An uncountable countable set
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From: Tony Orlow on 29 Sep 2006 14:32 stephen(a)nomail.com wrote: > Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: >> Virgil wrote: > >>> In article <d12a9$451b74ad$82a1e228$6053(a)news1.tudelft.nl>, >>> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: >>> >>>> Randy Poe wrote, about the Balls in a Vase problem: >>>> >>>>> It definitely empties, since every ball you put in is >>>>> later taken out. >>>> And _that_ individual calls himself a physicist? >>> Does Han claim that there is any ball put in that is not taken out? > >> Nonsense question. Noon doesn't exist in this problem. > > Yes it is a nonsense question, in the sense > that it is non-physical. You cannot actually perform > the "experiment". Just as choosing a number uniformly > from the set of all naturals is a non-physical nonsense > question. You cannot perform that experiment either. > > Stephen Yes, they both sound equally invalid, and it all goes back to omega, but Han has a point about the density of the set in the naturals throughout its range, and the overall statistical probability of selecting one of that subset from the naturals, even if having probabilities of 1/omega for each natural presents problems. Tony
From: Virgil on 29 Sep 2006 14:35 In article <4402e$451cc6d7$82a1e228$6256(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > stephen(a)nomail.com wrote: > > > So why is it okay to end up with zero balls, when you never remove > > any at all, but it is not okay to end up with zero balls when > > each ball is clearly removed at a definite time? > > Why is it not okay to approach the infinite otherwise than via the limit > concept? Applied to a _finite_ sequence of events? > > Han de Bruijn We know how finite sequences work rather better that how infinite sequences work, as illustrated by the present debate over properties of infinite sequences. And the limit definitions have been shown to be more reliable than intuition in determining patterns in infinite sequences.
From: Virgil on 29 Sep 2006 14:37 In article <6f37e$451cc732$82a1e228$6256(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > MoeBlee wrote: > > > Tony Orlow wrote: > > > >>You might want to expand your reading. > > > > That's rich coming from a guy who hasn't read a single book on > > mathematical logic or set theory. > > That's rich coming from a guy who hasn't read _anything else_ than books > on mathematical logic or set theory. > > Han de Bruijn When TO so obviously pontificates on what he knows nothing about, it is hardly surprising that many people suggest TO needs to study further in the area of his ignorance.
From: Virgil on 29 Sep 2006 14:40 In article <51de2$451cc7e9$82a1e228$6256(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > Virgil wrote: > > > In article <2e658$451b78ef$82a1e228$7519(a)news1.tudelft.nl>, > > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > > >>Tony Orlow wrote: > >> > >>>MoeBlee wrote: > >>> > >>>>Tony Orlow wrote: > >>>> > >>>>>Constructivism and Axiomatism are two sides of a coin. They can be > >>>>>reconciled in larger framework, I think. > >>>> > >>>>I don't know what your definition of 'axiomatism' is, but there are > >>>>axiomatic systems for constructive mathematics. > >>> > >>>I dunno. I was responding to Han's comment. I think he means > >>>constructive concepts vs. axiomatic declarations. > >> > >>It's a priorities issue. Do axioms have to dictate what constructivism > >>should be like? Should constructivism be tailored to the objectives of > >>axiomatics? I think not. > > > > But if you cannot clearly state what you are assuming/accepting as true, > > all you have is a morass of ambiguity. > > Ambiguity does not necessarily comprise a morass. > > Han de Bruijn How do constructivists deduce new things if there is nothing that they can say is true to start with?
From: Tony Orlow on 29 Sep 2006 14:40
Virgil wrote: > In article <451bec94(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Randy Poe wrote: >>> Tony Orlow wrote: >>>> Virgil wrote: >>>>> In article <451b3296(a)news2.lightlink.com>, >>>>> Tony Orlow <tony(a)lightlink.com> wrote: >>>>> >>>>>> Randy Poe wrote: >>>>>>> Tony Orlow wrote: >>>>>> You must have been a strange 10 year old, like that kid >>>>>> down the block that used to pull the legs off of roaches. >>>>> Only those that looked like TO. >>>>> >>>>>>>>> So the reason I don't say it's full "an infinitesimal time >>>>>>>>> before noon" or "some other time before noon" is that >>>>>>>>> I don't say it's full. >>>>>>>> But, you do say it's full or empty, right? >>>>> One can easily say that it is empty at any time at which every ball >>>>> that was put in has been taken out again. >>>>> >>>>> Does TO suggest that at any time after noon there is any ball that was >>>>> put in that was not also taken out? >>>> Yes, at any given time 9/10 of the balls inserted remain. >>> Which ball does not have a definite time at which it >>> is removed? >>> >> Any ball which does not have a definite time at which it is inserted. > > That excludes every ball, since each has a specific time of insertion > and an equally specific time of removal. At any of those specific times are there balls in the vase? If there are balls in the vase at every time before noon, but not at noon, then the final ball is removed at noon. Which ball is that? >>>>>>> So your conclusion from my statement that I would never >>>>>>> say it's full is that sometimes I would say it's full? >>>>>> Uh, you would say it contains an infinite number of balls in some >>>>>> circumstances, as I understand it. >>>>> Then you misunderstand it. >>>> No, your labels misconstrue the problem with your silly fixation on >>>> omega. Do I "misunderstand" that if you remove balls 1, then 11, then >>>> 21, etc, that the vase will NOT be empty? >>> We have different variants of the problem setup. Before >>> discussing too many details, we need to agree on >>> what EXACTLY are the starting assumptions. >> The subject is whether that makes any difference or not. It doesn't. > > Then TO should not object to any specific starting assumptions, as he > claims they make no difference. > >> Your dual gedankens imply that changing the labeling scheme after noon >> makes the balls all disappear. That's ridiculous. > > No labelling scheme can be allowed to change any label once the ball > with that label is inserted in the vase. Why not? Why can't I take out 1, 11, 21 etc, leave 9/10 of the balls in the vase, then switch all the labels and make them disappear? That would be neat. I am sure that's a theorem in transfinitology. It's got to be. Please say it is. ;) >>> But in general if: >>> (a) Every ball has a label n which is a finite natural number. >>> (b) Every ball has a time t_n at which it is removed. >>> (c) There exists a supremum T of the set {t_n, n in N} >>> then for any time t >= T, the vase is empty. >> What is this "supremum", in terms of iterations? > > It is the least upper bound of the times, and any set of times which is > bounded above must have a supremum. > But is that supremum WITHIN the set of times? Eh? > >> Here's one iteration: >> (a) 10 balls added AND >> (b) 1 ball removed IMPLIES >> (c) net 9 balls added >> >> How many iterations? n? Fine. 9n balls remain. > > Which ones? Balls n/10+1 through n. Duh. > > Unless one is postulating that the balls, like electrons, have no > individual identities, but are totally interchangeable at all times, > even when labelled to give them identities, the question of which ones > is relevant. Then why is the answer so obvious before you introduce your labels? The labeling scheme is just a parlor trick for entertaining girls. "See, ladies? Poof! The vase is empty. Tada!" Tony |