Cantor Confusion
in [Math]
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From: Virgil on 9 May 2007 21:43 In article <1178737478.076726.140210(a)n59g2000hsh.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > I did not say that I could name a path which does never branch off > from p. But there must be such a path. Why not call it p''. Why must there be any different path that is not also a separate path? > > > > <snip> > > > > >The proof in the tree is, that for > > > > every path p, and level M, there is another path *existing in the > > tree* > > which is not different from p at a level less than or equal > > to M. > > > Yes. But why only see the one side of the medal? For every path p > there is another path *existing in the tree* which is not different > from p at any node, because p is not single at any node. False!!! Given any path p and any path p', if there is no node in one but not the other, then p = p'. Even more: if there not infinitely many nodes in one but not in the other then p = p'.
From: Virgil on 9 May 2007 21:46 In article <1178737721.434981.282490(a)u30g2000hsc.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > 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. AS WM's dreams are not true in any set theory yet presented here, such as ZF or NBG, his allegations of contradiction do not hold in such as ZF or NBG either.
From: Dik T. Winter on 9 May 2007 21:40 In article <1178716585.114658.276830(a)h2g2000hsg.googlegroups.com> WM <mueckenh(a)rz.fh-augsburg.de> writes: .... > > > > There are subsets of a finitely definable set which are not > > > > finitely definitable. .... > 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. Try to find a finite definition of the set of Ulam's lucky numbers. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Virgil on 9 May 2007 21:59 In article <1178738274.909825.208790(a)e65g2000hsc.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > 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? I am reasonably certain that that question does not mean in standard English, what you intended to ask. The axioms of ZF tell how to specify sets in ZF. > > > > > > > > > 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. So a set van be defined by being a set! How marvelously circular! > > > > > > > > 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? There are a domain and range to a relation, but they are consequences of its definition, not antecedents to it.
From: Dik T. Winter on 9 May 2007 21:59
In article <1178721265.006218.171950(a)e51g2000hsg.googlegroups.com> WM <mueckenh(a)rz.fh-augsburg.de> writes: > On 9 Mai, 04:20, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > In article <1178463181.745792.46...(a)y80g2000hsf.googlegroups.com> WM <mueck...(a)rz.fh-augsburg.de> writes: > > > On 6 Mai, 04:40, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: .... > > > > > Theorem 2: The set of finitely defined numbers cannot be put into a > > > > > list. > > > > > > > > > > Proof: Assume such a list exists. This means, such a list has been > > > > > finitely defined. > > > > > > > > Wrong. Existence does *not* mean finitely definable. > > > > > > Marthematical existence does mean definable. What else should it mean? > > > Definitions can only be finite. > > > > Right you are. > > Nice to hear. Here are some mathematicians who will even accept > infinite definitions. There is a slight difference. Finite definitions do not imply finitely definable. Consider the computable numbers as they are presented by halting Turing machines. The definition is indeed finite. Nevertheless they are considered not to be finitely defined because it is impossible to find out whether a particular Turing machine indeed *does* halt. > > But that a set of entities is finitely definable does *not* > > mean that each element of the set is finitely definable. Consider Cauchy > > sequences. By (a finite) definition a Cauchy sequence denotes a real > > number. This is a *finite* definition of the set of real numbers. > > Yes, of the set. But it is only a definition of those real numbers for > which Cauchy sequences can be finitely defined. No. It is a definition of the set. It is not a definition of the real numbers at all. > > This does *not* > > mean that each real number is finitely definable. > > And it does not mean that every Cauchy sequence has a finite > definition. Yes, no problem with that. > I gave a bijection between the set of nodes and the set of branching- > offs of paths bunches. You did not. What node is bijected with 0.1010101010...? > As there can be not more results of branching- > offs than branching-offs, the task is done. As in each node the number of path bunches decreases I wonder about this statement. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |