Cantor Confusion
in [Math]
|
From: Carsten Schultz on 7 May 2007 18:17 Virgil schrieb: > In article <463F9D4F.7030904(a)osu.edu>, James Burns <burns.87(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&dq=Hrbace >> 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. So, is he dishonest or just clueless? -- Carsten Schultz (2:38, 33:47) http://carsten.codimi.de/ PGP/GPG key on the pgp.net key servers, fingerprint on my home page.
From: MoeBlee on 7 May 2007 18:30 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 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. And, moving past the informal descriptions and motivation, here is the explicit definition: > : * 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/. And there it is, exactly the EXPLICIT DEFINITION that is the very set theoretic definition I've told WM about, and the one that is of concern in formal Z set theory statements and proofs of Cantor's theorem. WM rested upon giving the authors' initial account of an informal notion in general mathematics as he failed to recognize the exact formal definition that I has already referred to. > : 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/.] The 'not defined' part can be handled differently though, if one adopts certain formulations. But that is not at issue here anyway. > : 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/. Yep, entirely usual all the way. In set theory, a function is a relation such that every member of the domain maps to exactly one thing. And, WM is completely, and as usual, incorrect as to even such basic matters of set theory when he claims that these authors represent that the set theoretic definition of 'function' is that of a formula, domain, and range. MoeBlee
From: MoeBlee on 7 May 2007 18:36 On May 7, 3:17 pm, Carsten Schultz <cars...(a)codimi.de> wrote: > Virgil schrieb: > > As usual, WM includes only the irrelevant bits and excludes the part > > that gives the formal definition, and, incidentally, proves him wrong. > > So, is he dishonest or just clueless? He strikes me as dishonestly clueless and cluelessly dishonest. MoeBlee
From: Virgil on 7 May 2007 22:25 In article <f1o8hh$5g4$1(a)news2.open-news-network.org>, Carsten Schultz <carsten(a)codimi.de> wrote: > Virgil schrieb: > > In article <463F9D4F.7030904(a)osu.edu>, James Burns <burns.87(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&dq=Hrb > >> ace > >> 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. > > So, is he dishonest or just clueless? WM has all the earmarks of one of Eric Hofer's "true believers", whom are, I suppose, closer to clueless than dishonest.
From: WM on 8 May 2007 08:20
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? Regards, WM |