An uncountable countable set
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From: Tony Orlow on 28 Oct 2006 13:35 imaginatorium(a)despammed.com wrote: > 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? I suppose that depends on whether he liked me. I don't think hed find my ideas objectionable. He was obviously an original thinker. 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? Why don't you tell me? Did Boole? Did Cantor? 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. Yes, the surreals. 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. > That's worth a try. > 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 > A function in poetry?
From: Tony Orlow on 28 Oct 2006 13:43 David Marcus wrote: > 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? > No, I don't care, but I'll do it anyway. :) Just kidding. Of course I care, or I wouldn't waste my time. I am beginning to realize just how much trouble the axiom of extensionality is causing here. That is what you're using, here, no? The sets are "equal" because they contain the same elements. That gives no measure of how the sets compare at any given point in their production. Sets as sets are considered static and complete. However, when talking about processes of adding and removing elements, the sets are not static, but changing with each event. When speaking about what is in the set at time t, use a function for that sum on t, assume t is continuous, and check the limit as t->0. Then you won't run into silly paradoxes and unicorns.
From: Tony Orlow on 28 Oct 2006 13:45 stephen(a)nomail.com wrote: > Tony Orlow <tony(a)lightlink.com> wrote: >> stephen(a)nomail.com wrote: >>> Tony Orlow <tony(a)lightlink.com> 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. >>>> Tony >>> What balls and vase? Are you really incapable of answering a simple >>> question about sets without bringing in total irrelevancies? >>> >>> Stephen > >> If the problem is irrelevant to your question, then your question is >> irrelevant to the problem, eh? > > I explained the purpose of my question. If you are not going to bother > to read what I write and to respond to what I write, you should > not bother responding at all. > > Stephen The purpose being to try to obscure details of the stated problem. Why should I entertain that in a discussion of this? The sequence matters, the limit matters, your completed potential infinity does not.
From: imaginatorium on 28 Oct 2006 13:49 Tony Orlow wrote: > 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? I'm agog to hear Ross's answer... > 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. Well, isn't 'iota' the "smallest positive Freal"? And isn't 2 also a Freal? So what is iota/2? > 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? No, no, concentrate on the poetry. "Why the question?" you ask - some might think it was just to make you look silly and ignorant, but in fact that's not necessary at all. Brian Chandler http://imaginatorium.org
From: Lester Zick on 28 Oct 2006 13:50
On Fri, 27 Oct 2006 18:47:55 +0000 (UTC), stephen(a)nomail.com wrote: >David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >> Lester Zick wrote: >>> On Fri, 27 Oct 2006 16:30:04 +0000 (UTC), stephen(a)nomail.com wrote: >>> >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. >>> >>> So non zero integers are not real? > >> That's a pretty impressive leap of illogic. > >Using Lester IllLogic it is easy to prove that 2 is not prime. >2 is the largest even prime integer. There is no largest prime >integer. Therefore 2 cannot be prime. Why prove that, Stephen, when I can rely on your proof that dr=v. ~v~~ |