Cantor Confusion
in [Math]
|
From: Virgil on 7 May 2007 17:06 In article <1178566715.079760.283530(a)e51g2000hsg.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 7 Mai, 18:53, Virgil <vir...(a)comcast.net> wrote: > > In article <1178537072.163162.176...(a)u30g2000hsc.googlegroups.com>, > > > > WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > On 7 Mai, 00:59, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > > > Look! Over there! A pink elephant! > > > > > That seems necessary to make us believe that a paths that shares > > > every node with another path does not share every node with another > > > path. But it is not sufficient. > > > > > Regards, WM > > > > If WM believes that one path in any tree can share every node with > > another and different path, he has passed beyond reason into some sort > > of fugue state. > > Can you explain, how every node of p is shared by other paths If p represents a path and p' a different path in a CIBT, then the set of shared nodes is finite but not empty, and will have a "last", or highest level, node in it. However, the set of ALL paths which pass through that last common node will biject with the set of all paths in the entire tree (just cut them off below that node so that node becomes the new root node). , but not > every node is shared by at least one path p'? Do the companion paths > alternate? E.g., p' for every even node and p'' for every odd node? > > Regards, WM
From: MoeBlee on 7 May 2007 17:08 On May 7, 12:45 pm, 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>, > > > > 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. I can see online the pages in that book defining 'ordered pair', 'binary relation', and other terms in the the completely usual set theoretic manner. But the online view cuts off before the section on functions. So I would like to know the exact quote in this book for the definition of 'function'. Moreover, no matter what definition this particular book gives, what I said, as quoted above and stated here again, stands: ['A formula, domain and range'] 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. MoeBlee
From: Virgil on 7 May 2007 17:13 In article <1178566976.513253.63820(a)p77g2000hsh.googlegroups.com>, WM <mueckenh(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.
From: James Burns on 7 May 2007 17:42 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=Hrbacek++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/. :
From: Virgil on 7 May 2007 17:50
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. |