Cantor Confusion
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From: mueckenh on 18 Dec 2006 08:52 Virgil schrieb: > In article <1166359875.644197.31300(a)n67g2000cwd.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Bob Kolker schrieb: > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > > > > > > > > > Do you believe that today's standards are sound by tomorrow's > > > > standards? > > > > > > Wait a day and we will find out. Hilbert's axiomatization of Euclidean > > > Geometry done back in 1899 is as good today as it ever was. Aristotelean > > > logic and semantics have held up very well over 2400 years. Too bad that > > > Aristotelean physics fell way short. > > > > > > The difference between today's math and tomorrows math will be mostly in > > > scope, not validity. The arithmetic argument that Euclid used to prove > > > the infinitude of primes > > > > He proved that there are more primes than given. > > In fact, more than any finite number. > > > > > is as valid today as it was 2200 years ago. > > > > The difference between today's math and tomorrows math will be that the > > physical and physiological foundations will be explored and taken into > > account. > > WM would, if he had the power, allow only some of the words and none of > the music. I would help to recognize all words which can be spelled out and to distiguish them from such which never can be said. It would purify present mathematics from superstition. Regards, WM
From: mueckenh on 18 Dec 2006 08:59 Virgil schrieb: > > > > > > Each line differs by 1 element from the preceding line. If the number > > > > > > of lines is actually infinite then the number of differences must be > > > > > > actually infinite too. > > > > > > > > > > So far so good. > > > > > > > > > Their sum is an infinite number. If all elements are there, then also > > > > all sums are there. > > > All of the /finite/ sums of an infinite series are always there , but > the infinite sums are only "there" when the series converges. > That does not prevent an infinite series from having infinitely many > finite terms and infinitely many finite partial sums, even when it does > not converge. So, where is the sum 1+1+1+...+1 for infinitely many ones? Where does the equation 1+1+1+...+1 = n cease to hold? There can be infinitely many finite right sides but not infinitely many finite left sides? Miraculous mathematics. > > > > If not all sums are there, not all elements are > > > > there. > > > > > > There are infinite sequences that do not converge. > > > > > > And 1 + 2 + 3 + ... is one of them. > > > > And 1 + 1 + 1 + ... also is one of them and, in addition, it is an > > element of the infinite set. Therefore, in an actually infinte set N, > > there must be an actually infinite number. > > Then 1 + 1/2 + 1/4 + 1/8 + ... must be equally infinite. That "conclusion" is neither logic nor mathematics. Regards, WM
From: mueckenh on 18 Dec 2006 09:13 Virgil schrieb: > In article <1166361248.951652.84540(a)l12g2000cwl.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Virgil schrieb: > > > > > In article <1166185295.562736.84050(a)j72g2000cwa.googlegroups.com>, > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > Virgil schrieb: > > > > > > > > > > > > > > > > > > > > No I simply state that the diagonal proof fails in case of the tree. > > > > > > > > > > AS the diagonal proof is not allied to trees, so what? > > > > > > > > > Its contradiction is. > > > > > > Show me! > > > > If you believe in a diagonal longer than any line of the matrix and in > > paths which have no beginning, then I can't show you anything. > > You have things backwards, as usual. The reason that you cannot show me > things is that you have not presented any axiom system on which to base > them. When you want to prove something, you merely pull assumptions out > of the air to serve as justifications. > > You cannot show me things within in ZFC or NBG that require you to > assume things that contradict the XFC or NBG axioms because in an axiom > system you are not allowed to make assumptions that contradict the > axioms already assumed. If you are forced to believe in a diagonal longer than any line of the matrix and in paths of the tree which have no beginning, then I can't show you anything. Anyhow, it is useless to discuss this topic further. I only wanted to point out the fact that you are forced to believe these things. That shows all I wanted to show. Regards, WM
From: mueckenh on 18 Dec 2006 09:18 Newberry schrieb: > Sorry that I joined a bit late. Doesn't matter. >Are you saying that (in an infinite > binary tree) the set of paths is uncountable but the set of edges is > countable? The set of edges is obviously countable by, e.g., 1, 2, 3,4,5,6, 7,... As no path can separate from another one without the existence of two more edges, the number of edges is an upper bound for the number of paths. Regards, WM
From: mueckenh on 18 Dec 2006 09:27
Franziska Neugebauer schrieb: > >> > > > All elements that can be shown to exist in the diagonal can be > >> > > > shown to exist in one single line. [(P1)] > > This proposition P1 has _not_ yet been proved (shown). This proposition is the definition of the diagonal. > > You misinterpret L_D. L_D is that line which contains all numbers > > contained in the diagonal. > > The Diagonal is unbounded thus any _assumed_ L_D is not bounded, too. Correct is: If the diagonal is unbounded then any _assumed_ L_D is not bounded, too. >From the finiteness of any L_D we obtain: The diagonal is not unbounded. > Hence L_D cannot be a line of the list (meaning: cannot be _in_ the > list) for any line in the list is bounded (proof by induction). > > Hence contradiction to P1. P1 must be dropped. > > > If your L_D does not contain them, then you have the wrong L_D. > > Upto now we have no line L_D at all since P1 has not been proved. > There is nothing to prove in a definition. I remember you said a definition cannot be wrong. So it cannot be proved. Regards, WM |