Cantor Confusion
in [Math]
|
Prev: Pi berechnen: Ramanujan oder BBP
Next: Group Theory
From: Dik T. Winter on 1 Nov 2006 21:29 In article <1162405329.073198.286680(a)h48g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: .... > > > Irrational numbers have no last digit. Therefore, with a sequence of > > > digits like the diagonal number is, one can never have a completed > > > number but only come as close as possible to any number --- or avoid to > > > do so. > > The problem is that: > For the diagonal number of Cantor's list it is not sufficient to come > arbitrarily close to a number which is different from any list number > --- or avoid to do so. . You lost me here. Numbers do not come arbitrarily close to each other. Numbers are fixed entities. Sequences can come arbitrarily close to each other. > > > Not so. Of course we talk about a fixed base like 10. > > > > Ah. In that case one or two, depending on the number involved. But > > the diagonal obviously depends on the actual representative chosen. > > So we have no arbitrary choice but , in case of irrational numbers, > exactly one representation. And this representation is *the limit* of > all the sequences of the due equivalence class. You are pretty wrong here. There are a lot of rational numbers for which there is only one representative in decimals. And, representatives are *not* limits. When considering the equivalence classes, most sequences have one limit: the equivalence class it is sitting in. I think that you are still thinking that *some* represenation defines a real number; but that is not the case. That is especially not the case when you consider only representations to some integral base. There are other methods to define numbers. You do not like to call them numbers, but ideas. But you can not prevent me to call something like sqrt(2) a (real) number. And in common mathematics that is just what it is. By definition, every sequence (use any definition, I know Cantor, Dedekind, Baudet and Weierstrass, they all lead to the same): {sum{k = 1...n} a_k/10^k} is a representative of a "number". It is just a sequence of rationals. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 1 Nov 2006 21:40 In article <1162405520.008395.100850(a)e64g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: .... > > Finally you give a definition. Why did it take so long? > > I thought that this was so clear that no explanation was required. I asked you for a definition quite a few times. I would have thought that that was enough indication that it was not clear at all. > > > > And in mathematics 1/oo is *not* defined. > > > > > > Not in mathematics. But in a theory which assumes omega to be a whole > > > number. > > > > Which theory? > > Set theory. > Cantor invented omega and defined omega as a whole number. > Who changed this standard meaning? > Why do you think this meaning was changed? > When do you think the contrary meaning became standard? > What is the contrary meaning? > Do you agree that A n: n < omega is incorrect? > If not, why do you complain about on-standard meaning on Cantor's > definition? A nice rant. Where did Cantor define 1/oo? A quote might be appropriate. > > > > > > Die Anzahl einer unendlichen Menge [ist] eine durch das Gesetz der > > > Z=E4hlung mitbestimmte unendliche ganze Zahl. (G. Cantor, Collected > > > Works p. 174) > > > ... kann also omega sowohl als eine gerade, wie als eine ungerade Zahl > > > aufgefa=DFt werden. (G. Cantor, Collected Works p. 178) > > > > Where in the above quote is 1/oo defined? > > It is defined that omega (which Cantor used later instead of oo) is a > number larger than any natural number n. Omega is the limit ordinal > number. Therefore 1/omega must be a number smaller than every fraction > 1/n. Why? As long as 1/omega is not defined you can not talk about it. You simply assume that 1/omega is a number. But that is not the case, it is not defined. > > > Therefore we have there limit ordinal numbers? > > > > Yes. So what? Limits are in general not defined. > > In particular the limit of all segments of N is not defined. But set > theory does it. No. And if so, show me where. > > Nonsense. Suppose I define an ordering relation on sets. How can I apply > > that knowledge to numbers? Defining something in one context does not > > make it immediately applicable in another context. > > Everything is a set (in ZF). Numbers are sets too. In the Von Neumann model of ZF, I would think. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: David Marcus on 2 Nov 2006 00:28 Dik T. Winter wrote: > In article <sjnfk256m5tau0bmk5u4vjdg53al1f2sc9(a)4ax.com> Lester Zick <dontbother(a)nowhere.net> writes: > > Yours are some of the very > > few that don't. And I've seen posts of other .nl correspondents which > > come through in color. > > It has nothing to do with the domain where you come from. It has everything > to do with the manner things are quoted and in the way *your* newsreader is > able to handle that. I know the reason your newsreader does not show colours. > It is because I prepend the '>' sign with a space when quoting. And I > have pretty good reasons to do that. (One of the reasons being that this > article would be rejected because there is more quotation than new text.) My newsreader lets me post articles that have more quotation than new text. -- David Marcus
From: Han de Bruijn on 2 Nov 2006 03:00 Virgil wrote: > In article <dec9e$454858d1$82a1e228$21143(a)news1.tudelft.nl>, > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: >>A large mass (as with the balls in a vase close to noon) surely _will_ >>halt time, according to the General Theory of Relativity. > > As those infinitely many balls are required to exist before any are put > in the vase, if they are going to stop time at all, they will stop it > before any of them are to be put into the vase. So the need for a > discontinuity at noon will be no problem. Ah, that's true! So time has stopped well before the experiment has even started. So the balls in a vase problem is ill-posed from the beginning, as I've always said. Han de Bruijn
From: Han de Bruijn on 2 Nov 2006 03:04
Virgil wrote: > In article <57d64$45485a34$82a1e228$21322(a)news1.tudelft.nl>, > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > >>Virgil wrote: >> >>>In article <bdc92$45476e9e$82a1e228$30478(a)news1.tudelft.nl>, >>> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: >>> >>>>It's easy to come up with a correct physical problem and >>>>solve it with the wrong mathematics. As you did. >>> >>>HdB has come up with a mathematical problem which he claims can end the >>>world: >>> >>>HdB claimed that a discontinuity in a mathematical function of time >>>causes time to stop. And he claimed this followed from physics. >>> >>>Thus, if HdB is right, the vase problem will cause the end of the >>>world. >> >>Yes. If it came into _existence_, it would cause the end of the world. >>Because an infinite mass would be no less than a Cosmic Disaster. > > Since it _exists_ only as a gedankenexperiment, a sort of mental game, > such fears are irrelevant. Other fears are more relevant. Such as the fear that infinities in mathematics may easily lead to irrelevancies in physics (like e.g. with Black Holes or String Theory) Han de Bruijn |