An uncountable countable set
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From: David Marcus on 28 Oct 2006 12:35 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". -- David Marcus
From: stephen on 28 Oct 2006 12:37 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
From: imaginatorium on 28 Oct 2006 12:41 Tony Orlow wrote: > imaginatorium(a)despammed.com wrote: > > Forgive me if I blunder in on Chas's carefully constructed argument, > > but... <snip> > > Here's something I don't understand. I believe, Tony, that you think > > that if every one of these pofnat-labelled balls is inserted one minute > > earlier (so *informally*, instead of a "sliver" tapering to zero width, > > we have an endless boomerang shape, with the width tending to 1 as you > > go ever up the y-direction), then at noon no balls are left. Presumably > > because once all the balls are IN (at 11:59), there is only removal, > > tick, tick, tick, ... and all are gone at noon. But why doesn't this > > stuff about "noon being incompatible" apply here too? Is there a > > *principled* way in which you determine which arguments apply at > > particular points? (I'm sure it appears to most non-cranks here that > > there isn't.) Taking your comments slightly out of order: > Which of my statements [below (now)] do you find objectionable, and why? That > would be helpful to know. "Objectionable" sounds slightly unpleasant. As poetry it's, well, not exactly very stimulating to me, since it's based on a rather pathetic inability to grasp some rather elementary concepts (like the unending set of finite counting-numbers). As mathematics, it's wrong where it isn't meaningless. > That's very simple, Brian. The limit of balls as n->noon is 0. Define your interpretation of "limit". (Oh, dear, we know you don't have the basic logical tools to do that.) > That's > not the case in the original problem. There, there is no limit. The sum > diverges, as it does in this case until 11:59. Wait a minute - there's a fragment there that's correct. > Those points of > infinitely quick iterations ultimately include an uncountable number of > iterations, unless their countability is specified, in which case they > do not reach those points of uncountability. > > Additionally, we have the fact that, if property p applies at all times > before time t, and does not change state at time t, then it continues to > apply at time t. In the case where we have a countably infinite number > of balls at t=-1, no matter how they got there, and start removing them > in Zeno fashion, we can conceptually empty the vase by time 0. Once time > 0 is there, nothing else happens. So if all balls have been removed by > then, that's the way it is at time 0. If all balls haven't been removed, > due to a condition of the problem under consideration, then it's not > empty. All balls have NOT been removed before noon in the gedanken. AT > noon, no balls are removed. The vase can only be empty after having been > non-empty if removals have occurred between those two times. Rest is just babble. 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? Here's another question - I' ve asked before, so I don't hold out much hope of an answer, but anyway: Suppose you could ask Abraham Robinson what he thought of your ideas: what do you suppose he would say? Do you think he might just latch onto the IFR, N^L=S (whaddeveritwas), your T'rrible numbers, the twilight zone, etc., or do you suppose he would dismiss it as total nonsense? Do you actually think you might ever find a real mathematician who thought there was anything at all of merit in what you have to say? Here's a suggestion: Robinson died in 1974, at a rather early age, but Conway is very much still alive - he's a very helpful person (I've seen him laboriously explaining something rather elementary on a geometry list I think it was), and he also created another non-standard collection of numbers, with what's more, constructions like omega/2 in it. So send him an email, of not more than say 200 words, setting out your most basic ideas; ask him if he thinks you're wasting your time? Don't forget to mention that you are quite sure the set of natural numbers is not infinite. Meanwhile, I've got just a bit tired of asking the same unanswered questions over and over again. I mean, what is a _function_ in poetry? Brian Chandler http://imaginatorium.org
