Cantor Confusion
in [Math]
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From: WM on 8 May 2007 08:56 On 7 Mai, 22:59, Virgil <vir...(a)comcast.net> wrote: > In article <1178565971.031862.318...(a)y5g2000hsa.googlegroups.com>, > > > > > > WM <mueck...(a)rz.fh-augsburg.de> wrote: > > On 7 Mai, 18:06, Virgil <vir...(a)comcast.net> wrote: > > > In article <1178533278.892846.274...(a)w5g2000hsg.googlegroups.com>, > > > > > The number of separated paths is less than the number of nodes at each > > > > finite level. What happen after each finite level is irrelevant. > > > > The number of paths, or more properly uncountable sets of paths, > > > distinguishable at any level is, like the level itself, finite, but > > > there are infinitely many levels, and what happens at any one level is > > > not the end. In order to consider the whole tree one must consider all > > > infinitely many levels. > > > Yes. But if you consider two infinite sequences, namely the sequence > > 1,2,3... and the sequence 2,4,6,..., and put the tems in bijection, > > 1-2, 2-4, 3-6, ..., then you can wait and wait and wait: The first > > term will never get larger than the second. Even in the infinite this > > will not happen. > > What you are saying has no relevance to paths in my infinite trees. Oh yes, it is the same arguing. You hope "the infinite" may fulfill your requirements. But it won't because it can't. Regards, WM
From: William Hughes on 8 May 2007 08:56 On May 8, 8:46 am, Carsten Schultz <cars...(a)codimi.de> wrote: > William Hughes schrieb:> On May 8, 8:20 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > [The usual stupid stuff that Mückenhein writes] > > > Here is a very simple, countable example. > > > Let p be the path 000... > > > Let p'(1) be the path 0111... > > Let p'(2) be the path 00111... > > Let p'(3) be the path 000111... > > ... > > > let P' be the set of all the p'(n) > > > In general p'(n) has a n 0 nodes, followed by only 1 nodes. > > (Note that there is no last p'(n)) > > [...] > > You mean the set of all integers exists, and really and actually exists, > and still does not have a largest element? Wow, that must be hard to > swallow for Mückenheim. As usual Muekenheim breaths hot and cold on the existence of infinite sets. He treats them as existing when it suites him, but when cornered, says that they do not exist (not that he then stops talking about infinite sets and their properties). However, the example can be phrased in terms of potentially infinite paths and having P' a potentially infinite set of potentially infinite paths. It is not necessary for the set of integers to "actually exist". - William Hughes
From: WM on 8 May 2007 09:03 On 7 Mai, 23:13, Virgil <vir...(a)comcast.net> wrote: > In article <1178566976.513253.63...(a)p77g2000hsh.googlegroups.com>, > > WM <mueck...(a)rz.fh-augsburg.de> wrote: > > On 7 Mai, 18:50, Virgil <vir...(a)comcast.net> wrote: > > > In article <1178536779.509069.124...(a)l77g2000hsb.googlegroups.com>, > > > > There are all sorts of things in naive set theory texts that formal set > > > theories do not allow. > > > I am not interested in formal set theory but only in the question: Are > > there uncountably many real numbers? > > There are uncountably many binary strings, and only countably many of > them map to duplicates of other strings when interpreted as reals in > [0,1], so, in ZF or NBG, yes! > > > > > > > Fn = (2-1-1) > > > > It is the number of separated path coming out of the n-th element of > > > > the tree minus the number of ingoing separated paths minus the number > > > > of nodes of this element. > > > > In that case, it is irrelevant. > > > Why? > > Because you never are counting anything but "bunches" each of which > represents a set of uncountably many paths, so the number of bunches is > not relevant to the number of paths. Why are you never countaing paths? Perhaps because there are no paths? I there were paths, then they could be counted. Regards, WM
From: WM on 8 May 2007 09:07 On 7 Mai, 23:50, Virgil <vir...(a)comcast.net> wrote: > In article <463F9D4F.7030...(a)osu.edu>, James Burns <burns...(a)osu.edu> > wrote: > > > > > > > Carsten Schultz wrote: > > > WM schrieb: > > > >>On 7 Mai, 18:50, Virgil <vir...(a)comcast.net> wrote: > > > >>>In article <1178536779.509069.124...(a)l77g2000hsb.googlegroups.com>, > > > >>>>>That is a very common NON-formal definition and notion of function > > >>>>>found in a great amount of mathematics. However, it is NOT the set > > >>>>>theoretic defintion that is being used in formal Z set theory and is > > >>>>>NOT the definition that is used in formal Z set theory to prove > > >>>>>Cantor's theorem that there is no function from a set onto its power > > >>>>>set. > > > >>>>It is a definition from a book on set theory, called "Introduction to > > >>>>Set Theory". So it gives basic set theory. > > > >>>It may very well be a very naive set theory for non-mathematicians. > > >>>Who wrote it, and for what sort of students is it supposed to be an > > >>>introduction? > > > >>Karel Hrbacek and Thomas Jech: "Introduction to Set Theory" > > >>Marcel Dekker Inc., New York, 1984, 2nd edition. 250 pages. > > >>For students of set theory. > > > > Now please, before we rush to look it up: I guess that everyone here > > > agrees that a function is a special kind of relation, where a relation > > > is a subset of a cartesian product of two sets. What is debated is that > > > there is an additional requirement that a function has to assign values > > > according to a rule in a sense which implies computation, definability > > > or something similar. So what exactly is it that the book you mention > > > says? And when you answer, please distinguish between a technical > > > definition and illustrative prose. > > > Too late. I'm afraid I've already rushed to look it up. > > I only found the 1999 edition of Hrbacek and Jech, but perhaps > > that will be a close approximation of the wording in the > > 1984 edition. I searched Google Books with > > Hrbacek Jech set theory function > > and got the definition of function in the 1999 edition. > > (Unfortunately, I can't copy and paste from there. Pardon > > the typos, please.) > > > Oddly enough, the definition of function there looks essentially > > identical to every other (mathematical) definition of function I > > have ever seen. > > > Jim Burns > > >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 and excludes the part > that gives the formal definition, and, incidentally, proves him wrong.- Zitierten Text ausblenden - So 3.1 proves wrong what a function is understood in mathematics? Why do you think Hrbacek and Jech do so? Regards, WM
From: William Hughes on 8 May 2007 09:09
On May 8, 8:27 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > On 7 Mai, 21:49, William Hughes <wpihug...(a)hotmail.com> wrote: > > > Since there is no last path which branches off, this list never ends. > > So every node of p is shared by other paths. > > However, every path branches of somewhere, so there > > is no path p' which shares every node of p. > > You state: > (1) Every node of p is shared by other paths. > (2) Every path branches off somewhere. > > (1) ==> Not every path has banched off somewhere (because there must > remain at least one "other path" and "is shared" is the opposite of > has branched off). > (2) ==> Every path has branched off somewhere. > > So your claim can be summarized: > Not every path has branched off somewhere. > Every path has branched off somewhere. The statement "Every path branches off somewhere" is true. However. this implies "Every path has branched off somewhere" only after the last path has branched off. You have not been doing your exercises. - William Hughes |