An uncountable countable set
in [Math]
|
Prev: integral problem
Next: Prime numbers
From: Mike Kelly on 5 Oct 2006 04:43 Han de Bruijn wrote: > Randy Poe wrote: > > > Math has to be logical. It doesn't have to be physically > > realizable. > > Wrong. Math has to be an idealization which can be materialized again > into something that is physically realizable. Why? >Quote: Physicists also > > > realize that things can exist in mathematics that aren't even > > approximations of a physical realizable. That aren't physically > > sensible in other words. > > That's only true for non-disciplinary mathematics. What is non-disciplinary mathematics? Is it not mathematics? -- mike.
From: Mike Kelly on 5 Oct 2006 04:49 Han de Bruijn wrote: > Virgil wrote: > > > In article <b8b9$45236b93$82a1e228$13353(a)news2.tudelft.nl>, > > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > > >>Virgil wrote: > >> > >>>That the "series" diverges means that there is no such thing as a limit, > >>>so that method does not say anything about the result. > >> > >>That method says there is no result at noon. > > > > Not so. That method merely says that IT cannot tell what happens at noon. > > Don't know who IT is. > > > It definitely does NOT say that there is no other method that can tell > > what happens at noon. > > There is no other method. Rubbish. You mean to say there is no other method that you approve of. That isn't the same thing at all; you are certainly not the arbiter of what is and is not mathematical. Pinpoint your beef with the following "method" : Problem : at one minute to noon, balls 1 thru 10 are added to the vase and ball 1 is removed. At half a minute to noon balls 11 thru 20 are added and ball 2 is removed. etc. Let noon = 0 and "one minute to noon" = -1. Let A(n,t) be 1 if the ball n is in the vase at time t, 0 if it is not in the vase at time t. Let B(n) be the time that the nth ball is added to the vase and C(n) be the time that it is removed. B(n) = -1/(2^(floor((n-1)/10))) C(n) = -1/(2^(n-1)) Note that B(n) and C(n) are strictly less than 0. Now A(n,t) = { 1 if B(n) <= t < C(n) 0 otherwise } Note that A(n,0) = 0. Let S(t) be the number of balls in the vase at time t. Then S(t) = { sum(n=1..) A(n,t) } Then S(0) = { sum (n=1..) A(n,0) } = { sum (n=1..) 0 } = 0 QED. > >>You can find this in any > >>first year calculus text book. > > > > I have looked in several calculus books, starting with Apostol's, and > > found no such thing in any of them. They are all careful to say that, > > absent convergence, limit definitions say nothing about what happens. > > There are no other definitions of the infinite than limit definitions. There obviously *are* other definitions, you just do not approve of them. Why should anyone particularly care for your likes and dislikes? -- mike.
From: Mike Kelly on 5 Oct 2006 04:51 Han de Bruijn wrote: > Virgil wrote: > > > In article <161ca$4523bb80$82a1e228$8996(a)news2.tudelft.nl>, > > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > > >>imaginatorium(a)despammed.com wrote: > >> > >>>Han de Bruijn wrote: > >>> > >>>>The question is: how many balls are there in the vase at noon. > >>>>This question is meaningless, because noon is never reached. > >>> > >>>Really? When's lunch, then? > >> > >>Time is _suggested_, but not present, in the Balls in a Vase problem. > > > > Without any time to put balls into the vase, the vase is empty. > > In real time there are no singularities. With the Balls in a Vase there > is a singularity at noon. Therefore the quantity called "time" resembles > real time sometimes, but not always. Which is neither here nor there as the Balls in a Vase is not (OBVIOUSLY not) intended to be a "real world" problem. -- mike.
