An uncountable countable set
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From: Tony Orlow on 28 Oct 2006 09:27 David Marcus wrote: > Tony Orlow wrote: >> MoeBlee wrote: >>> Tony Orlow wrote: >>>> Eat me. Do you maintain that the two theories are compatible with each >>>> other? Is there, and also not, a smallest infinity. >>> They're not in conflict, becuase 'smallest infinite' means something >>> DIFFERENT in the different contexts. How many times will I say that >>> while you STILL refuse to hear it? >> So, either smallest has two meanings, or infinite has tow meanings, or >> both. Would you like to elucidate the matter by enumerating the various >> definitions of "small" and "infinite"? A table might be nice... > > As many have said, "infinite" has many meanings. I'm afraid it isn't > practical to produce a table. > How about a list? ;)
From: Tony Orlow on 28 Oct 2006 09:51 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? > > 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. However, lim(t->0: sum(B_n| B_n(t)=1))=oo. Why do you conveniently forget that fact? >>> 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. >
From: Tony Orlow on 28 Oct 2006 10:00 Virgil wrote: > 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. Yeah. Uh, I'll try harder.....
From: Tony Orlow on 28 Oct 2006 10:10 RLG wrote: > "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 > > > Yes, I am well aware of that. Apparently you are not aware of my position on the subject. Bijections alone do not prove equinumerosity for infinite sets. Cardinality is a rough measure of equivalence class, not a precise measure of the size of a set. In order to precisely compare such infinite sets of values, one must measure over a common infinite value range formulaically. Then we easily get that half the naturals are even, and other pleasant, intuitive notions. Express the number of balls as a function on n, and it's f(n)=9n. As n->oo, f(n)->oo. n(t)=floor(1/(0-t)). As t->0, n(t)->oo, and f(n(t))->oo. It's that simple. The axiom of extensionality cannot be applied here. It provides no measure with respect to t. T
From: Tony Orlow on 28 Oct 2006 10:17
Mike Kelly wrote: > 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? Conceptually, sure. > > 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? Nothing is allowed to happen at noon in either experiment. They both end up with countably many balls in the vase at noon. The experiment's stated sequence logically precludes that the vase become empty. > >> 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. > Very good! There are an unboundedly large, but finite, number of finite natural numbers. That's what "countable" infinity means: no element is infinitely far from any other. I'm impressed, Mike! Skip right to the front of the class and take a bow. You get to clean the chalkboard erasers! :) Tony |