>> : /b_2/. In other words, a binary relation /F/ is a function > >>>>> : if and only if for every /a/ from dom /F/ there is exactly > >>>>> : one /b/ such that /aFb/. This unique /b/ is called > >>>>> : /the value of F at a/ and is denoted /F(a)/ or /F_a/. > >>>>> : [F(a) is not defined if /a [not in] dom F/.] If /F/ is > >>>>> : a function with /dom F = A/ and /ran /F/ [subset] B/, > >>>>> : it is customary to use the notations /F: A -> B/, > >>>>> : /<F(a)| a [in] A>/, /<F_a>_a[in]A/ for the function /F/. > >>>>> : The range of the function /F/ can then be denoted > >>>>> : /{F(a)| a [in] A}/ or /{F_a}_a[in]A/. > >>>>> : > >>>> As usual, WM includes only the irrelevant bits > >>> I only wanted to avoid typing infinite definitions. > >>>> and excludes the part > >>>> that gives the formal definition, and, incidentally, proves him wrong > >>> The second paragraph proves the first one wrong, in your opinion? > >> A formal definition always REPLACES any informal ones, and governs the > >> meaning and usage of the thing defined. > > > But the formal definition does not specify how a and b are related > > other than by mentioning F. This F however is undefined unless you > > know from the first paragraph that it is a procedure or rule. > > No, it says that F is a binary relation. Do you know how a binary > relation is defined? By at least one set and a rule. > > > Further definition 3.1 contains: It is customary to use the notations / > > F: A -> B/. Why do you think A and B were not two sets and F was not a > > formula, as I said (and as anybody says who ever studied some > > mathematics)? > > Why don't you go to a university, take a course in set theory, and come > back once you have mastered the basics? Because it seems you have done this. And we see the deterrent result. Regards, WM |