Cantor Confusion
in [Math]
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From: WM on 8 May 2007 09:11 On 8 Mai, 00:30, MoeBlee <jazzm...(a)hotmail.com> wrote: > On May 7, 2:42 pm, James Burns <burns...(a)osu.edu> wrote: > > > 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 > > Thank you for this. > > My remarks in this post are upon the supposition that, between the > older and more recent editions, there is not a material difference on > this matter of functions.. > > >http://books.google.com/books?id=Er1r0n7VoSEC&pg=PA23&ots=1afi1e6mts&... > > /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/. > > Even in that informal description, the authors do not mention a > "formula" as did WM. A procedure or rule is a frmula. > > > 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/. > > Right, a function is a relation. A function is not itself, as WM > claimed the authors to represent, a formula, domain, and range. 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. > > And, moving past the informal descriptions and motivation, here is the > explicit definition: which does not contradict the former. Regards, WM
From: WM on 8 May 2007 11:00 On 8 Mai, 14:41, William Hughes <wpihug...(a)hotmail.com> wrote: > On May 8, 8:20 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > On 7 Mai, 14:10, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > > > Every node, n, of path p is shared with another path > > > > > p'(n). Note that p'(n) can be different for every n. > > > > > Note that p'(n) need not diffrent for every n. > > > > However, it cannot be the same for all n. > > > For which n must it change? > > > Could you explain, please, how you define a set of at least two > > different paths, p' and p'' (or even more) which are required to > > accompany p for all n? > > 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)) > > A: for any node k, of p, there are an infinite number of paths > in P' which contain node k 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... ? None of them is *required* to have a company of p for all n. And obviously none of them does accompany p for all n, because already p'(4) = 0000111... outperforms them all. I asked for at least two paths which cannot be substituted by other paths or eve one path. Could you explain, please, how you define a set of at least two different paths, p' and p'' (or even more) which are required to accompany p for all n? Regards, WM
From: WM on 8 May 2007 11:02 On 7 Mai, 21:36, Virgil <vir...(a)comcast.net> wrote: > In article <5hJ%h.160241$kJ4.150...(a)reader1.news.saunalahti.fi>, > Aatu Koskensilta <aatu.koskensi...(a)xortec.fi> wrote: > > > On 2007-05-07, in sci.math, Virgil wrote: > > > Some definitions of pi are infinite but quite well understood. > > > Really? What definitions would these be? > > The ones that define it as the limit of an infinite sequence, as such > processes are, in theory, infinite, but usually in practice soon close > enough for all practical purposes. > > While the process can, in some senses, be "finitely defined", the exact > value cannot be finitely determined. Unless the process was, in any sense, finitely defined, we could not apply it. But now I can fully appreciate that and why you believe in the infinite. Regards, WM
From: WM on 8 May 2007 11:04 On 7 Mai, 23:42, James Burns <burns...(a)osu.edu> wrote: > 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. I completely agree. And there is no contradiction between the two paragraphs they wrote. Regards, WM
From: WM on 8 May 2007 11:07
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? Regards, WM |