Cantor Confusion
in [Math]
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From: WM on 9 May 2007 15:08 On 9 Mai, 20:10, Virgil <vir...(a)comcast.net> wrote: > In article <1178707429.230677.318...(a)n59g2000hsh.googlegroups.com>, > > WM <mueck...(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 subset 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 which 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. That Virgil chooses to ignore the fact that the set of infinite paths is in bijection with the countable set of branching-offs will not make the contradiction in set theory disappear. Regards, WM
From: Virgil on 9 May 2007 15:12 In article <1178724189.164861.276200(a)e65g2000hsc.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > The infinite path {0.000...} is the union of all finite paths with > only 0's, namely {0.} U { 0.0} U {0.00} U ... > accepted? In mathematics, {0.} U {0.0} U {0.00} U ... = {0., 0.0, 0.00, ...} Now if Wm means {0.} U { 0., 0.0} U {0., 0.0, 0.00} U ... where each numeral specifies a node, and each set of nodes a path, ...
From: Virgil on 9 May 2007 15:13 In article <1178724386.739343.150080(a)y80g2000hsf.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > > Time does not play a role in set theory but to branch off is a > temporal act. > > Regards, WM And we know that WM has branched way off from mathematics.
From: WM on 9 May 2007 15:17 On 9 Mai, 20:16, Virgil <vir...(a)comcast.net> wrote: > In article <1178707943.860551.269...(a)h2g2000hsg.googlegroups.com>, > > > 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". How do you define a set, unless you cannot specify the elements belonging to it? > > > > > 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. A formula is a rule or prescription specifying or determining which element of the domain is mapped on which element of the range. The formula in its widest sense can even consist of several expressions. In case every element of the domain has to be mapped by its own expression, the formula can even be a catalogue or set. > > > > > 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. Do you know what a relation is? Do you know that there is also the domain and the range required? Regards, WM
From: WM on 9 May 2007 15:30
On 9 Mai, 20:21, Virgil <vir...(a)comcast.net> wrote: > In article <1178716585.114658.276...(a)h2g2000hsg.googlegroups.com>, > > > > > > WM <mueck...(a)rz.fh-augsburg.de> wrote: > > 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"? > > 'Countability' is an attribute of some sets. A set is countable when > there is a surjective function (in the strictly set theoretic meaning of > function) from N to that set/ We can state: There are sets which are said countable but which do *not* allow for a bijection or one-to-one correspondence with the set N of all natral numbers. > > > > > > > > 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 is finitely > > definable. > > Perhaps not in WM's mathematics. If you cannot finitely define an entity in mathematics, then it does not exist. The form of this definition does not play a role. You must be able to identify the entity by means of a finite number of words. > > > 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. > > Where does that definition of 'definition' occur? Certainly not in any > English language nor mathematical dictionary. Do you so gossly underestimate English textbooks that you think they state: "A definition must not have an end, i.e., there must not be a last word." Or do you think, "definition" should not be defined at all? Or do you disregard tertium non datur? Or do you refrain from any thinking except that set theory is correct? Your postings suggest the last. Do you probably translate de-finition as non-finishing? But to answer your question, it is me who says: A definition must be finite. And yes, students in Germany have to learn that. Regards, WM |