Cantor Confusion
in [Math]
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From: WM on 8 May 2007 17:34 On 8 Mai, 17:48, Carsten Schultz <cars...(a)codimi.de> wrote: > WM schrieb: > > > > > > > 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? > > The second paragraph proves your interpretation of the first paragraph > wrong. Note that `procedure' and `rule' in the first paragraph are > undefined, which is why axiomatization of set theory is a good thing. Note that formula n my text is as undefined as rule in H&J. Why don't you ask a mathematician how to interpret text, if you can't read? Regards, WM
From: Virgil on 8 May 2007 18:12 In article <1178659617.233805.68040(a)e51g2000hsg.googlegroups.com>, WM <mueckenh(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. > > > 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.
From: Virgil on 8 May 2007 18:20 In article <1178659949.553077.14050(a)h2g2000hsg.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 8 Mai, 17:28, William Hughes <wpihug...(a)hotmail.com> wrote: > > On May 8, 11:09 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > > > > > > > > > > On 8 Mai, 15:09, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > > > 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. > > > > > If "*there is* a set such that every path branches off somewhere", > > > then every path *has branched off* somewhere. > > > > No. 'every path branches off somewhere' needs only existence, no > > order (temporal or otherwise) is needed > > To branch off is a temporal act. Maybe in WM's world mathematical trees are dynamic, changing with time, , but in ZF and NBG, they are static, and branching is a matter of which nodes are linked by an edge to which other nodes. > > > > 'every path *has branched off* somewhere' needs some sort of > > order (temporal or otherwise) > > Yes, factual existence and completion of the generaton process. > > > > There are ordered sets that do not have a last element. > > Why do you stress this? Pinky Billy? Because it distinguishes ZF and NBG from WM's world where "sets" apparently always have "last elements" though those "last elements" are indeterminate and ephemeral. > > Regards, WM
From: Virgil on 8 May 2007 18:23 In article <1178660072.618678.192000(a)w5g2000hsg.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 8 Mai, 17:48, Carsten Schultz <cars...(a)codimi.de> wrote: > > WM schrieb: > > > > > > > > > > > > > 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? > > > > The second paragraph proves your interpretation of the first paragraph > > wrong. Note that `procedure' and `rule' in the first paragraph are > > undefined, which is why axiomatization of set theory is a good thing. > > Note that formula n my text is as undefined as rule in H&J. Therefore anything depending on "formula" in your text is equally undefined. > > Why don't you ask a mathematician how to interpret text, if you can't > read? Do not, as some ungracious pastors do, Show me the steep and thorny way to heaven, Whiles, like a puff'd and reckless libertine, Himself the primrose path of dalliance treads And recks not his own rede.
From: William Hughes on 8 May 2007 18:56
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... M: 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... M: WAHH! WAHH! WAHH! I'm not listening. - William Hughes |