Cantor Confusion
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From: Virgil on 17 Oct 2006 15:36 In article <1161079802.120515.175530(a)i3g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > > 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. No argument of "continuity" has been applied in (1) or (2), so there are no "same arguments" to fail. Given the timings of (1) and of (2), the only conclusion for (3) is N\N = {}. > > This is the contradiction. The "contradiction" appears to be that "Mueckenh" assumes things not in evidence and then finds contradictions between his extraneous assumptions and the actual problem as stated. No matter what the result (3) may be. > > Regards, WM
From: Virgil on 17 Oct 2006 15:40 In article <1161079883.263059.285400(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > > 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. "Mueckenh" again manages to ignore the equally important and concomitant decumulation of those balls.
From: Virgil on 17 Oct 2006 15:46 In article <1161080228.311432.261510(a)b28g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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. Do you encourage your students to divide up wholes into halves and quarters and so on when asked to compare the number of whole items in two collections? The problem at hand it to compare the "number" of whole edges to the "number" of whole paths. If we are to divide either edges or paths into parts we must, in fairness, divide both into parts in the same way.
From: Virgil on 17 Oct 2006 15:51 In article <1161080347.779369.135780(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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? Because 18 cm is not greater than 5 m. Unless the 18 and the 5 represent the same units, the inequality need not be as trivial as "Mueckenh" claims. > But if it bothers you, you may denote the shares of the edges. For that > sake we have fractions and letters. For any valid comparison of the "number" of edges with the "number" of paths, we need only whole edges and only whole paths.
From: Virgil on 17 Oct 2006 15:59
In article <78269$4534cb2a$82a1e228$21528(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > stephen(a)nomail.com wrote: > > > You have not answered the question about how one determines > > if a thing in the real world is true. I can guess you > > will say something about measurements, but how does one > > know that your measurements are "true", or that they truly > > correspond to "a thing in the real world", and so on. > > It is a big ugly kettle of philosophical fish. > > Not only philosophical fish. Also religious fish. And political fish. > And scientific fish. Actually everyday's life fish. You are right! > > > I agree that it is sensible to assume that the Universe > > is consistent, but given how strange and unintuitive > > the Universe can be, who knows. > > I think we agree on the above. But it doesn't mean that we cannot answer > _part_ of the question: do INFINITIES exist or not. Are they true or are > they false? And IMO _that_ can be decided _now_, without rocket science. Do physical infinities exist? Probably not. Do physical triangles exist? Almost certainly not. Therefore, according to HdB's thesis, we should abolish trigonometry. On the other hand, do mental or imagined triangles exist and be manipulated? Absolutely! Can mental or imagined infinities exist and be manipulated? Absolutely! Does HdB's mental dyspepsia about this state of affairs bother anyone else? No! |