Cantor Confusion
in [Math]
|
Prev: Pi berechnen: Ramanujan oder BBP
Next: Group Theory
From: georgie on 13 Oct 2006 08:51 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?
From: stephen on 13 Oct 2006 08:58 Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > David Marcus wrote: >> Han de Bruijn wrote: >> >>>Dik T. Winter wrote: >>> >>>>In article <1160647755.398538.36170(a)b28g2000cwb.googlegroups.com> >>>>mueckenh(a)rz.fh-augsburg.de writes: >>>>... >>>> > It is not >>>> > contradictory to say that in a finite set of numbers there need not be >>>> > a largest. >>>> >>>>It contradicts the definition of "finite set". But I know that you are >>>>not interested in definitions. >>> >>>Set Theory is simply not very useful. The main problem being that finite >>>sets in your axiom system are STATIC. They can not grow. Which is quite >>>contrary to common sense. (I wouldn't imagine the situation that a table >>>in a database would have to be redefined, every time when a new row has >>>to be inserted, updated or deleted ...) >> >> 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. > Han de Bruijn What does "physically inconsistent" mean? Can you give an example where set theory is "physically inconsistent"? The balls and the vase problem is not such an example, as it is not physically realizable. Yes, set theory can model unphysical things, but so can any mathematics. For example, suppose you have an acceleration of 10m/s^2. To determine your velocity after n seconds you calculate / n | 10 dt / 0 Of course this is wrong if n is 30000. Does that make calculus physically inconsistent? Or is it just the case that calculus can be used to describe unphysical situations? Stephen
From: Dik T. Winter on 13 Oct 2006 09:06 In article <b2f47$452f51eb$82a1e228$2726(a)news2.tudelft.nl> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: .... > b(t) diverges at noon. Thus b(noon) is undefined. Thus t is not time. > > It's impossible in a renormalized mathematics that limits are different > from the actual values of functions at that place. "Diverges at noon" is strange wording in my opinion. I would say b(t) diverges when going to noon. And so the limit does not exist. But this is not in contradiction with b(noon) = 0. However, let's have a look at the entier function. lim(x from 0 -> 1) entier(x) = 0, but entier(1) = 1. It seems that the limit is different from the actual value of the function. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: William Hughes on 13 Oct 2006 09:11 mueckenh(a)rz.fh-augsburg.de wrote: > I did not > claim that one letter depends on the other. [...] > And that is obvious. Take just M = N. - William Hughes
From: Dik T. Winter on 13 Oct 2006 09:12
In article <a1364$452f53c9$82a1e228$2828(a)news2.tudelft.nl> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: .... > I don't know about Mueckenheim's opinion in these matters, but an old > Greek philosopher - Heraclitos - has said: "panta rei kai ouden menei" > (everything flows and nothing remains the same). And I agree with that. Pray warn me when 2 has changed sufficiently to be the square of a rational. I would not like to miss that moment. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |