An uncountable countable set
in [Math]
|
Prev: integral problem
Next: Prime numbers
From: Virgil on 27 Oct 2006 23:26 In article <4542ab5d(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <45422a2f(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > >> And, what about those rare occasions when I agree with Virgil? Uh oh. > > > > It actually has happened that TO and I agree on something. > > > > It does not happen boringly often but it does happen. > > Does that mean that my assent discredits anything that you have to say? No! TO is not wrong reliably enough so that one may always assume the opposite of what he claims.
From: Virgil on 27 Oct 2006 23:28 In article <4542abbc(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Well, it's good to know that there is no mapping from the naturals to > the evens by the formulaic relation f(x)=2x. That clears up a lot of > problems.... TO again assumes things not said.
From: Virgil on 27 Oct 2006 23:32 In article <4542abf4(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <454238e4(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > > > >>> I think it is an answer. Just to be sure, please confirm that you agree > >>> that, with the definitions above, V(0) = 0. Is that correct? > >>> > >> Sure, all finite balls are gone at noon. > > > > And in any system compatible with ZF or NBG there aren't any others. > > When did I claim my ideas were consistent with those "theories"? But we claim that the original gedankenexperiment must be compatible with some standard set theory unless it explicitly specifies some other set theory, and ZF and NBG are the only ones around. If TO claims some other set theory, he has yet to specify which one, so we can also assume that he is referring to one of the standards.
From: RLG on 28 Oct 2006 02:23 "Virgil" <virgil(a)comcast.net> wrote in message news:virgil-0C72C0.16562227102006(a)comcast.dca.giganews.com... > In article <454229fa(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> > Or, since one ball is removed each time ten more are added, we should >> > write: >> > >> > >> > 10 + (10-1) + (10-1) + (10-1) + (10-1) + ... = Infinite. >> > >> > >> > Now, this divergent series is conditionally convergent. That means we >> > can >> > make the sum equal any value we like depending on how the terms are >> > arranged. So if we choose 0 for the sum that is perfectly valid: >> > >> > >> > 10 + (10-1) + (10-1) + (10-1) + (10-1) + ... = 0. >> > >> >> No, we went through this in another thread. The only way to get a sum of >> 0 is by rearranging the terms and grouping so you have ten -1's for >> every +10. But, the sequence of events is specified NOT to be in that >> order. No ball can be removed without having ten inserted immediately >> before. No Tony. You can form a bijection between two function that allows you to re-group the terms in the form (10-1-1-1-1-1-1-1-1-1-1). Consider two functions, f(n) and g(n), such that f(n) tallies the total number of times a ball labeled with the natural number n enter the vase (before noon) and g(n) tallies the total number of times a ball labeled with the natural number n leave the vase (again before noon). It is easy to see that we can form the below bijection since f(n)=1 and g(n)=1 for all n: f(n) <----> g(n). -R
From: Mike Kelly on 28 Oct 2006 05:03
Tony Orlow wrote: > Mike Kelly wrote: <snip> > > Now correct me if I'm wrong, but I think you agreed that every > > "specific" ball has been removed before noon. And indeed the problem > > statement doesn't mention any "non-specific" balls, so it seems that > > the vase must be empty. However, you believe that in order to "reach > > noon" one must have iterations where "non specific" balls without > > natural numbers are inserted into the vase and thus, if the problem > > makes sense and "noon" is meaningful, the vase is non-empty at noon. Is > > this a fair summary of your position? > > > > If so, I'd like to make clear that I have no idea in the world why you > > hold such a notion. It seems utterly illogical to me and it baffles me > > why you hold to it so doggedly. So, I'd like to try and understand why > > you think that it is the case. If you can explain it cogently, maybe > > I'll be convinced that you make sense. And maybe if you can't explain, > > you'll admit that you might be wrong? > > > > Let's start simply so there is less room for mutual incomprehension. > > Let's imagine a new experiment. In this experiment, we have the same > > infinite vase and the same infinite set of balls with natural numbers > > on them. Let's call the time one minute to noon -1 and noon 0. Note > > that time is a real-valued variable that can have any real value. At > > time -1/n we insert ball n into the vase. > > > > My question : what do you think is in the vase at noon? > > > > A countable infinity of balls. 1) It's not clear to me what you mean by that phrase but I'll assume the standard definition. Still, the question remains of which balls you think are in the vase? Does every natural number, n, have a ball in the vase labelled with that n? 2) How come noon "exists" in this experiment but it didn't exist in the original experiment? Or did you give up on claiming noon doesn't "exist"? What does that mean, anyway? > This is very simple. Everything that occurs is either an addition of ten > balls or a removal of 1, and occurs a finite amount of time before noon. > At the time of each event, balls remain. At noon, no balls are inserted > or removed. The vase can only become empty through the removal of balls, > so if no balls are removed, the vase cannot become empty at noon. It was > not empty before noon, therefore it is not empty at noon. Nothing can > happen at noon, since that would involve a ball n such that 1/n=0. Well, we're skipping ahead here much faster than I think will prove productive. Let's see.... By this logic, there is not a countable infinity of balls in the vase at noon in the new experiment I proposed. Everything that occurs is an addition of a single ball. At the time of each event, a finite number of balls are in the vase. At noon, no balls are inserted. If no balls are inserted at noon, the vase has the same state as before noon - a finite number of balls. -- mike. |