An uncountable countable set
in [Math]
|
Prev: integral problem
Next: Prime numbers
From: Tony Orlow on 26 Oct 2006 23:13 David Marcus wrote: > Tony Orlow wrote: >> David Marcus wrote: >>> Tony Orlow wrote: >>>> David Marcus wrote: >>>>> Tony Orlow wrote: >>>>>> So, David, you think the fact that balls leave the vase only by being >>>>>> removed one at a time, and the fact that at all times before noon there >>>>>> are balls in the vase, and the fact that at noon there are no balls in >>>>>> the vase, is consistent with the fact that no balls are removed at noon? >>>>>> How can you not see the logical inconsistency of an infinitude of balls >>>>>> disappearing, not just in a moment, but at no possible moment? Are you >>>>>> so steeped in set theory that you cannot see that an unending sequence >>>>>> of +10-1 amounts to an unending series of +9's which diverges? What is >>>>>> illogical about that? >>>>>> In your set-theoretic interpretation of the experiment there is a >>>>>> problem which makes your conclusion incompatible with conclusions drawn >>>>>> from infinite series, and other basic logical approaches. >>>>> I gave a Freshman Calculus interpretation/translation of the problem (no >>>>> set theory required). Here is a suitable version: >>>>> >>>>> For n = 1,2,..., define >>>>> >>>>> A_n = -1/floor((n+9)/10), >>>>> R_n = -1/n. >>>>> >>>>> 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}^infty B_n(t). What is V(0)? >>>>> >>>>> I suppose you either disagree with this interpretation/translation or >>>>> you disagree that for this interpretatin V(0) = 0. Which is it? >>>> t=0 is precluded by n e N and t(n) = -1/n. >>> Sorry, I don't follow. Were you answering my question? I gave you a >>> choice: >>> >>> 1. Disagree with the interpretation/translation >>> 2. Agree with the interpretation/translation, but disagree that V(0) = 0 >>> >>> Are you picking #1 or #2? >> I'll choose #2 on the grounds that 0 does not exist in the experiment >> and that V(0) is therefore without meaning. >> >>>>> Given my interpretation/translation of the problem into Mathematics (see >>>>> above) and given that the "moment the vase becomes empty" means the >>>>> first time t >= -1 that V(t) is zero, then it follows that the "vase >>>>> becomes empty" at t = 0 (i.e., noon). >>>> Yes, now, when nothing occurs at noon, and no balls are removed, what >>>> else causes the vase to become empty? >>> No balls are added or removed at noon, but the vase becomes empty at >>> noon. >> Through some other mechanism than ball removal? >> >>> If you consider the vase becoming empty to be "something" rather than >>> "nothing", then it is not true that nothing occurs at noon. If by >>> "nothing occurs at noon", you mean no balls are added or removed, then >>> it is true that nohting occurs at noon. >> And, if no balls are moved at noon, what causes the vase to become empty >> at noon? Evaporation? A black hole? >> >>> The cause of the vase becoming empty at noon is that all balls are >>> removed before noon, but at all times between one minute before noon and >>> noon, there are balls in the vase. >> The fact that there are balls at all times before noon and that no balls >> are removed at noon imply that there are balls in the vase at noon, if >> it exists in the experiment at all to begin with. >> >>> Let me ask you the same question regarding the following problem. >>> >>> Problem: For n = 1,2,..., let >>> >>> A_n = -1/floor((n+9)/10), >>> R_n = -1/n. >>> >>> 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 B_n(t). What is V(0)? Answer: V(0) = 0. >>> >>> Considering that for all n we have A_n <> 0 and B_n <> 0 and that V(t) >>> is approaching infinity as t approaches zero from the left, what causes >>> V(0) to be zero? >> The fact that you have no upper bound to the naturals. This is the same >> technique, essentially, which equates the naturals with, say, the evens, >> or squares of naturals, even though those are proper subsets of the >> naturals. You can draw a 1-1 correspondence between the balls in and >> out, sure. There's a bijection there. Infinite bijections do not given >> any notion of measure unless they are parameterized. Here, you can look >> at number of balls in the vase as a function of n or of t. In either >> case, the sum diverges. It is only in trying to consider the unbounded >> set as completed that you come to this silly conclusion. > > Let's see if I understand what you are saying. Consider this math: > > -------------------------- > For