An uncountable countable set
in [Math]
|
Prev: integral problem
Next: Prime numbers
From: David Marcus on 28 Oct 2006 13:09 Tony Orlow wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> David Marcus wrote: > >>> Tony Orlow wrote: > >>>> David Marcus wrote: > >>>>> Tony Orlow wrote: > >>>>>> David Marcus wrote: > >>>>>>> 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? > >>>>>> On the surface, you math appears correct, but that doesn't mend the > >>>>>> obvious contradiction in having an event occur in a time continuum > >>>>>> without occupying at least one moment. It doesn't explain how a > >>>>>> divergent sum converges to 0. Basically, what you prove, if V(0)=0, is > >>>>>> that all finite naturals are removed by noon. I never disagreed with > >>>>>> that. However, to actually reach noon requires infinite naturals. Sure, > >>>>>> if V is defined as the sum of all finite balls, V(0)=0. But, I've > >>>>>> already said that, several times, haven't I? Isn't that an answer to > >>>>>> your question? > >>>>> 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. > >>> Please note that there are no balls or time in the above mathematics > >>> problem. However, I'll take your "Sure" as agreement that V(0) = 0. > >> > >> Okay. > >> > >>> Let me ask you a question about this mathematics problem. Please answer > >>> without using the words "balls", "vase", "time", or "noon" (since these > >>> words do not occur in the problem). > >> I'll try. > >> > >>> First some discussion: For each n, B_n(0) = 0 and B_n is continuous at > >>> zero. > >> What??? How do you conclude that anything besides time is continuous at > >> 0, where yo have an ordinal discontinuity???? Please explain. > > > > I thought we agreed above to not use the word "time" in discussing this > > mathematics problem? > > If that's what you want, then why don't you remove 't' from all of your > equations? It is just a letter. It stands for a real number. Would you prefer "x"? I'll switch to "x". > > As for your question, let's look at B_2 (the argument is similar for the > > other B_n). > > > > B_2(t) = 1 if A_2 <= t < R_2, > > 0 if t < A_2 or t >= R_2. > > > > Now, A_2 = -1 and R_2 = -1/2. So, > > > > B_2(t) = 1 if -1 <= t < -1/2, > > 0 if t < -1 or t >= -1/2. > > > > In particular, B_2(t) = 0 for t >= -1/2. So, the value of B_2 at zero is > > zero and the limit as we approach zero is zero. So, B_2 is continuous at > > zero. > > Oh. For each ball, nothing is happening at 0 and B_n(0)=0. That's for > each finite ball that one can specify. I thought we agreed to not use the word "ball" in discussing this mathematics problem? Do you want me to change the letter "B" to a different letter, too? > However, lim(t->0: sum(B_n| B_n(t)=1))=oo. Why do you conveniently > forget that fact? Your notation is nonstandard, so I'm not sure what you mean. Do you mean to write lim_{x -> 0-} sum_n B_n(x) = oo ? If so, I don't understand why you think I've forgotten this fact. If you look in my previous post (or below), you will see that I wrote, "Now, V is the sum of the B_n. As t approaches zero from the left, V(t) grows without bound. In fact, given any large number M, there is an e < 0 such that for e < t < 0, V(t) > M." > >>> In fact, for a given n, there is an e < 0 such that B_n(t) = 0 for > >>> e < t <= 0. > >> There is no e<0 such that e<t and B_n(t)=0. That's simply false. > > > > Let's look at B_2 again. We can take e = -1/2. Then B_2(t) = 0 for e < t > > <= 0. Similarly, for any other given B_n, we can find an e that does > > what I wrote. > > Yes, okay, I misread that. Sorry. For each ball B_n that's true. For the > sum of balls n such that B_n(t)=1, it diverges as t->0. > > >>> In other words, B_n is not changing near zero. > > > >> Infinitely more quickly but not. That's logical. And wrong. > > > > Not sure what you mean. > > The sum increases without bound. > > >>> Now, V is the > >>> sum of the B_n. As t approaches zero from the left, V(t) grows without > >>> bound. In fact, given any large number M, there is an e < 0 such that > >>> for e < t < 0, V(t) > M. We also have that V(0) = 0 (as you agreed). > >>> > >>> Now the question: How do you explain the fact that V(t) goes from being > >>> very large for t a little less than zero to being zero when t equals > >>> zero even though none of the B_n are changing near zero? > >> I'll consider answering that when you correct the errors above. Sorry. I believe we now agree that what I wrote is correct. So, let me repeat my question: How do you explain the fact that V(x) goes from being very large for x a little less than zero to being zero when x equals zero even though none of the functions B_n are changing near zero? -- David Marcus