From: Han de Bruijn on 5 Oct 2006 05:11 Mike Kelly wrote: > Han de Bruijn wrote: > >>Quote [ Randy Poe ] : Physicists also >> >>>realize that things can exist in mathematics that aren't even >>>approximations of a physical realizable. That aren't physically >>>sensible in other words. >> >>That's only true for non-disciplinary mathematics. > > What is non-disciplinary mathematics? Is it not mathematics? It is mathematics without the discipline. Han de Bruijn
From: Mike Kelly on 5 Oct 2006 05:19
Tony Orlow wrote: > Mike Kelly wrote: > > mueckenh(a)rz.fh-augsburg.de wrote: > >> Han de Bruijn schrieb: > >> > >>> stephen(a)nomail.com wrote: > >>> > >>>> Han.deBruijn(a)dto.tudelft.nl wrote: > >>>> > >>>>> Worse. I have fundamentally changed the mathematics. Such that it shall > >>>>> no longer claim to have the "right" answer to an ill posed question. > >>>> Changed the mathematics? What does that mean? > >>>> > >>>> The mathematics used in the balls and vase problem > >>>> is trivial. Each ball is put into the vase at a specific > >>>> time before noon, and each ball is removed from the vase at > >>>> a specific time before noon. Pick any arbitrary ball, > >>>> and we know exactly when it was added, and exactly when it > >>>> was removed, and every ball is removed. > >>>> > >>>> Consider this rephrasing of the question: > >>>> > >>>> you have a set of n balls labelled 0...n-1. > >>>> > >>>> ball #m is added to the vase at time 1/2^(m/10) minutes > >>>> before noon. > >>>> > >>>> ball #m is removed from the vase at time 1/2^m minutes > >>>> before noon. > >>>> > >>>> how many balls are in the vase at noon? > >>>> > >>>> What does your "mathematics" say the answer to this > >>>> question is, in the "limit" as n approaches infinity? > >>> My mathematics says that it is an ill-posed question. And it doesn't > >>> give an answer to ill-posed questions. > >> You are right, but the illness does not begin with the vase, it beginns > >> already with the assumption that meaningful results could be obtained > >> under the premise that infinie sets like |N did actually exist. > > > > The meaningful result is that if you allow "|N exists" then the vase > > empties at noon. Even if you don't allow that in your mathematics, you > > can surely accept the logical conclusion that IF you allow that THEN > > the vase is empty at noon. No? > > Only if you change the order of events, In the original problem, every ball that is added is added at a time before noon and has a removal time before noon. To note this isn't changing the order of events. It's just to point out the utter absurdity of your position. For ball n, as stated in the original problem : Insertion time = -1/(2^(floor((n-1)/10))) Removal time = -1/(2^(n-1)) >or refuse to say when the vase > empties or how. Any "|N" aside, the problem clearly states that ten > balls are added and then one removed, per iteration, so if the vase > emptied, it could only be with the removal of that 1 ball, not with the > addition of the ten balls, since that would require that there had been > -10 balls in the vase. But, for there to be 1 ball left, which when > removed left an empty vase, ten would have been inserted right > beforehand, meaning there had to have been -9 balls in the vase. Neither > negative count is possible, therefore the vase could not have emptied. There is no last step in an infinite sequence of events. You still don't understand this a year later. Pathetic, really. The first point in time after the experiment begins at which the vase is empty is noon. There is no iteration at which the vase becomes empty. The vase does not become empty at an iteration. It becomes empty at noon and not before.Every iteration is a time before noon. There is no last iteration before noon. > When you come to two logical conclusions given two lines of thought, how > do you resolve that? By noting that your conclusion is not logical; it is a kludge based on your errant intuition and vomiting vacant verbiage that you think sounds mathematical. You think to maniuplate limits in your argumentst. But how do you know when you are using them in a valid way? You have *no idea* as all you know about limits is gleaned from reading this newgroup and skimming online sources like Wikipedia and Mathworld. lim(n->oo,S(T(n))) = S(lim(n->oo),T(n)) is true when S is continuous Did you know this? I highly doubt it. So how can anyone expect to trust your arguments based on limits, if you don't know when it's valid to apply them? You don't do actual mathematics, you just like to waffle about vague ideas you have based on your misconceptions of what mathematics is. This isn't going to be productive. Unless I am very much mistaken, a large part of the whole *point* of the ball and vase problem is to demonstrate that one cannot always mess around with limits willy-nilly and expect to obtain valid results. Of course, many will stumble when first seeing it because they are used to informal manipulation of limits to obtain what feels like the "right" result. It's supposed to be a prod to make you realise that it's important to be very careful that any limit manipulation you do is based on a rigorous bedrock. But you don't even know what the rigorous rules governing when limits are a valid tool *are*. I'm expecting you to respond with a comment that you have your own version of limits and you can use them as a magic wand to deduce whatever your intuition tells you is the "right answer". But you don't have a mathematical tool. You have your vague ideas which are backed up by nothing more than "This feels right to Tony Orlow". You're attacking the problem with a blamanche when what you need are precision tools. -- mike. |