Cantor Confusion
in [Math]
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From: WM on 9 May 2007 06:43 On 8 Mai, 19:48, Virgil <vir...(a)comcast.net> wrote: > > You cannot determine or define or construct or write down or tell me a > > bijection between the set of numbers which are defined by a definition > > identifying them uniquely and the set of natural numbers. > > That is the whole point, nobody can. > The same is true for the paths in the tree. They form a subet of a countable set but we cannot find a bijection with N. Regards, WM
From: WM on 9 May 2007 06:52 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. 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)? > > And the formal definition requires a function to be a set of ordered > pairs (a relation), so that it is first of all a SET, which is quite > different from being merely a rule.- Of course it is not merely a rule. It is a rule and two sets, which are connected by this rule. Regards, WM
From: Carsten Schultz on 9 May 2007 07:05 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? > 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? >> And the formal definition requires a function to be a set of ordered >> pairs (a relation), so that it is first of all a SET, which is quite >> different from being merely a rule.- > > Of course it is not merely a rule. It is a rule and two sets, which > are connected by this rule. Have you read definition 3.1? -- 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: WM on 9 May 2007 09:16 On 9 Mai, 00:12, Virgil <vir...(a)comcast.net> wrote: > In article <1178659617.233805.68...(a)e51g2000hsg.googlegroups.com>, > > 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 subsets of countable sets are countable. What means "countable"? > > > > > > There are subsets of a finitely definable set which are not > > > finitely definitable. > > > And they do allow for a bijection with N? > > Yup, but those bijections are not finitely defineable. There does not exist anyting in mathematics, unless it isfinitely definable. In particular, any infinite definition is not a definition, because there is a definition of "definition" which says: Definition: A definition has an end, i.e., a last word and a point. Regards, WM
From: WM on 9 May 2007 09:21
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. 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. Reards, WM |