From: Tony Orlow on 28 Oct 2006 13:27 David Marcus wrote: > Tony Orlow wrote: >> David Marcus wrote: >>> Tony Orlow wrote: >>>> David Marcus wrote: >>>>> Tony Orlow wrote: >>>>>> David Marcus wrote: >>>>>>> Tony Orlow wrote: >>>>>>>> David Marcus wrote: >>>>>>>>> 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. >>>>>>>> Not really, but there is a big difference between this and the original >>>>>>>> experiment. If noon did exist here as the time of any event (insertion), >>>>>>>> then you would have an UNcountably infinite set of balls. Presumably, >>>>>>>> given only naturals, such that nothing is inserted at noon, by noon all >>>>>>>> naturals have been inserted, for the countable infinity. Then insertions >>>>>>>> stop, and the vase has what it has. The issue with the original problem >>>>>>>> is that, if it empties, it has to have done it before noon, because >>>>>>>> nothing happens at noon. You conclude there is a change of state when >>>>>>>> nothing happens. I conclude there is not. >>>>>>> So, noon doesn't exist in this case either? >>>>>> Nothing happens at noon, and as long as there is no claim that anything >>>>>> happens at noon, then there is no problem. Before noon there was an >>>>>> unboundedly large but finite number of balls. At noon, it is the same. >>>>> So, noon does exist in this case? >>>> Since the existence of noon does not require any further events, it's a >>>> moot point. As I think about it, no, noon does not exist in this problem >>>> either, as the time of any event, since nothing is removed at noon. It >>>> is also not required for any conclusion, except perhaps that there are >>>> uncountably many balls, rather than only countably many. But, there are >>>> only countably many balls, so, no, noon is not part of the problem here. >>>> As we approach noon, the limit is 0. We don't reach noon. >>> To recap, we add ball n at time -1/n. We don't remove any balls. With >>> this setup, you conclude that noon does not exist. Is this correct? >> I conclude that nothing occurs at noon in the vase, and there are >> countably, that is, potentially but not actually, infinitely many balls >> in the vase. No n in N completes N. > > Sorry, but I'm not sure what you are saying. Are you saying that what I > wrote is correct or are you saying it is not correct? I'll repeat the > question: > > We add ball n at time -1/n. We don't remove any balls. With > this setup, you conclude that noon does not exist. Is this correct? > Please answer "yes" or "no". > What do YOU mean by "exist"? Does anything happen which is proscribed if noon DOES arrive? No, not in this case. So, noon case "exist" or not. In the other case, the vase also does not empty before noon, and nothing happens at noon. So, then, why do you conjecture that it's empty AT noon? This version doesn't include any contingencies between insertions and removals. The original does. That's why it's a paradox. That's why it doesn't make sense, and why there is a logical error in it which must be resolved. See?
From: Tony Orlow on 28 Oct 2006 13:28 David Marcus wrote: > Tony Orlow wrote: >> David Marcus wrote: >>> Tony Orlow wrote: >>>> David Marcus wrote: >>>>> Tony Orlow wrote: >>>>>> David Marcus wrote: >>>>>>> Tony Orlow wrote: >>>>>>>> David Marcus wrote: >>>>>>>>> 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. >>>>>>>> Not really, but there is a big difference between this and the original >>>>>>>> experiment. If noon did exist here as the time of any event (insertion), >>>>>>>> then you would have an UNcountably infinite set of balls. Presumably, >>>>>>>> given only naturals, such that nothing is inserted at noon, by noon all >>>>>>>> naturals have been inserted, for the countable infinity. Then insertions >>>>>>>> stop, and the vase has what it has. The issue with the original problem >>>>>>>> is that, if it empties, it has to have done it before noon, because >>>>>>>> nothing happens at noon. You conclude there is a change of state when >>>>>>>> nothing happens. I conclude there is not. >>>>>>> So, noon doesn't exist in this case either? >>>>>> Nothing happens at noon, and as long as there is no claim that anything >>>>>> happens at noon, then there is no problem. Before noon there was an >>>>>> unboundedly large but finite number of balls. At noon, it is the same. >>>>> So, noon does exist in this case? >>>> Since the existence of noon does not require any further events, it's a >>>> moot point. As I think about it, no, noon does not exist in this problem >>>> either, as the time of any event, since nothing is removed at noon. It >>>> is also not required for any conclusion, except perhaps that there are >>>> uncountably many balls, rather than only countably many. But, there are >>>> only countably many balls, so, no, noon is not part of the problem here. >>>> As we approach noon, the limit is 0. We don't reach noon. >>> To recap, we add ball n at time -1/n. We don't remove any balls. With >>> this setup, you conclude that noon does not exist. Is this correct? >> I conclude that nothing occurs at noon in the vase, and there are >> countably, that is, potentially but not actually, infinitely many balls >> in the vase. No n in N completes N. > > Sorry, but I'm not sure what you are saying. Are you saying that what I > wrote is correct or are you saying it is not correct? I'll repeat the > question: > > We add ball n at time -1/n. We don't remove any balls. With > this setup, you conclude that noon does not exist. Is this correct? > Please answer "yes" or "no". > By the way, yes and no.
