Cantor Confusion
in [Math]
|
Prev: Pi berechnen: Ramanujan oder BBP
Next: Group Theory
From: Han.deBruijn on 16 Oct 2006 16:06 mueckenh(a)rz.fh-augsburg.de schreef: > Dik T. Winter schrieb: > > > In article <1160933229.072292.316580(a)e3g2000cwe.googlegroups.com> Han.deBruijn(a)DTO.TUDelft.NL writes: > > > Dik T. Winter schreef: > > > > In article <1160856895.115824.134080(a)b28g2000cwb.googlegroups.com> > > > > Han.deBruijn(a)DTO.TUDelft.NL writes: > > ... > > > > > Come on, guys! You all know that, in the world of approximations, > > > > > 2 _is_ the square of a rational and the circle _is_ squared. > > > > > > > > I thought you were talking mathematics? > > > > > > I thought approximations were a part of mathematics? > > > > They are. Numerical mathematics in particular. But also in them, 2 is > > *not* the square of a rational number. The best you can state is that > > there is a rational number whose square approximates 2 with a certain > > precision. > > Correct. Doing better is impossible. I cannot improve on your response. Han de Bruijn
From: mueckenh on 16 Oct 2006 16:06 Alan Morgan schrieb: > >> As I have inductively gone through the entire list of balls introduced > >> into the vase and found that each of them has been removed before noon, > >> why should stating that trivial fact be considered a joke? > > > >But you cannot go inductively through the cardinal numbers of the sets > >of balls in the vase? They are 9, 18, 27, ..., and, above all, we can > >show inductively, that this function can never decrease. > > You think that's bad? I have an even simpler situation! Add one ball > at 1 minute to noon, another ball at half a minute to noon, another > at 1/4 minute to noon, and so on. The number of balls in the vase before > noon is always finite, but somehow, miraculously, at noon the number of > balls in the vase becomes infinite. When, oh when, does that transition > from finite to infinite happen? > > I submit that this is just as wierd a result as the original problem. Weird is that adding 9 balls instead of 1 per transaction leads to zero balls. Weird is that taking off 1 ball per transaction leads to all balls taken off and no ball remaining, if the enumeration is 1,2,3,... but to infinitely many balls remaining, if the enumeration is 10, 20, 30, .. . This in particular is weird because there is a simple bijection between 1,2,3,... and 10, 20, 30, ... Regards, WM
From: William Hughes on 16 Oct 2006 16:08 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > > > > But the end time of the problem (noon) does not correspond to > > > > an integer (neither in standard mathematics, nor in your > > > > system, whether or not you interpret the problem as dealing > > > > with infinite integers as well as finite integers). So the function > > > > 9n does not have a value at noon. There is no way > > > > it can be continuous at noon. And since there is no > > > > value of n that corresponds to noon, 9n cannot be used > > > > to determine the number of balls in the vase at noon. > > > > > > But the function n can be used to determine the number of balls removed > > > from the vase at noon? > > > > > > > Nope. [There are no balls removed from the vase at noon] > > Arbitrary misunderstanding? > > > The function 9n has nothing to do with the number of > > balls in the vase at noon. > > But the function n can be used to determine the number of balls having > been removed > from the vase at noon? No. There are no balls removed from the vase at noon. Note, that there is no time "just before noon". At any time before noon there remain an infinite number of steps. So no value of n is close to the end. The balls are removed during an infinite number of steps. - William Hughes.
From: Han.deBruijn on 16 Oct 2006 16:09 David Marcus schreef: > stephen(a)nomail.com wrote: > > Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > > > stephen(a)nomail.com wrote: > > > > >> Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > > >> > > >>>Dik T. Winter wrote: > > >> > > >>>>In article <1160857746.680029.319340(a)m7g2000cwm.googlegroups.com> > > >>>>Han.deBruijn(a)DTO.TUDelft.NL writes: > > >>>> > Virgil schreef: > > >>>>... > > >>>> > > I do not object to the constraints of the mathematics of physics when > > >>>> > > doing physics, but why should I be so constrained when not doing physics? > > >>>> > > > >>>> > Because (empirical) physics is an absolute guarantee for consistency? > > >>>> > > >>>>Can you prove that? > > >> > > >>>Is it possible to live in a (physical) world that is inconsistent? > > >> > > >> Perhaps. How could we know? > > > > > How can we know, heh? Can things in the real world be true AND false > > > (: definition of inconsistency) at the same time? > > > > > Han de Bruijn > > > > What does it mean for a thing in the real world to be true? > > How do you know if a thing in the real world is true? > > > > Consider the twin slit experiment. Is the fact that none of > > the following accurately describe the situation an inconsistency? > > a) the photon goes through one slit > > b) the photon goes through both slits > > c) the photon goes through neither slit > > In Bohmian Mechanics (and similar theories), the photon goes through > only one slit. Physicists could learn something about logical thinking > from mathematicians. Sure, theories. Can't you talk about something else but "theories"? Han de Bruijn
From: David Marcus on 16 Oct 2006 16:10
mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > By the way: Every means to draw conclusions and to calculate results is > > > mathematics. There is no need to prefer a certain language (unless > > > there is someone who cannot speak another one). Would you assert > > > Archimedes did not do mathematics, because he used only the Greek > > > language and had not yet special symbols but Greek letters to denote > > > numbers? > > > > The language of Mathematics has evolved over time. > > And it is going on to evolve, like mathematics itself. > > > > If you assert that the ball and vase problem shows that modern > > Mathematics contains a contradiction, then please state the problem > > using the language of modern Mathematics. "Balls" and "vases" are not > > part of Mathematics, although people may use such language to talk > > informally about Mathematics. > > I did mention already that "balls" is an abbreviation for "natural > numbers" but that I prefer to use balls in order not to intermingle > these numbers with the natural numbers used for the transactions t. > > > So, please state the ball and vase problem > > using the language of Mathematics (e.g., "sets", "functions", > > "integers", "reals") so that we can see what mathematical problem you > > are talking about. > > > > What you wrote above (i.e., X(t), Y(t), Z(t)) uses the language of > > Mathematics, but (as I pointed out above) is not the statement of a > > problem since it doesn't end with a question. > > It is not the statement of a problem but the proof of a contradiction. > Therefore I need no question mark. > > The result, written in mathematical language, is: lim {t-->oo} 9t = 0. It is really very difficult to follow you if you jump from topic to topic. First you state that the balls and vase problem shows a contradiction. Now you state that knowing the balls and vase problem is not relevant. Are you now saying that you have a proof of lim_{t->infty} 9t = 0 in standard mathematics? If so, please give the proof (and just the proof). -- David Marcus |