Cantor Confusion
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From: mueckenh on 17 Oct 2006 06:08 MoeBlee schrieb: > 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. > You haven't yet noticed it? Each digit of the infinitely many digits of the diagonal number has the same weight or importance for the proof. In mathematics, the weight of the digits of reals is 10^(-n). Infinite sequences of digits with equal weight are undefined and devoid of meaning. Regards, WM
From: mueckenh on 17 Oct 2006 06:10 MoeBlee schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > David Marcus schrieb: > > > ZFC does not talk about "vases". If you say that ZFC is inconsistent, > > > please give a statement using the language of ZFC. > > > > All mathematics is built on ZFC and derived from it. From ZFC we can > > derive the result of the vase-balls problem too. Hence we can criticize > > ZFC if its results are contradictory. Mathematics based on ZFC says > > that the vase at noon is empty or not empty (by the way, what was your > > result V(12)?). > > > > But whatever your result may be: Both statemets are in contradiction > > with the foundations of ZFC. If the vase is empty then mathematics of > > limits as derived from ZFC is wrong. If the vase is not empty then the > > assumption is wrong all natural numbers could be put into a bijection > > with each other. > > You have an interpretation of a thought experiment that differs from > the interpretation of other people. That doesn't make set theory > inconsistent. It just makes set theory not suitable for your intuitions > regarding the thought experiment. The inconsistency is that 1) For the balls inserted until noon, you can find the result: It is the set N. 2) For the balls removed until noon, you can find the result: It is the set N. 3) For the balls remaining at noon, the same arguments of continuity which lead to (1) and (2) cannot apply. This is the contradiction. No matter what the result (3) may be. Regards, WM
From: mueckenh on 17 Oct 2006 06:11 MoeBlee schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > David Marcus schrieb: > > > ZFC does not talk about "vases". If you say that ZFC is inconsistent, > > > please give a statement using the language of ZFC. > > > > All mathematics is built on ZFC and derived from it. From ZFC we can > > derive the result of the vase-balls problem too. Hence we can criticize > > ZFC if its results are contradictory. Mathematics based on ZFC says > > that the vase at noon is empty or not empty (by the way, what was your > > result V(12)?). > > > > But whatever your result may be: Both statemets are in contradiction > > with the foundations of ZFC. If the vase is empty then mathematics of > > limits as derived from ZFC is wrong. If the vase is not empty then the > > assumption is wrong all natural numbers could be put into a bijection > > with each other. > > You have an interpretation of a thought experiment that differs from > the interpretation of other people. That doesn't make set theory > inconsistent. It just makes set theory not suitable for your intuitions > regarding the thought experiment. It is a rather silly accusation to speak of "intuition" in connection with the observation that continuously accumulating numbers cannot lead to an empty set. Regards, WM
From: mueckenh on 17 Oct 2006 06:17 Virgil schrieb: > In article <1161001547.844210.170720(a)k70g2000cwa.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Virgil schrieb: > > > > > > Every edge must have its own label if they are to be referred to by > > > label. > > > > Of course, but what I did is only to give an example how the edges and > > their parts are inherited. For my proof I do not need to refer to every > > single edge but only to count the shares. > > Once you start split up edge,s one might equally well split up paths to > have uncountably many parts of each path for each edge. > > In counting one does not split things up but only counts them as wholes. > > > > > > > How about L and R a labels for the left and right branches at the root > > > node, LL and LR for the left and right branches at the left node and RL > > > RR for the rightmost pair at the right node , > > > then LLL, LLR; LRL, LRR; RLL, RLR; RRL, RRR for edges at the next level, > > > and so on ad infinitum. > > > > > > That gives every edge in the entire tree a unique label by which it may > > > be referenced. > > > > Yes, if necessary, one could do so. > > It is necessary to deal with whole edges, not mere fractions of them, so > it is necessary to have a unique identity for each edge and for each > path if one is to compare the set sizes. In advanced mathematics (10 years and elder pupils) they also count halves and quarters and so on. Regards, WM
From: mueckenh on 17 Oct 2006 06:19
Virgil schrieb: > > It is necessary to deal with whole edges, not mere fractions of them, so > it is necessary to have a unique identity for each edge and for each > path if one is to compare the set sizes. Why should I need a unique identity for the units in order to find 18 > 5? But if it bothers you, you may denote the shares of the edges. For that sake we have fractions and letters. Regards, WM |