An uncountable countable set
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From: Tony Orlow on 14 Oct 2006 11:51 David R Tribble wrote: > Tony Orlow wrote: >>> then there are naturals which are >>> the result of infinite increments, which must have infinite value. > > David R Tribble wrote: >>> Where's your proof? >>> What is an "infinite increment" (or "infinite successor")? > > Tony Orlow wrote: >> If there is a !number! n of successors, there exists a successor n steps >> ahead. If there are an infinite !number! n of successors, there is a >> successor n, an infinite number of steps ahead. >> >> If you increment a natural n times, you have added n to it. If successor >> is increment, and there are an infinite !number! of such increments, you >> have added this infinite number to your starting value. Adding an >> infinite number to a finite yields an infinite. Therefore, the infinite >> set includes infinite values. > > So how do you know when you've reached the point of adding an > infinite number of increments? Is there some way of counting all > the increments? > When it is greater than any finite number.
From: Tony Orlow on 14 Oct 2006 11:53 David R Tribble wrote: > Tony Orlow wrote: >>> That doesn't seem "real", and the axiom of choice aside, I don't see >>> there being any well ordering of the reals. The closest one can come is >>> the H-riffic numbers. :) > > David R Tribble wrote: >>> Hardly. The H-riffics are a simple countable subset of the reals. >>> Anyone mathematically inclined can come up with such a set. > > Tony Orlow wrote: >>> You never paid enough attention to understand them. They cover the reals. > > David R Tribble wrote: >>> They omit an uncountable number of reals. Any power of 3, for example, >>> which you never showed as being a member of them. Show us how 3 fits >>> into the set, then we'll talk about "covering the reals". > > Tony Orlow wrote: >> 3 is an unending string, just like 1/3 is in base-10. Rusin confirmed >> that about two years ago. But, you're right, I need to construct a >> formal proof of the equivalence between the H-riffics and the reals. > > Your definition of your H-riffic numbers excludes unending strings. Since when? Do the digital reals exclude unending strings? > So 3 can't be a valid H-riffic, and neither can any of its successors. Nice fantasy, but that's all it is. I suppose 1/3 doesn't exist in decimal either. > I know you don't get this, but go back and read your own definition. > Every H-riffic corresponds to a node in an infinite, but countable, > binary tree. No, like the reals, it corresponds to a path in the tree. > > The H-riffics is only a countable subset of the reals, and omits an > uncountable number of reals. Just like all finite-length reals. That is only a countable set. So, the digital reals are not the reals? Tell it to Cantor the Diagonal. >
From: Tony Orlow on 14 Oct 2006 11:56 David R Tribble wrote: > Tony Orlow wrote: >>> The sequence of events consists of adding 10 and removing 1, an infinite >>> number of times. In other words, it's an infinite series of (+10-1). > > Virgil wrote: >>> That deliberately and specifically omits the requirement of identifying >>> and tracking each ball individually as required in the originally stated >>> problem, in which each ball is uniquely identified and tracked. > > Tony Orlow wrote: >> The original statement contrasted two situations which both matched this >> scenario. The difference between them was the label on the ball removed >> at each iteration, and yet, that's not relevant to how many balls are in >> the vase at, or before, noon. > > How about a slightly different, but equivalent, approach to the > problem? > > At each moment 2^-n sec prior to noon, add 10 balls (10n+1, 10n+2, > ..., 10n+10) to the vase. (Tony, the numbering works out if you start > with n=0.) Obviously, the insertions stop by noon. > > Now at each moment 2^-n sec before 1:00pm, remove ball n from > the vase. (Tony, the numbering works if you start with n=1.) > Obviously, the removals stop by 1:00pm. > > So how many balls are left in the vase at 1:00pm? > If you paid attention to the various subthreads, you'd know I just answered that. Where the insertions and removals are so decoupled, there is no problem. Where the removal of a ball is immediately preceded and succeeded by insertions of 10, the vase never empties.
From: David Marcus on 14 Oct 2006 14:09 Tony Orlow wrote: > David Marcus wrote: > > Han de Bruijn wrote: > >> Virgil wrote: > >> > >>> In article <66a8$452f4298$82a1e228$30886(a)news2.tudelft.nl>, > >>> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > >>> > >>>> Randy Poe wrote about the Balls in a Vase problem: > >>>> > >>>>> Tony Orlow wrote: > >>>>>> Specifically, that for every ball removed, 10 are inserted. > >>>>> All of which are eventually removed. Every single one. > >>>> All of which are eventually inserted. Every single one. > >>> None are reinserted after being removed but,each is removed after having > >>> been inserted, so that leaves them all outside the vase at noon. > >> Huh! Then reverse the process: first remove 1, then insert 10. It must > >> be no problem in your "counter intuitive" mathematics to start with -1 > >> balls in that vase. > > > > Consider this situation: Start with an empty vase. Add a ball at time 5. > > Remove it at time 6. > > > > How you would translate that into Mathematics? > > > > What happens between times 5 and 6? Nothing. > Are there other balls involved? No. -- David Marcus
From: David Marcus on 14 Oct 2006 14:15
Tony Orlow wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> David Marcus wrote: > >>> How about this problem: Start with an empty vase. Add a ball to a vase > >>> at time 5. Remove it at time 6. How many balls are in the vase at time > >>> 10? > >>> > >>> Is this a nonsensical question? > >> Not if that's all that happens. However, that doesn't relate to the ruse > >> in the vase problem under discussion. So, what's your point? > > > > Is this a reasonable translation into Mathematics of the above problem? > > I gave you the translation, to the last iteration of which you did not > respond. > > "Let 1 signify that the ball is in the vase. Let 0 signify that it is > > not. Let A(t) signify the location of the ball at time t. The number of > > 'balls in the vase' at time t is A(t). Let > > > > A(t) = 1 if 5 < t < 6; 0 otherwise. > > > > What is A(10)?" > > Think in terms on n, rather than t, and you'll slap yourself awake. Sorry, but perhaps I wasn't clear. I stated a problem above in English with one ball and you agreed it was a sensible problem. Then I asked if the translation above is a reasonable translation of the one-ball problem into Mathematics. If you gave your translation of the one-ball problem, I missed it. Regardless, my question is whether the translation above is acceptable. So, is the translation above for the one-ball problem reasonable/acceptable? -- David Marcus |