Cantor Confusion
in [Math]
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From: Virgil on 8 May 2007 13:02 In article <1178626806.864579.326190(a)h2g2000hsg.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > 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? For every two paths, p and p',there is a first node, at some smallest level n, at which one path branches left and the other branches right. Then p(n+1) =/= p'(n+1). > > 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? It is only in WM's fantasy world where any such behavior is even expected. For any actual tree, there will be for every two distinct paths, a last (highest level) node that they have in common and from which they branch differently.
From: MoeBlee on 8 May 2007 13:05 On May 8, 6:07 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > So 3.1 proves wrong what a function is understood in mathematics? Why > do you think Hrbacek and Jech do so? No, blockhead, they provide a rigorous definition that allows for a formalization of a more general informal notion. That is the kind of action you will find ubiquitous in mathematics. It's time you understood it. MoeBlee
From: Virgil on 8 May 2007 13:12 In article <1178627275.254033.313390(a)u30g2000hsc.googlegroups.com>, WM <mueckenh(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). The only nodes of any path in any tree that are not shared by other paths are leaf nodes (from which no further edges extend), so that WM is claiming that there are leaf nodes in infinite trees. That does not happen in in our infinite trees. > (2) ==> Every path has branched off somewhere. > > So your claim can be summarized: > Not every path has branched off somewhere. False translation. What Hughes says, and means, is that there is no 'last' node in a infinite path by which all other paths have branched off. There are always further nodes and further branchings. > Every path has branched off somewhere. But there is never a last node after which no further branchings can occur, except in finite trees. > > Regards, WM
From: MoeBlee on 8 May 2007 13:15 On May 8, 6:11 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > On 8 Mai, 00:30, MoeBlee <jazzm...(a)hotmail.com> wrote: > > >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 procedure or rule is a frmula. That is arguable. > > > 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. > > 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. And THAT is NOT the DEFINITION the authors give. The DEFINITION is given NEXT, as it is indeed MARKED "DEFINITION". > > And, moving past the informal descriptions and motivation, here is the > > explicit definition: > > which does not contradict the former. The former is not a rigorous mathematical definition. It's not even a matter of contradicting it. By DEFINITION, a function is a certain kind of set of ordered pairs. The set theoretic DEFINITION is NOT that of a formula (nor a procedure), domain, and range. You are in terrible need of understanding the difference between stating a general informal mathematical notion as BACKGROUND for giving a rigorous definition and the rigorous definition itself. MoeBlee
From: Virgil on 8 May 2007 13:48
In article <1178628813.339084.150020(a)q75g2000hsh.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 7 Mai, 22:55, Virgil <vir...(a)comcast.net> wrote: > > In article <1178565586.844949.42...(a)n59g2000hsh.googlegroups.com>, > > > > > > You are the one pushing "finitely definable", whatever that means. > > > > > That means: 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. > > > > In Zf, every set, including N, has a power set, and there is provably no > > surjection from any set in ZF to its power set, but trivially an > > injection, so that the cardinality of any set is less that that of its > > power set. > > > > The paths in a complete infinite binary tree in ZF are easily seen to > > biject with the power set of N, so that set of paths has cardinality > > greater than N. > > > > And nothing that WM can say can change that. > We know your opinion. But it is out of topic. What is remarkable is > the following: > > 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. We know it cannot be done, but it is the essence of WMism that it CAN be done. |