Cantor Confusion
in [Math]
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From: Carsten Schultz on 9 May 2007 12:50 WM schrieb: > 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"? How about a proof as a basis for a discussion? Theorem (ZF). Let X be a countable set, and Y an infinite subset of X. Then Y is countable. Proof (sketch): Let a: N->X be a bijection. We define a function k:N->N recursively by k(n) = min {n in N\{k(m) | m<n} | a(n) in Y}. The set on the right hand side is never empty, since Y is infinite. The composition aËk is a bijection from N to Y. >>>> 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. We say that an object x with property P exists, if the statement "there exists an x with P(x)" can be proven. No further definition of x is necessary. > 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. -- 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 13:30 On 8 Mai, 20:00, Virgil <vir...(a)comcast.net> wrote: > And perhaps, as a non-mathematician, one might even say an > anti-mathematician, WM is incompetent to judge the value of mathematical > definitions to mathematics. If you call yourself a mathematician, then I will agree to the honour of being an anti-mathematician. > > I attended Harvard and Oxford, among others. Not studying mathematics there, I hope. At least not learning it. Or did they teach you infinite definitions there? Probably they did not get ready yet? > > You need it taught by anyone at all with any more mathematical > competence than you have yourself. Only the ignorant laugh at their own > ignorance. Then try to stop it. Do you know the definition of a binary relation by H&J? 2.1 Definition A set R is a binary relation if all elements of R are ordered pairs, i.e., if for any z Î R there exist x and y such that z = (x, y). 2.3 Definition Let R be a binary relation. (a) The set of all x which are in relation R with some y is called the domain of R and denoted by dom R = {x | there exists y such that xRy}. dom R is the set of all first coordinates of ordered pairs in R. (b) The set of all y such that, for some x, x is in relation R with y is called the range of R, denoted by ran R. So ran R = {y | there exists x such that xRy}. Can you find the words domain and range in this text? Regards, WM
From: Virgil on 9 May 2007 14:05 In article <1178706968.438955.310770(a)u30g2000hsc.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > So it is in vain to exchange any further argument with you. > > Bye. > > WM If that is a promise, Keep it! mathematics does not need your sort of ignorance.
From: Virgil on 9 May 2007 14:10 In article <1178707429.230677.318770(a)n59g2000hsh.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > 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. As the set of paths in any infinite tree is bijectable with the uncountable set of all subsets of N, it is quite true that it is not bijectable with N, but there is no superset of the set of paths wish is bijectable with N. That WM chooses to ignore the bijection between that set of paths and the power set of N will not make the bijection disappear.
From: Virgil on 9 May 2007 14:16
In article <1178707943.860551.269780(a)h2g2000hsg.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > 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. If it is a set, it is defined as sets are defined, which does not require anything as undefined as "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 Define "formula". Until it is defined, it has no mathematical significance, and it hasn't yet been defined. > > > > > 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. Not in mathematics. In mathematics, it is a relation with certain special properties. > > Regards, WM |