An uncountable countable set
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From: Virgil on 14 Oct 2006 15:31 In article <453105b4(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <452fbf62(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Virgil wrote: > > > >>>> which is supposed to be the limit of this > >>>> sequence? > >>> Why is it the limit of any sequence? > >>> And since the set of balls removed by noon includes every ball, how > >>> does TO come up with any balls still waiting to be removed at noon? > >> You tell me how many were removed, and I'll tell you how many remain. > > > > Card(N) were removed including the first numbered ball and each > > successively numbered ball. > > That is not a number. It is in ZF and NBG, which is good enough for me.
From: Virgil on 14 Oct 2006 15:40 In article <453105fb(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > > Then how is it that in your analysis, by putting the balls in earlier, > > but taking them out at the original times, one ends with fewer in the > > vase? Now THAT is prime hogwash. > > Because, during the period that the balls are being removed none are > being inserted. If all balls that are removed are still removed at the same times as before, how can varying the times of insertion affect the ultimate result? The only necessary constraint on insertions of balls into the vase and removals of balls from the vase is that each ball that is to be removed must be inserted before it can be removed, and, subject only to that constraint, the set of balls remaining in the vase at the end of all removals is independent of both the times of insertion and of the times of removal. To argue otherwise is to misrepresent the problem.
From: Virgil on 14 Oct 2006 15:43 In article <45310688(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <452fbf0e(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > So let us leave them coupled but merely change the coupling so that the > > nth ball is inserted, say , 1/2^n minutes before it is removed. Both the > > insertions and the removals are still all completed before noon, and it > > is obvious that the vase is empty at noon. > > > > > > > Then you are inserting balls one at a time, and removing them as you > insert the next. What does that have to do with the original problem? The only necessary constraint on insertions of balls into the vase and removals of balls from the vase is that each ball that is to be removed must be inserted before it can be removed, and, subject only to that constraint, the set of balls remaining in the vase at the end of all removals is independent of both the times of insertion and of the times of removal. To argue otherwise is to misrepresent the problem. > > When infinitely many are inserted and all of them removed, what is > > obvious to TO is false to logic. > > Your take on logic is very, shall we say, provincial. You may say what you like, however it remains correct.
From: Virgil on 14 Oct 2006 15:48 In article <45310817(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > 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? They exclude all strings not having a most significant digit. All such digital reals are expressible as strings having a first, or leading, non-zero digit.
From: Virgil on 14 Oct 2006 15:51
In article <453108b5(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > > 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. It may that it never "empties", but at noon, and thereafter, it has "become empty". The only necessary constraint on insertions of balls into the vase and removals of balls from the vase is that each ball that is to be removed must be inserted before it can be removed, and, subject only to that constraint, the set of balls remaining in the vase at the end of all removals is independent of both the times of insertion and of the times of removal. To argue otherwise is to misrepresent the problem. |