Cantor Confusion
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From: David Marcus on 13 Oct 2006 21:07 Han de Bruijn wrote: > David Marcus wrote: > > Is your claim only that set theory is not useful or is contrary to > > common sense? Or, are you claiming something more, e.g., that set theory > > is mathematically inconsistent? > > I said that set theory is not *very* useful. I have developed (limited) > set theoretic applications myself, so I don't say it is useless. > Yes, a great deal of set theory is contrary to common sense. Especially > the infinitary part of it (: say cardinals, ordinals, aleph_0). > > I'm not interested in the question whether set theory is mathematically > inconsistent. What bothers me is whether it is _physically_ inconsistent > and I think - worse: I know - that it is. What does "physically inconsistent" mean? Wouldn't your comments be better posted to sci.physics? Most people in sci.math are (or at least think they are) discussing mathematics. -- David Marcus
From: David Marcus on 13 Oct 2006 21:17 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. 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. -- David Marcus
From: David Marcus on 13 Oct 2006 21:18 georgie wrote: > > Tony Orlow wrote: > > > So, they can represent a third kind of number, which is not finite, nor > > infinite? Pray tell, what sort be's such a number? > > Cantor proved there are transfinite numbers. Why > can't there be subinfintie ones? There can be if you define them and construct them. -- David Marcus
From: MoeBlee on 13 Oct 2006 22:34 David Marcus wrote: > 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. 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. MoeBlee
From: mueckenh on 14 Oct 2006 04:07
David Marcus schrieb: > Dik T. Winter wrote: > > In article <1160649760.068206.172400(a)b28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > To inform the set theorist about the possible existence of sets with > > > finite cardinality but without a largest number. > > > > Interesting but in contradiction with the definition of the concept of > > "finite set". So you are talking about something else than "finite > > sets". > > It would seem he is. I don't understand why people use words in non- > standard ways without explaining what they mean. They are guaranteeing > that no one will understand them. A finite set is a set with a number of elements, which is smaller than some natural number. As far as I know this notion is covered by the standard meaning of words. I talked about a set with less than 100 elements. Therefore I used the word "finite set". Regards, WM |