Cantor Confusion
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From: Franziska Neugebauer on 5 Jan 2007 09:37 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: >> mueckenh(a)rz.fh-augsburg.de wrote: >> >> > Franziska Neugebauer schrieb: >> > >> > It is without any value to follow your text. >> >> LOL. Head-in-the-sand? > > Lack of interest in useless sophisms. So you don't want to know where you are wrong? > If you are interested in meaningful discussion, The discussion of your "theories" which reveals your errors in reasoning _is_ a meaningful discussion. F. N. -- xyz
From: Dave Seaman on 5 Jan 2007 09:53 On Fri, 05 Jan 2007 09:42:37 EST, Andy Smith wrote: > But anyway, what integer does your mapping map 1/3 (a real in [0,1]) > to? > It would map to ..0101010 ! That's not an integer. -- Dave Seaman U.S. Court of Appeals to review three issues concerning case of Mumia Abu-Jamal. <http://www.mumia2000.org/>
From: William Hughes on 5 Jan 2007 10:00 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > > > I do not assume that 0.111... exists. I only assume that > > > > given that 0.111...1 exists you can show that 0.111...11 exists. > > > > > > > That position is correct. It is called potential infinity. > > > > > > > > > > That is the same with the lines. Why should the diagonal exist actually > > > > > but the system of lines should not exist actually? (1/9 = 0.111...) > > > > > > > > Indeed. It is the same with lines. It is always possible to find > > > > another line. However, iii says nothing about whether the > > > > diagonal "actally exists" (or equivalently whether the > > > > system of lines "acutally exists"). > > > > > > But you always assert that the the complete diagonal exists, i.e., a > > > diagonal which could not be extended. Then you must also accept a line > > > system which cannot be extendend. > > > > No. At no time do I assume that the complete diagonal exists. > > > > > > > > > > > > > > > iv: Not all the elements of the diagonal exist > > > > > > > > > > Not all natural numbers in unary representation 0.1, 0.11, 0.111, ... > > > > > exist. If all elements of the diagonal exist (which are the last digits > > > > > of the unary numbers) then these unary numbers must exist too, as far > > > > > as I understand existence. > > > > > > > > Any discussion of iv is completely irrelevant. iv is neither needed > > > > nor used. > > > > > > Then drop it. > > > > I have never used it ("neither needed nor used"). > > > > > > > > You used to use ~iv: all the elements of the diagonal exist > > > > > > > Only to point out that I neither need nor use the > > assumption ~iv. > > > > > > The contradiction is: > > > > > > > > A: L_D must have the property that given any set of elements > > > > that exist in L_D one can always find another element of > > > > L_D > > > > [This follows immediately from i ii and iii] > > > > > > It is not different for the lines. > > > > > > > > B: L_D has a largest element. [this follows from the fact the > > > > L_D > > > > is a line] > > > > > > This follows from the assumption (iv) of complete existence of L_D, > > > which implies complete existence of the system of lines. If you drop > > > it, then the contradiction vanishes. > > > > No, it follows from the fact that L_D is a line. A line has a largest > > element. > > L_D has a largest element? Yes. L_D is a line. - William Hughes
From: Andy Smith on 5 Jan 2007 00:37 >>> But anyway, what integer does your mapping map 1/3 (a real in [0,1]) >>> to? >> It would map to ..0101010 ! >That's not an integer It is the natural successor to ..0101001 ! But I agree it is hard to see how one can count up to ..0101010 from 0. As I said in my earlier post, I am suspicious of anything to do with infinite sets. But what is wrong with my systematic technique for counting the reals, and how does it differ substantivelly from mapping the rationals onto the natural numbers? I should say that I am not sufficently barking to think that Cantor is wrong, just trying to get my head straight, and informed advice is welcome ...
From: Dave Seaman on 5 Jan 2007 11:16
On Fri, 05 Jan 2007 10:37:01 EST, Andy Smith wrote: >>>> But anyway, what integer does your mapping map 1/3 (a real in [0,1]) >>>> to? >>> It would map to ..0101010 ! >>That's not an integer > It is the natural successor to ..0101001 ! That's not an integer either. > But I agree it is hard to see how one can count up to ..0101010 from 0. As I said in my earlier post, I am suspicious of anything to do with infinite sets. But what is wrong with my systematic technique for counting the reals, and how does it differ substantivelly from mapping the rationals onto the natural numbers? I thought that was already answered. If your method actually works, then it should map some integer to 1/3. Which one is it? > I should say that I am not sufficently barking to think that Cantor is wrong, just trying to get my head straight, and informed advice is welcome ... My advice is, once an error in your reasoning has been pointed out, don't simply ignore it and continue asking where the error is. -- Dave Seaman U.S. Court of Appeals to review three issues concerning case of Mumia Abu-Jamal. <http://www.mumia2000.org/> |