From: Tony Orlow on 28 Oct 2006 13:29 stephen(a)nomail.com wrote: > Tony Orlow <tony(a)lightlink.com> wrote: >> stephen(a)nomail.com wrote: >>> Tony Orlow <tony(a)lightlink.com> wrote: >>>> stephen(a)nomail.com wrote: >>>>> David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >>>>>> Tony Orlow wrote: >>>>>>> David Marcus wrote: >>>>>>>> Your question "Is there a smallest infinite number?" lacks context. You >>>>>>>> need to state what "numbers" you are considering. Lots of things can be >>>>>>>> constructed/defined that people refer to as "numbers". However, these >>>>>>>> "numbers" differ in many details. If you assume that all subjects that >>>>>>>> use the word "number" are talking about the same thing, then it is >>>>>>>> hardly surprising that you would become confused. >>>>>>> I don't consider transfinite "numbers" to be real numbers at all. I'm >>>>>>> not interested in that nonsense, to be honest. I see it as a dead end. >>>>>>> >>>>>>> If there is a definition for "number" in general, and for "infinite", >>>>>>> then there cannot both be a smallest infinite number and not be. >>>>>> A moot point, since there is no definition for "'number' in general", as >>>>>> I just said. >>>>>> -- >>>>>> David Marcus >>>>> A very simple example is that there exists a smallest positive >>>>> non-zero integer, but there does not exist a smallest positive >>>>> non-zero real. If someone were to ask "does there exist a smallest >>>>> positive non-zero number?", the answer depends on what sort >>>>> of "numbers" you are talking about. >>>>> >>>>> Stephen >>>> Like, perhaps, the Finlayson Numbers? :) >>> If they were sensibly defined then sure you could talk about them. >>> Nothing Ross has ever said has made any sense to me, and >>> I severely doubt there is any sense to it, but I could be wrong. >>> The point is, there are different types of numbers, and statements >>> that are true of one type of number need not be true of other >>> types of numbers. >>> >>> Stephen > >> Well, then, you must be of the opinion that set theory is NOT the >> foundation for all mathematics, but only some particular system of >> numbers and ideas: a theory. That's good. > > You do not understand what people mean when they say set theory > is the foundation for all mathematics, do you? Set theory > provides a set of primitives which can be used to describe > mathematics. Integers, real numbers, hyperreal numbers, imaginary > numbers, polynomicals, limits, functions, can all be described in > terms of set theory. Set theory is like assembly language. You can > use it to build up higher level concepts. Is it the only possible > foundation for mathematics? Of course not, but it currently appears to > be the best. > > Stephen > "Best" in which respect? Most ridiculous when it comes to oo? Yes, it's a riot.
From: David Marcus on 28 Oct 2006 13:31
imaginatorium(a)despammed.com wrote: > Anyway, I'm getting a giggle from hearing about you "reading" Robinson; > makes me wonder if that's how you can find anything of merit in > Lester's endless drivel - you just cruise through looking for an > attractive sentence here or there? If you don't realize that the words are supposed to convey rigorous mathematics, you can read a math book the same way that you do a novel. I fear that most undergraduates who are not math majors read their math books this way. -- David Marcus |