An uncountable countable set
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From: RLG on 25 Oct 2006 01:57 "Tony Orlow" <tony(a)lightlink.com> wrote in message news:453e4a3f(a)news2.lightlink.com... > > Yes, David. What else happens in this experiment besides insertions and > removals of naturals at finite times before noon? If the infinite sequence > of events is actually allowed to continue until t=0, then you are talking > about events not indexed with natural numbers, so you're not talking about > the same experiment. If noon is not allowed, and all times in the > experiment are finitely before noon, well, at none of those times does the > vase empty, as we all agree. This is why I am asking when this occurs. It > can't, given the constraints of the problem. From my reading of this issue the vase is empty at noon, as David Marcus says. But Tony, I have a question for you. Suppose we put one more ball into the vase, at any time before noon, and that ball is labeled "oo". At exactly noon that ball is removed from the vase. At noon is the vase empty or does it contain the ball labeled "oo"? -R
From: David Marcus on 25 Oct 2006 14:18 imaginatorium(a)despammed.com wrote: > David Marcus wrote: > > Tony Orlow wrote: > > > David Marcus wrote: > > > > Tony Orlow wrote: > > > >> As each ball n is removed, how many remain? > > > > > > > > 9n. > > > > > > > >> Can any be removed and leave an empty vase? > > > > > > > > Not sure what you are asking. > > > > > > If, for all n e N, n>0, the number of balls remaining after n's removal > > > is 9n, does there exist any n e N which, after its removal, leaves 0? > > > > I don't know what you mean by "after its removal"? > > Oh, I think this is clear, actually. Tony means: is there a ball (call > it ball P) such that after the removal of ball P, zero balls remain. > > The answer is "No", obviously. If there were, it would be a > contradiction (following the stated rules of the experiment for the > moment) with the fact that ball P must have a pofnat p written on it, > and the pofnat 10p (or similar) must be inserted at the moment ball P > is removed. I agree. If Tony means is there a ball P, removed at time t_P, such that the number of balls at time t_P is zero, then the answer is no. After all, I just agreed that the number of balls at the time when ball n is removed is 9n, and this is not zero for any n. > Now to you and me, this is all obvious, and no "problem" whatsoever, > because if ball P existed it would have to be the "last natural > number", and there is no last natural number. > > Tony has a strange problem with this, causing him to write mangled > versions of Om mani padme hum, and protest that this is a "Greatest > natural objection". For some reason he seems to accept that there is no > greatest natural number, yet feels that appealing to this fact in an > argument is somehow unfair. The vase problem violates Tony's mental picture of a vase filling with water. If we are steadily adding more water than is draining out, how can all the water go poof at noon? Mental pictures are very useful, but sometimes you have to modify your mental picture to match the mathematics. Of course, when doing physics, we modify our mathematics to match the experiment, but the vase problem originates in mathematics land, so you should modify your mental picture to match the mathematics. -- David Marcus
From: Tony Orlow on 25 Oct 2006 14:25 RLG wrote: > "Tony Orlow" <tony(a)lightlink.com> wrote in message > news:453e4a3f(a)news2.lightlink.com... >> Yes, David. What else happens in this experiment besides insertions and >> removals of naturals at finite times before noon? If the infinite sequence >> of events is actually allowed to continue until t=0, then you are talking >> about events not indexed with natural numbers, so you're not talking about >> the same experiment. If noon is not allowed, and all times in the >> experiment are finitely before noon, well, at none of those times does the >> vase empty, as we all agree. This is why I am asking when this occurs. It >> can't, given the constraints of the problem. > > From my reading of this issue the vase is empty at noon, as David Marcus > says. But Tony, I have a question for you. Suppose we put one more ball > into the vase, at any time before noon, and that ball is labeled "oo". At > exactly noon that ball is removed from the vase. At noon is the vase empty > or does it contain the ball labeled "oo"? > > -R > > Hi RLG. Welcome to the conversation. According to the experiment, all balls inserted and removed are finite, so that doesn't really apply. Every ball n is inserted at time -1/n (or -1/2^n originally, but it doesn't matter), so ball oo cannot be inserted before noon. But, if you want to entertain the idea of inserting an extra ball named "oo", or "Bill", or "RLG", the addition of a ball is not going to make the vase any more empty. I wonder what logical implication you think that has.... TO
