An uncountable countable set
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From: David Marcus on 26 Oct 2006 14:39 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, Nonstandard analysis is built on mathematical logic as is 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. > 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. -- David Marcus
From: David Marcus on 26 Oct 2006 14:50 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? -- David Marcus
From: MoeBlee on 26 Oct 2006 15:03 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! 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. 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. 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... > 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'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: Randy Poe on 26 Oct 2006 15:08 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? > 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
From: David Marcus on 26 Oct 2006 15:13
Tony Orlow wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> David Marcus wrote: > >>> 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. > >> Fine for each ball n. > >> > >>> Let V(t) = sum{n=1}^infty B_n(t). Let L = lim_{t -> 0-} V(t). Let S = > >>> V(0). Let T be the number of balls that you say are in the vase at > >>> noon. > >> You are summing B_n(t) to oo? > > > > The sum is over all positive integers. There is no B_oo. I'm sticking to > > standard Calculus notation. Does that change your answers below? > > Not really, but it's hard to tell with that notation whether you are > including noon or not. It is standard Calculus notation. Have you ever taken a Calculus course? What math courses have you taken? > >>> Problem 1. For n = 1,2,..., define > >>> > >>> A_n = -1/floor((n+9)/10), > >>> R_n = -1/n. > >>> > >>> Then L = infinity, S = 0, and T = undefined. > >> I say that if noon exists, there are an infinite number of balls in the > >> vase. n=oo -> L=T. > > > > By "noon exists" do you mean there is a ball B_oo? There isn't. > > No kidding. That's why noon cannot be part of the experiment. So, what is your answer for T in this problem? > >>> Problem 2. For n = 1,2,..., define > >>> > >>> A_n = -1/n, > >>> R_n = -1/(n+1). > >>> > >>> Then L = 1, S = 0, T = 1. > >> Yeah, L=T again. > >> > >>> Problem 3. For n = 1,2,..., define > >>> > >>> A_n = -1, > >>> R_n = -1/n. > >>> > >>> Then L = infinity, S = 0, T = 0. > >> L=lim(x->oo: oo-x) = 0 <> oo > > > > L is the limit of V(t) as t approaches zero from the left. So, L = oo. > > > > I don't know what you mean by "lim(x->oo: oo-x)". We don't normally > > define things like oo-x, for x an integer, in Calculus. Of course, the > > most natural definition would be for oo-x to equal oo, in which case > > your limit would also be oo. But, you say it is zero. > > If oo=oo then oo-oo=0. You wrote "lim". You don't evaluate a limit by plugging in the limiting value. So, What is the value of T, and how do you calculate it? > >> L=T > >> > >>> Tony, can you give us a general procedure to let us determine T given > >>> the A_n's and B_n's? > >> You can keep track of the points between your time vortexes and what's > >> going on during those periods, for starters. > > > > I'm afraid I don't know what I'm supposed to keep track of. In truth, I > > thought that by calculating V, I was keeping track. But, neither of the > > two quantities I can get from V, i.e., L and S, seem to consistently > > match your value. > > You are supposed to keep in mind the coupling of 10 additions with each > subtraction, and note that this sum of balls cannot converge to 0 no > matter how long you keep it up. That's what L does. But, you don't seem to agree with the value of L all the time (although you agree with it more than you do with S). -- David Marcus |