Cantor Confusion
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From: David Marcus on 23 Oct 2006 22:16 Dik T. Winter wrote: > In article <1161377915.999210.39660(a)m7g2000cwm.googlegroups.com> "MoeBlee" <jazzmobe(a)hotmail.com> writes: > > mueck...(a)rz.fh-augsburg.de wrote: > ... > > Okay, now that I asked for a definition of the relation you mentioned, > > you're not giving that definition, but instead giving a > > combinatorical/numerical argument with more terminology. What is a > > "load of edges"? What is the definition of "a path carries a load of > > edges"? If this is standard terminology in graph theory, then please > > forgive my ignorance and supply me with the standard definition. If it > > is not standard terminology, then please give me your own definition. > > By this time you ought to know that Mueckenheim *never* gives definitions. > Or actually states that he is not able to give a definition for a > particular term. Asking for definitions from Mueckenheim is as useful as > talking to an eel. This does seem to be true. I suspect he doesn't know what the word "definition" means in mathematics. So, he probably sincerely believes he is giving "definitions". -- David Marcus
From: David Marcus on 23 Oct 2006 22:24 MoeBlee wrote: > mueckenh(a)rz.fh-augsburg.de wrote: > > If you claim that what you call modern set theory has a deviating > > definition of infinity, then I am not interested in your theory. > > This is not a matter of defining 'is infinite'. And if you're not > interested in Z set theory, then fine. But then you don't claim that > your argument about trees in Z set theory? I take it that your argument > about trees is in your informal understanding of pre-formal Cantorian > set theory. And I am not interested in your informal understanding of > pre-formal Cantorian set theory as if your informal understanding of > pre-formal Cantorian set theory has anything to do with formal > mathematics. That is rather remarkable. On the one hand, Mueckenheim claims standard set theory is inconsistent, but on the other he makes it clear that what he calls "set theory" is not standard set theory. Do you think he really thinks that standard set theory is inconsistent despite admitting that he hasn't read anything written on the subject since Cantor? -- David Marcus
From: David Marcus on 23 Oct 2006 22:30 georgie wrote: > > David Marcus wrote: > > georgie wrote: > > > Virgil wrote: > > > > In article <1160675643.344464.88130(a)e3g2000cwe.googlegroups.com>, > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > With the diagonal proof you cannot show anything for infinite sets. > > > > > > > > Maybe "Mueckenh" can't but may others can. > > > > > > Only a very very small group of self-proclaimed experts better known > > > as the mathematics community think they can. But they do so > > > with circular arguments as this thread shows. > > > > > > The only explanation to the OP from the math community so far: > > > > > > #2 is not self-referential because #1 says ANY. > > > #1 is correct in saying ANY because #2 holds. > > > > The OP said that a definition was invalid because it was self- > > referential. However, there is no rule against self-reference in modern > > mathematics, so the OP's objection is not valid. > > So the set of sets containing themselves is ok by you. I don't see where I said that. I said there is no rule against self- reference. The rules are specified by the axioms of ZFC. > > Almost a century ago, Russell and Whitehead attempted to develop such > > rules as a way of avoiding the paradoxes, but their approach was too > > cumbersome. So, ZFC avoids the paradoxes in a different way. The > > diagonal argument follows the rules of ZFC. If you want more details on > > what the rules are, there are quite a few good books on the subject. > > This is a bunch of BS. There are no references to ZFC in Cantor's > proof. All the steps of Cantor's argument can be justified within ZFC. People don't normally trace their arguments all the way back to the axioms. The whole point of Mathematics is that we build definitions and theorems and then use those to build more. Which books on set theory have you read? -- David Marcus
From: David Marcus on 23 Oct 2006 22:40 Han de Bruijn wrote: > MoeBlee wrote: > > Han de Bruijn wrote: > >>David Marcus wrote: > >>>Dik T. Winter wrote: > >>> > >>>>I think that, compared to Cantor, in modern set theory potential and actual > >>>>infinity are split up again. The contents of the set N form only a potential > >>>>infinity, on the other hand, the *size* is an actual infinity. > >>> > >>>I've never seen "potential infinity" or "actual infinity" in any > >>>textbook I've used. > >> > >>True. And you haven't seen any binary tree either. > > > > What are you talking about? Graphs, edges, paths, trees, binary trees, > > et. al are all discussed in many textbooks and in many advanced > > textbooks in which set theory and graph theory intersect. > > Yes. That's why I'm surprised that David Marcus hasn't met them before. Where did I say or imply that I haven't seen binary trees? Are you just making stuff up? -- David Marcus
From: David Marcus on 23 Oct 2006 22:48
mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > Therefore, in the limit {2,4,6,...} there are infinitely many finite > > > natural numbers m > |{2,4,6,...}| > > > > Once again you use words and phrases that are not part of standard > > mathematics, i.e., "in the limit {2,4,6,...} there are infinitely many > > finite natural numbers m > |{2,4,6,...}|". What does "in the limit > > {2,4,6,...}" mean? > > > You said (repeatedly) that standard mathematics contains a > > contradiction, so please state the contradiction in standard > > mathematics. If you use new terms, please define them. > > Sorry, I do not know what you state of knowledge is. I already mentioned several books that you could use. However, it appears you haven't actually read any math books published in the last 100 years. > {2,4,6,...} means > obviously "all natural numbers". That is the usual notation in > mathematics. What happened to 1, 3, and 5? They aren't natural numbers? > Binary Tree > > Unfortunately, it was described in a way that I can't understand it. A > > wild guess on my part is that you mean to set up a correspondence > > between edges and sets of paths. > > I am sorry, but if you need a wild guess to understand this text, then > we should better finish discussion. It isn't really a discussion since you are not speaking in English. > Observe just how the discussion > runs with all those who understood it, like Han, William, jpale. > Perhaps you will step by step understand it. I suspect there is nothing to understand. > > In standard mathematics, a finite set of natural numbers has a largest > > element. > > Please prove that a set of elements consisting of 100 bits has a > largest element. First you will have to define what the phrase "elements consisting of 100 bits" means, since I didn't say that. What I said was "a finite set of natural numbers has a largest element". > > So, what are you actually saying? Are you saying that you don't > > like standard mathematics? > > I am saying that standard mathematics is false. As I have shown there > are finite sets in mathematics which have no largest elements. What does "false" mean in this context? > > I've never seen "potential infinity" or "actual infinity" in any > > textbook I've used. > > So you read not the right books or too few. Or, you are making things up again. Feel free to tell us what book uses these terms. > > Bohmian Mechanics is a deterministic theory that avoids the measurement > > problem, satisfies Bell's Inequality (as do all theories of quantum > > mechanics), agrees with all experiments, and doesn't produce negative > > probabilities. So, it seems to be a better theory than the one you > > constructed. > > > And, Bohmian Mechanics is non-relativistic. > > Therefore I did not adopt it. My theory is even a > local-hidden-variables theory, which satisfies Bell's inequlities, a > goal never matched by any other theory. Unfortunately, experiments show that nature violates Bell's inequality. So, your theory may be very nice, but it doesn't agree with experiment. So, your theory is false. -- David Marcus |