An uncountable countable set
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From: Han de Bruijn on 5 Oct 2006 10:41 Randy Poe wrote: > 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. > > That's a property of the vase, not the clock. The clock > ticks on independently of the imaginary process > we're timing with it. No. The clock ticks because there is no Balls in a Vase in the universe. Han de Bruijn
From: Han de Bruijn on 5 Oct 2006 10:42 Randy Poe wrote: > Han de Bruijn wrote: > >>Mike Kelly wrote: >> >>>Han de Bruijn wrote: >>> >>>>Mike Kelly wrote: >>>> >>>>>Appeals to dead authorities aside, you're choosing not to explain what >>>>>is meant by "mathematics without the discipline". Why? >>>> >>>>Why not? >>> >>>It seems very silly to post seemingly silly messages to usenet and then >>>refuse to explain what non-silly meaning they have behind them. People >>>will think you utterly silly. >> >>No: What's the beef in explaining something that is self explanatory? > > If it's "self-explanatory" then wouldn't people who read > it understand it? People? Yes. > Do you think anyone reading this thread, beside you, > understands what you mean by that phrase? Sure. Han de Bruijn
From: Tony Orlow on 5 Oct 2006 10:48 Virgil wrote: > In article <4523c954$1(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> David R Tribble wrote: >>> Tony Orlow wrote: >>>>> On the other hand >>>>> I don't know why I said "neither can the reals". In any case, the only >>>>> way the ordinals manage to be "well ordered" is because they're defined >>>>> with predecessor discontinuities at the limit ordinals, including 0. >>>>> That doesn't seem "real" >>> Virgil wrote: >>>>> In what sense of "real". There are subsets of the reals which are order >>>>> isomorphic to every countable ordinal, including those with limit >>>>> ordinals, so until one posits uncountable ordinals there are no problems. >>> Tony Orlow wrote: >>>> The real line is a line, with >>>> each point touching two others. >>> That's a neat trick, considering that between any two points there is >>> always another point. An infinite number of points between any two, >>> in fact. So how do you choose two points in the real number line >>> that "touch"? >>> >> They have to be infinitely close, so actually, they have an >> infinitesimal segment between them. :) > > But any "infinitesimal segment" within the reals is bisectable. Within the standard reals, it's one number, if it's closer than any finite distance of a that number.
From: Tony Orlow on 5 Oct 2006 10:51 Virgil wrote: > In article <4523cb30(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> 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, 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 > > It also says precisely which numbered balls are added at which times and > which numbered balls are removed at which times. Absent that > information, one has a different puzzle which has an indeterminant > result. > It also says which balls remain when each is taken out, namely, when ball n is removed, balls n+1 through 10n remain. > > >> 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. > > That assumes that there would have to be a "last ball", which equally > assumes that there would have to be a "last natural number", which > destroys TO's analysis. Uh, no, the very conclusion that the vase empties, when at most one ball is removed at a time, implies that there is a last ball removed, but that's impossible as I've shown.
From: Randy Poe on 5 Oct 2006 11:14
Han de Bruijn wrote: > Randy Poe wrote: > > > Han de Bruijn wrote: > > > >>Mike Kelly wrote: > >> > >>>Han de Bruijn wrote: > >>> > >>>>Mike Kelly wrote: > >>>> > >>>>>Appeals to dead authorities aside, you're choosing not to explain what > >>>>>is meant by "mathematics without the discipline". Why? > >>>> > >>>>Why not? > >>> > >>>It seems very silly to post seemingly silly messages to usenet and then > >>>refuse to explain what non-silly meaning they have behind them. People > >>>will think you utterly silly. > >> > >>No: What's the beef in explaining something that is self explanatory? > > > > If it's "self-explanatory" then wouldn't people who read > > it understand it? > > People? Yes. > > > Do you think anyone reading this thread, beside you, > > understands what you mean by that phrase? > > Sure. Who? OK, if there's anyone out there reading this who knows what Han means by "non-disciplinary mathematics", could you please explain since Han is unable to? If *I* were to characterize undisciplined approaches to mathematics, I would include something like "introduction of terms which the author is unable to define but nevertheless says 'the meaning is obvious'" - Randy |