n = 1,2,..., let > > A_n = -1/floor((n+9)/10), > R_n = -1/n. > > For n = 1,2,..., define a function B_n: R -> R 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 B_n(t). > -------------------------- > > Are you saying that V(0) is not equal to zero? > (sigh) I already answered this. Are you just trying to test for consistency in my statements? That gets a little tiresome. I am saying 0 doesn't happen in this experiment. All events are before 0. Any events occurring at 0, by the constraints of the experiment, must have an index n in the sequence such that 1/n=0, but this n cannot exist in the sequence of finite natural numbers. Therefore, nothing happens at t=0. If it did, all finite balls would be gone, but they would be replaced by an uncountable number of infinitely-numbered balls. At the time of each and every event before 0, without exception, more balls are left in the vase than were there before the event, and during (-1,0), the vase is never empty. Combine these two facts, and you get that the vase did not become empty, since nothing happens at noon to the balls to change the state of the vase, and the desired state does not occur before noon. It still has balls. And including noon doesn't change that, but only pushes the potential, countable infinity to the actual, uncountable point. Tony
From: David Marcus on 26 Oct 2006 23:20 Tony Orlow wrote: > Mike Kelly wrote: > > 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. So, "noon exists" in this case, even though nothing happens at noon. > 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. -- David Marcus
From: Tony Orlow on 26 Oct 2006 23:28 MoeBlee wrote: > Tony Orlow wrote: >> MoeBlee wrote: >>> Tony Orlow wrote: >>>>> But none of Robinson's non-standard numbers are cardinalities. >>>> No kidding. They actually make sense. >>> You said you have not properly studied chapter II in the book - the one >>> that includes mathematical logic, model theory, and set theory (does it >>> not? I'll stand corrected if it doesn't). What are you going to say >>> when you find out that what you say makes sense rests on a foundation >>> of set theory that you say doesn't make sense? Or, if I'm incorrect >>> that Robinson's work in non-standard analysis doesn't presuppose basic >>> mathematical logic, model theory, and set theory, then I'll benefit by >>> being corrected in my admittedly cursory understanding of the matter. >>> >>> MoeBlee >>> >> Uh, if Robinson's thesis is built upon transfinite set theory, then that >> is evidence right there that it's inconsistent, since you have a >> smallest infinity, omega, but Robinson has no smallest infinity. > > We JUST agreed that 'smallest infinity' means two different things when > referring to ordinals and when referring to certain kinds of other > orderings! It is AMAZING to me that even though I took special care to > make sure this was clear, and then you agreeed, you NOW come back to > conflate the two ANYWAY! Ahem. I said that Robinson's analysis seems to have nothing to do with transfinitology. They appear to be unrelated. However, they cme to two very different conclusions regarding a basic question: is there a smallest infinite number? It seems clear to me there is not, for the very same reason that Robinson uses: if there is an infinite number, you can subtract 1 and get a different, smaller infinite number. It's the same logic y'all use to argue that there's no largest finite. It's correct. The Twilight Zone between finite and infinite CANNOT really be pinpointed that way. So, that's a verrry basic discrepancy. There is clearly a contradiction between the two theories. They can't both be right about that, can they? Is there, and at the same time is not, a smallest infinite number? Tach me more about mathematical logic and consistency. > > No, it is NOT a contradiction with set theory and there being a > smallest infinite ordinal and smallest infinite cardinal that there are > also non-standard orderings (which are NOT cardinality or ordinal > ordering, as even YOU recognized) that have what are CALLED 'infinite > elements' but with no least one. In transfinitology? Why would anyone argue against me saying there is no smallest infinite? Sorry, you're not meshing. > > Even though I WARNED you, and you RECOGNIZED, you still got yourself > mixed up by thinking that the word 'infinite' means the same thing in > two different contexts. And that happened because you're an arrogant > ignoramus who thinks