From: David Marcus on 28 Oct 2006 12:47 Tony Orlow wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> David Marcus wrote: > >>> Tony Orlow wrote: > >>>> David Marcus wrote: > >>>>> Tony Orlow wrote: > >>>>>> stephen(a)nomail.com wrote: > >>>>>>> What are you talking about? I defined two sets. There are no > >>>>>>> balls or vases. There are simply the two sets > >>>>>>> > >>>>>>> IN = { n | -1/(2^floor(n/10)) < 0 } > >>>>>>> OUT = { n | -1/(2^n) < 0 } > >>>>>> For each n e N, IN(n)=10*OUT(n). > >>>>> Stephen defined sets IN and OUT. He didn't define sets "IN(n)" and "OUT > >>>>> (n)". So, you seem to be answering a question he didn't ask. Given > >>>>> Stephen's definitions of IN and OUT, is IN = OUT? > >>>> Yes, all elements are the same n, which are finite n. There is a simple > >>>> bijection. But, as in all infinite bijections, the formulaic > >>>> relationship between the sets is lost. > >>> Just to be clear, you are saying that |IN - OUT| = 0. Is that correct? > >>> (The vertical lines denote "cardinality".) > >> Um, before I answer that question, I think you need to define what you > >> mean by "|IN - OUT|" =0. How are you measuring IN and OUT, and how do > >> you define '-' on these "numbers"? > > > > IN and OUT are sets, not "numbers". For any two sets A and B, the > > difference, denoted by A - B, is defined to be the set of elements in A > > that are not in B. Formally, > > > > A - B := {x| x in A and x not in B} > > > > Note that the difference of two sets is again a set. For any set, the > > notation |A| means the cardinality of A. So, saying that |A| = 0 is > > equivalent to saying that A is the empty set. In particular, for any set > > A, we have |A - A| = 0. > > Sure, in the sense of containing the same n's, they are the same set. > That entirely ignores the rates at which those sets are processed over > time, which is expressed in your floor(n/10), causing ten times as many > in IN as in OUT, for any given value range of the two functions defining > them. If you have n balls in, that took n/10 steps. If you have n balls > out, that took n steps. The vase accumulates more balls at every step. > So, the axiom of extensionality doesn't address this matter of measure > in the sequence, but tries to cover it up in typical set theoretic fashion. Let me recap the discussion: Stephen suggested the following problem (which may or may not have some relationship to any other problem that anyone has ever considered): Define the following sets of natural numbers. IN = { n | -1/(2^floor(n/10)) < 0 }, OUT = { n | -1/(2^n) < 0 }. What is |IN\OUT|? Stephen suggested that this problem would "not cause any fuss at all", i.e., everyone would agree what the answer is. In reply, you wrote, "It would still be inductively provable in my system that IN=OUT*10." We all took this to mean that you disagreed that |IN\OUT| = 0. Now, you seem to be saying that you agree that |IN\OUT| = 0. Care to clear up this confusion? -- David Marcus
From: Tony Orlow on 28 Oct 2006 12:52
imaginatorium(a)despammed.com wrote: > Tony Orlow wrote: >> Virgil wrote: >>> In article <454286e8(a)news2.lightlink.com>, >>> 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? :) >>> Any set of numbers whose properties are known. Are the properties of >>> "Finlayson Numbers" known to anyone except Ross himself? >> Uh, yeah, I think I understand what his numbers are. Perhaps you've seen >> our recent exchange on the matter? They are discrete infinitesimals such >> that the sequence of them within the unit interval maps to the naturals >> or integers on the real line. Is that about right, Ross? > > Do they form a field? > > Brian Chandler > http://imaginatorium.org > Good question. Ross? What says you to this? Here's what Wolfram says applies to fields: http://mathworld.wolfram.com/FieldAxioms.html My understanding, looking at each of these axioms, is that they apply to this system, and that it's a field. I suppose you would want proof of each such fact, but perhaps you could move the process along by suggesting which of the ten axioms you think the Finlayson Numbers might violate? After all, if you find only one, then you've proved your point. Not that I am necessarily concerned with whether they form a ring or a field or whatever, until that becomes important. Is it? Why the question? Tony |