From: David Marcus on 25 Oct 2006 14:31 RLG wrote: > "Tony Orlow" <tony(a)lightlink.com> wrote in message > news:453e4a3f(a)news2.lightlink.com... > > > Yes, David. What else happens in this experiment besides insertions and > > removals of naturals at finite times before noon? If the infinite sequence > > of events is actually allowed to continue until t=0, then you are talking > > about events not indexed with natural numbers, so you're not talking about > > the same experiment. If noon is not allowed, and all times in the > > experiment are finitely before noon, well, at none of those times does the > > vase empty, as we all agree. This is why I am asking when this occurs. It > > can't, given the constraints of the problem. > > From my reading of this issue the vase is empty at noon, as David Marcus > says. But Tony, I have a question for you. Suppose we put one more ball > into the vase, at any time before noon, and that ball is labeled "oo". At > exactly noon that ball is removed from the vase. At noon is the vase empty > or does it contain the ball labeled "oo"? I'll add that one to my list: For n = 1,2,..., suppose we have numbers A_n and R_N (the addition and removal times of ball n where time is measured in minutes before noon). For n = 1,2,..., define a function B_n by B_n(t) = 1 if A_n <= t < R_n, 0 if t < A_n or t >= R_n. Let V(t) = sum{n=1}^{oo} B_n(t). Let L = lim_{t -> 0-} V(t). Let S = V(0). Let T be the number of balls that Tony says are in the vase at noon. Problem 1. For n = 1,2,..., let A_n = -1/floor((n+9)/10), R_n = -1/n. Then L = oo, S = 0, T = undefined. Problem 2. For n = 1,2,..., let A_n = -1/n, R_n = -1/(n+1). Then L = 1, S = 0, T = 1. Problem 3. For n = 1,2,..., let A_n = -1, R_n = -1/n. Then L = oo, S = 0, T = 0. Problem 3. For n = 1,2,..., let A_n = -1, R_n = -1/n. Then L = oo, S = 0, T = 0. Problem 4. For n = 1,2,..., let A_n = -1/floor((n+9)/10), R_n = -1/n. Let A_oo = -1, R_oo = 0. Let B_oo(t) = 1 if A_oo <= t < R_oo, 0 if t < A_oo or t >= R_oo. Let V(t) = B_oo(t) + sum{n=1}^{oo} B_n(t). Then L = oo, S = 0, T = ?. -- David Marcus
From: stephen on 25 Oct 2006 14:31
David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > imaginatorium(a)despammed.com wrote: >> David Marcus wrote: >> > Tony Orlow wrote: >> > > David Marcus wrote: >> > > > Tony Orlow wrote: >> > > >> As each ball n is removed, how many remain? >> > > > >> > > > 9n. >> > > > >> > > >> Can any be removed and leave an empty vase? >> > > > >> > > > Not sure what you are asking. >> > > >> > > If, for all n e N, n>0, the number of balls remaining after n's removal >> > > is 9n, does there exist any n e N which, after its removal, leaves 0? >> > >> > I don't know what you mean by "after its removal"? >> >> Oh, I think this is clear, actually. Tony means: is there a ball (call >> it ball P) such that after the removal of ball P, zero balls remain. >> >> The answer is "No", obviously. If there were, it would be a >> contradiction (following the stated rules of the experiment for the >> moment) with the fact that ball P must have a pofnat p written on it, >> and the pofnat 10p (or similar) must be inserted at the moment ball P >> is removed. > I agree. If Tony means is there a ball P, removed at time t_P, such that > the number of balls at time t_P is zero, then the answer is no. After > all, I just agreed that the number of balls at the time when ball n is > removed is 9n, and this is not zero for any n. >> Now to you and me, this is all obvious, and no "problem" whatsoever, >> because if ball P existed it would have to be the "last natural >> number", and there is no last natural number. >> >> Tony has a strange problem with this, causing him to write mangled >> versions of Om mani padme hum, and protest that this is a "Greatest >> natural objection". For some reason he seems to accept that there is no >> greatest natural number, yet feels that appealing to this fact in an >> argument is somehow unfair. > The vase problem violates Tony's mental picture of a vase filling with > water. If we are steadily adding more water than is draining out, how > can all the water go poof at noon? Mental pictures are very useful, but > sometimes you have to modify your mental picture to match the > mathematics. Of course, when doing physics, we modify our mathematics to > match the experiment, but the vase problem originates in mathematics > land, so you should modify your mental picture to match the mathematics. As someone else has pointed out, the "balls" and "vase" are just an attempt to make this sound like a physical problem, which it clearly is not, because you cannot physically move an infinite number of balls in a finite time. It is just a distraction. As you say, the problem originates in mathematics. Any attempt to impose physical constraints on inherently unphysical problem is just silly. The problem could have been worded as follows: Let IN = { n | -1/(2^floor(n/10) < 0 } Let OUT = { n | -1/(2^n) } What is | IN - OUT | ? But that would not cause any fuss at all. Stephen |