he can spout on the Internet about mathematical > developments that are VERY specific and technical and require very > specific and technical understanding of basics (and even ADVANCED model > theory) that you ignore. Eat me. Do you maintain that the two theories are compatible with each other? Is there, and also not, a smallest infinity. Can one, while they also can't, order infinite sets with precision in many cases? Are we allowed to note that lim(n=1->oo: 9) = 9*oo, and not play stupid label games? > > How frustrating it is trying to have a conversation with you. You > pretty much skip the part of Robinson's book that talks about > mathematical logic and set theory, so you don't at all understand the > basis of what he's doing. Then I WARN you not to confuse 'least > infinite' in the sense of ordinals with 'least infinite' in the sense > of a certain ordering in a non-standard model. And even though you > agreed that the non-standard ordering is not a cardinality ordering, > you come back to conflate a cardinality ordering with the non-standard > ordering anyway! I should have known you'd do that. I should have > known... > Geeze, calm down. I am not conflating them as if they were the same thing. I am clearly stating that they are obviously mutually incompatible, and you're too flustered by the disintegration of this magical theory to see that simple point. They can't coexist in a consistent universe. I don't opt for transfinitology, but for more measure-oriented means of thought, when dealing with numbers. >> Robinson doesn't use ordinals or cardinals that I've seen. He basically >> defines what a well-formed formula is in his system, which is a little >> more restrictive that some others, it seems, and uses the language to >> extend what can be said about finite n in N to include infinite n in *N. > > He uses model theory for models that have infinite universes, does he > not? And from STANDARD models, through the compactness theorem, he > proves the existence of non-standard models, does he not? In this basic > sense, it's not a question of ordinals and cardinals so much (in this > PARTICULAR regard) as it is of there being countable and uncountable > universes and applications of the compactness theorem and a whole bunch > of other mathematical logic and set theoretic model theory that is > applied. > > This is stupid for me to even be trying to talk to you about this. You > need to read and UNDERSTAND that damn first chapter in his book that > you're skipping. (And you'd understand it MUCH more easily if you first > read a book on mathematical logic and one on set theory). Otherwise, > you are oblivous to the BASIS of what he's doing. Sheesh. I admit that > I haven't read Robinson's original work, but at least I have > familiarized myself with well written summaries such as in Enderton's > book. And there is no way in heck that you're going to understand any > of this without getting a good basic understanding of the mathematical > logic and set theory that are the context and basis. You need to relax. I worked on the first chapter for a bit, and got most of the way through, but started to get bogged down, so I skipped ahead to see what the next chapter held, and it did. So, sue me. At least I didn't run to the Cliff Notes... > > You're a damn fool, fooling only yourself, by thinking that you can > shortcut the basics of the subject while you spout ignorantly on the > Internet about the subject. > > MoeBlee
From: Tony Orlow on 26 Oct 2006 23:30 Randy Poe wrote: > Tony Orlow wrote: >> Randy Poe wrote: >>> Tony Orlow wrote: >>>> t=0 is precluded by n e N and t(n) = -1/n. >>> Really? >>> >>> I hope you will accept as true that noon occurred yesterday. >>> >>> Let's define noon yesterday as t=0. Now let's define a set of values >>> t_n = -1/n seconds for n=1, 2, 3, ... , that is, for all FINITE >>> natural numbers n. >>> >>> Has my giving these names to those times somehow >>> precluded noon yesterday from occurring? Retroactively? >>> >> Do you live in the gedanken? Oy. Nothing happens at noon. > > Did noon occur? Not within the constraints of the experiment. Nothing is allowed to happen at noon. > >> Your desired result does not happen before noon. > > What desired result? I didn't have an experiment, I > just named a bunch of times. Is noon "precluded" by > my defining that countable set of variables? > > - Randy > Can anything happen at noon? What can change, in the vase, at noon? What is the state before noon?
