Cantor Confusion
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From: MoeBlee on 16 Oct 2006 13:22 mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > Virgil wrote: > > > Where in ZFC or NBG does "Mueckenh"find any definition of any such limit? > > > > Or, in what book does mueckenh find this? > > The book I > recommend is the collected works of Cantor. Just to be clear, you do understand that the works of Cantor do not govern ZFC or NBG, right? MoeBlee
From: MoeBlee on 16 Oct 2006 13:35 mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > > Han de Bruijn wrote: > > > Set Theory is simply not very useful. The main problem being that finite > > > sets in your axiom system are STATIC. They can not grow. > > > > Set theory provides for capturing the notion of mathematical growth. > > Sets don't grow, but growth is expressible in set theory. If there is a > > mathematical notion that set theory cannot express, then please say > > what it is. > > Obviously the notion of "rational relation" as used in the binary tree > cannot be expressed by mathematical notion: > Consider the binary tree which has (no finite paths but only) infinite > paths representing the real numbers between 0 and 1. The edges (like a, > b, and c below) connect the nodes, i.e., the binary digits. The set of > edges is countable, because we can enumerate them > > 0. > /a \ > 0 1 > /b \c / \ > 0 1 0 1 > ............. > > Now we set up a relation between paths and edges. Relate edge a to all > paths which begin with 0.0. Relate edge b to all paths which begin with > 0.00 and relate edge c to all paths which begin with 0.01. Half of edge > a is inherited by all paths which begin with 0.00, the other half of > edge a is inherited by all paths which begin with 0.01. Continuing in > this manner in infinity, we see that every single infinite path is > related to 1 + 1/2 + 1/ 4 + ... = 2 edges, which are not related to any > other path. The set of paths is uncountable, but as we have seen, it > contains less elements than the set of edges. Cantor's diagonal > argument does not apply in this case, because the tree contains all > representations of real numbers of [0, 1], some of them even twice, > like 1.000... and 0.111... . Therefore we have a contradiction: > > Card(R) >> Card(N) > || || > Card(paths) =< Card(edges) What I see above is a lot of mathematical terms that are used in set theory with precise definitions, but for which I do not know your own personal definitions. The contradiction you claim is not shown by you to be a contradiction in set theory. You seem to think there is some mathematics that leads to some contradiction (?) but that set theory cannot expess this (?). Why don't you just state your axioms for a mathematical theory? You may not agree with the axioms of set theory, but at least by giving you axioms, we tell you right up front precisely what mathematical sentences are assumed in our mathematical arguments. But if you don't give us axioms, then we can never tell what mathematical assumptions are behind your arguments. MoeBlee
From: MoeBlee on 16 Oct 2006 13:41 mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > > If we're still talking about the diagonal argument for the > > uncountability of the reals, then there's no "self-reference" anyway. > > > > The proof is good from an effectively decidable set of axioms using > > effectively decidable rules of inference. So if one claims that there > > is anything objectionable in the proof, then one should just say which > > axioms and/or rules of inference one rejects. Any other dispute with > > the mechanics or details of the proof is mindlessness. > > The rules to cover up with infinity are objectionable. > In decimal representations of irrational numbers the infinite string of > digits leads to an undefined result unless the factors 10^(-n) are > applied. In Cantor's diagonal proof each of the elements of the > infinite string is required with equal weight. There is no "equal weight" in the proof. Again, you need to say exactly which axioms or rules of inference you reject. Bringing in your own notions of "equal weight" or any other notion not in the axioms or rules of inference is irrelevent. Just say which axioms and rules of inference you reject, and then we'll at least have a clear objection from you. MoeBlee
From: MoeBlee on 16 Oct 2006 13:52 mueckenh(a)rz.fh-augsburg.de wrote: > Yes. The finite parts are useful, the transfinite part is useless. Except that you offer not a shadow of a hint of a clue as to axioms for calculus that don't provide for infinite sets. MoeBlee
From: MoeBlee on 16 Oct 2006 14:03
Tony Orlow wrote: > You fellers sure are dense. Han's simply saying that the universe is > consistent, or it doesn't exist at all. Clearly, it exists, so obviously > it's not self-contradictory. Where there is a conflict between notions, > where there is paradox, there is also explanation. One has to choose the > most consistent path, while integrating the widest array of ideas possible. You are welcome to define 'consistent' so that it applies to such things as the universe. Meanwhile, usually in mathematics, it is sets of sentences that are consistent or not. MoeBlee |