An uncountable countable set
in [Math]
|
Prev: integral problem
Next: Prime numbers
From: Randy Poe on 28 Sep 2006 09:42 Han de Bruijn wrote: > Randy Poe wrote about the Balls in a Vase problem: > > > Tony Orlow wrote: > > > >>So, it definitely empties...... > > > > Yes. > >> > >>Wow, that's deep. Math is cool. > > > > Cool? Yes. > > > > Deep? I dunno. As I said, this is reasoning that I didn't > > have a major struggle with at the age of about 10. > > And, at that time, Randy didn't have a major struggle with Santa Claus > as well. Alas, when cranks claim telepathic powers they are generally wrong. This is an incorrect statement. - Randy
From: stephen on 28 Sep 2006 10:10 Randy Poe <poespam-trap(a)yahoo.com> wrote: <snip> > What is the number of the ball which, when removed, > makes the vase empty? > I know the kind of nonsense you will spout in answer to > those questions, but the answers within our axiom system > are: (1) there is no t<noon which is the moment just > before noon. For any t<noon, there is t < t' < noon. > (2) There is no such ball. > Here are the Tony gobbledgook answers: > (1) noon - 1/oo > (2) Ball number omega > In TO-matics, one can confidently give an answer like > number 2 despite the fact that one can also agree > that no ball numbered omega is ever put into the > vase. In TO-matics, it is also possible to end up with an empty vase by simply adding balls. According to TO-matics ..1111111111 = 1 + 1 + 1 + 1 + ... and ..1111111111 + 1 = 0 So if you just keep on adding balls one at a time, at some point, the number of balls becomes zero. You have to add just the right number of balls. It is not clear what that number is, but it is clear that it exists in TO-matics. > But in mathematics and logic, we don't get to > keep a set of self-contradictory assumptions around, > only using the ones we want as needed. > - Randy Where's the fun in that? :) Stephen
From: Randy Poe on 28 Sep 2006 11:16 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? > >>> 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. 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. > >> If you say it empties, then you would agree that it either fills or it > >> empties. When does it empty? You say, not before noon. You also say > >> this does not occur at noon, but after noon there are no balls left. So > >> when does this occur? > > > > When every ball that was put in has also been taken out again. > > At noon or before noon? You're skirting the issue. In some of our setups, noon is a supremum, and no time before noon is a supremum. Therefore there is no time before noon when the vase is empty, and for every time at noon or after, the vase is empty. - Randy
From: Tony Orlow on 28 Sep 2006 11:21 Randy Poe wrote: > Tony Orlow wrote: >> Randy Poe wrote: >>> On rereading this, I think there was some confusion. Whether >>> through Tony deliberately misquoting, or a misunderstanding >>> on my part or Tony's, I'm not sure which. But Tony seems to >>> have conflated a statement I made about emptying with one >>> about filling. >>> >>> Tony Orlow wrote: >>>> Randy Poe wrote: >>> This was about emptying: >>> >>>>> It definitely empties, since every ball you put in is >>>>> later taken out. >>>> So, it definitely empties...... >>>> >>>>>> And, at the same time you >>>>>> say it does not do so at noon, nor does it do so before noon. When does >>>>>> this occur? >>> This was about "filling" when I said it: >>> >>>>> It doesn't. >>>> ...but it doesn't! >>> What I would say about emptying is that the vase is empty >>> at noon, but there is no identifiable time before noon at which >>> we can say "the last ball was taken out then". >>> >>> At any time before noon, there are balls in the vase. There >>> is no time we can say "there goes the last ball out" since there >>> is no last ball in. >>> >> If the vase is empty at noon, but not before, how can that not be the >> moment that it becomes empty? > > "Becomes empty" implies a transition from not-empty to > empty. And, in every finite case, it is non-empty, but you claim in the infinite case it is empty. So, indeed, you claim this "transition". > > What time was it in the moment just before noon? You tell me. Does it occur at noon or before? Do you believe in infinitesimals or not? > > What is the number of the ball which, when removed, > makes the vase empty? > There is no such number, since for each ball n removed, balls n+1 through balls n*10 remain. I have maintained throughout that, despite your "labeling schemes", 9/10 of the balls remain, if you add 10 and remove 1 repeatedly. It is precisely like adding 10 gallons and removing 1 per minute. The ocean will never empty. Think measure. > I know the kind of nonsense you will spout in answer to > those questions, but the answers within our axiom system > are: (1) there is no t<noon which is the moment just > before noon. For any t<noon, there is t < t' < noon. > (2) There is no such ball. > Your axiom system is a farse. > Here are the Tony gobbledgook answers: > (1) noon - 1/oo For given specific oo, we have a specific infinite number of balls. > (2) Ball number omega Omega is a concoction of standard limit ordinal nonsense. But, if you want to claim that's "how many" naturals there are, then balls numbered omega+1 through omega*10 remain. > > In TO-matics, one can confidently give an answer like > number 2 despite the fact that one can also agree > that no ball numbered omega is ever put into the > vase. A statement like that is very much like number 2 indeed. > > But in mathematics and logic, we don't get to > keep a set of self-contradictory assumptions around, > only using the ones we want as needed. > > - Randy > No, you only have whole theories whoch entirely contradict the rules of other theories, without any integration into a larger whole truth. Take a lesson from geometry. - Tony
From: Tony Orlow on 28 Sep 2006 11:24
stephen(a)nomail.com wrote: > Randy Poe <poespam-trap(a)yahoo.com> wrote: > > <snip> > >> What is the number of the ball which, when removed, >> makes the vase empty? > >> I know the kind of nonsense you will spout in answer to >> those questions, but the answers within our axiom system >> are: (1) there is no t<noon which is the moment just >> before noon. For any t<noon, there is t < t' < noon. >> (2) There is no such ball. > >> Here are the Tony gobbledgook answers: >> (1) noon - 1/oo >> (2) Ball number omega > >> In TO-matics, one can confidently give an answer like >> number 2 despite the fact that one can also agree >> that no ball numbered omega is ever put into the >> vase. > > In TO-matics, it is also possible to end up with > an empty vase by simply adding balls. According to TO-matics > > ..1111111111 = 1 + 1 + 1 + 1 + ... > > and > ..1111111111 + 1 = 0 > > So if you just keep on adding balls one at a time, > at some point, the number of balls becomes zero. > You have to add just the right number of balls. It is not > clear what that number is, but it is clear that it > exists in TO-matics. > >> But in mathematics and logic, we don't get to >> keep a set of self-contradictory assumptions around, >> only using the ones we want as needed. > >> - Randy > > Where's the fun in that? :) > > Stephen You drew that from my suggestion of the number circle, and that ...11111 could be considered equal to -1. Since then, I looked it up. I'm not the first to think that. It's one of two perspectives on the number line. It's either really straight, or circular with infinite radius, making it infinitesimally straight. The latter describes the finite universe, and the former, the limit. But, you knew that, and are just trying to have fun. Tony |