From: David Marcus on 26 Oct 2006 23:32
Tony Orlow wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> David Marcus wrote: > >>> Tony Orlow wrote: > >>>> David Marcus wrote: > >>>>> I gave a Freshman Calculus interpretation/translation of the problem (no > >>>>> set theory required). Here is a suitable version: > >>>>> > >>>>> For n = 1,2,..., define > >>>>> > >>>>> A_n = -1/floor((n+9)/10), > >>>>> R_n = -1/n. > >>>>> > >>>>> 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}^infty B_n(t). What is V(0)? > >>>>> > >>>>> I suppose you either disagree with this interpretation/translation or > >>>>> you disagree that for this interpretatin V(0) = 0. Which is it? > >>>> t=0 is precluded by n e N and t(n) = -1/n. > >>> Sorry, I don't follow. Were you answering my question? I gave you a > >>> choice: > >>> > >>> 1. Disagree with the interpretation/translation > >>> 2. Agree with the interpretation/translation, but disagree that V(0) = 0 > >>> > >>> Are you picking #1 or #2? > >> I'll choose #2 on the grounds that 0 does not exist in the experiment > >> and that V(0) is therefore without meaning. > >> > >>>>> Given my interpretation/translation of the problem into Mathematics (see > >>>>> above) and given that the "moment the vase becomes empty" means the > >>>>> first time t >= -1 that V(t) is zero, then it follows that the "vase > >>>>> becomes empty" at t = 0 (i.e., noon). > >>>> Yes, now, when nothing occurs at noon, and no balls are removed, what > >>>> else causes the vase to become empty? > >>> No balls are added or removed at noon, but the vase becomes empty at > >>> noon. > >> Through some other mechanism than ball removal? > >> > >>> If you consider the vase becoming empty to be "something" rather than > >>> "nothing", then it is not true that nothing occurs at noon. If by > >>> "nothing occurs at noon", you mean no balls are added or removed, then > >>> it is true that nohting occurs at noon. > >> And, if no balls are moved at noon, what causes the vase to become empty > >> at noon? Evaporation? A black hole? > >> > >>> The cause of the vase becoming empty at noon is that all balls are > >>> removed before noon, but at all times between one minute before noon and > >>> noon, there are balls in the vase. > >> The fact that there are balls at all times before noon and that no balls > >> are removed at noon imply that there are balls in the vase at noon, if > >> it exists in the experiment at all to begin with. > >> > >>> Let me ask you the same question regarding the following problem. > >>> > >>> Problem: For n = 1,2,..., let > >>> > >>> A_n = -1/floor((n+9)/10), > >>> R_n = -1/n. > >>> > >>> 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 B_n(t). What is V(0)? Answer: V(0) = 0. > >>> > >>> Considering that for all n we have A_n <> 0 and B_n <> 0 and that V(t) > >>> is approaching infinity as t approaches zero from the left, what causes > >>> V(0) to be zero? > >> The fact that you have no upper bound to the naturals. This is the same > >> technique, essentially, which equates the naturals with, say, the evens, > >> or squares of naturals, even though those are proper subsets of the > >> naturals. You can draw a 1-1 correspondence between the balls in and > >> out, sure. There's a bijection there. Infinite bijections do not given > >> any notion of measure unless they are parameterized. Here, you can look > >> at number of balls in the vase as a function of n or of t. In either > >> case, the sum diverges. It is only in trying to consider the unbounded > >> set as completed that you come to this silly conclusion. > > > > Let's see if I understand what you are saying. Consider this math: > > > > -------------------------- > > For n = 1,2,..., let > > > > A_n = -1/floor((n+9)/10), > > R_n = -1/n. > > > > For n = 1,2,..., define a function B_n: R -> R 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 B_n(t). > > -------------------------- > > > > Are you saying that V(0) is not equal to zero? > > (sigh) I already answered this. Are you just trying to test for > consistency in my statements? That gets a little tiresome. Sorry, but I think you answered a different question than I asked. > I am saying 0 doesn't happen in this experiment. All events are before > 0. Any events occurring at 0, by the constraints of the experiment, must > have an index n in the sequence such that 1/n=0, but this n cannot exist > in the sequence of finite natural numbers. Therefore, nothing happens at > t=0. If it did, all finite balls would be gone, but they would be > replaced by an uncountable number of infinitely-numbered balls. > > At the time of each and every event before 0, without exception, more > balls are left in the vase than were there before the event, and during > (-1,0), the vase is never empty. > > Combine these two facts, and you get that the vase did not become empty, > since nothing happens at noon to the balls to change the state of the > vase, and the desired state does not occur before noon. It still has > balls. And including noon doesn't change that, but only pushes the > potential, countable infinity to the actual, uncountable point. You are mentioning balls and time and a vase. But, what I'm asking is completely separate from that. I'm just asking about a math problem. Please just consider the following mathematical definitions and completely ignore that they may or may not be relevant/related/similar to the vase and balls problem: -------------------------- For n = 1,2,..., let A_n = -1/floor((n+9)/10), R_n = -1/n. For n = 1,2,..., define a function B_n: R -> R 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 B_n(t). -------------------------- Just looking at these definitions of sequences and functions from R (the real numbers) to R, and assuming that the sum is defined as it would be in a Freshman Calculus class, are you saying that V(0) is not equal to 0? -- David Marcus |