An uncountable countable set
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From: Lester Zick on 28 Oct 2006 15:06 On Sat, 28 Oct 2006 16:37:39 +0000 (UTC), 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. You know I will claim to be an expert in IBM Assembly Language and I find the analogy with modern math set analytical techniques extremely offensive. If there were anything like the consistency in modern set math required of Assembly and machine language design and implementation, modern math set analytical techniques wouldn't be nearly so fucked up. ~v~~
From: Lester Zick on 28 Oct 2006 15:12 On Fri, 27 Oct 2006 20:13:36 -0400, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: >> On Fri, 27 Oct 2006 00:07:16 -0400, Tony Orlow <tony(a)lightlink.com> >> wrote: >> >>> MoeBlee wrote: >>>> Tony Orlow wrote: >>>>> I share your and Godel's concerns about point set theory >>>> Oh how rich. How veddy veddy scholarly Mr. Orlow sounds when he says >>>> such things, "I share Godel's concerns about point set theory." Too bad >>>> Mr. Orlow doesn't know a single ding dang thing about Godel, or Godel's >>>> concerns, or mathematical logic, or set theory, or point set topology, >>>> or topology. >>>> >>>> MoeBlee >>>> >>> Wow, Lester's really getting under your skin, isn't he? He cracks me up. :) >> >> Of course a little levity, like motions to adjourn, is always in >> order, Tony. Mathematikers take all this drollery way too seriously. >> They get all huffy and self righteous when forced to actually explain >> things they're used to outsourcing to books. Thanks for the comment. >> >> ~v~~ > >A motion to adjourn!! Now that's an idea! Thanks, Lester! Damn court!! Once used the technique just after a particularly contentious condo board meeting election to limit debate. We won the vote and I just stood up and said "motion to adjourn" and the motion carried by acclamation. Very, very, useful. ~v~~
From: Tony Orlow on 28 Oct 2006 15:20 imaginatorium(a)despammed.com wrote: > Tony Orlow wrote: >> imaginatorium(a)despammed.com wrote: > > <snip> > >>> 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. > > What do you mean by "objectionable"? That he would object to any of my conclusions, or even my logic. How can whether you like someone > affect whether you think they are making mathematical sense or not? Emotions and intuition play a big role in "rational" decisions, whether you like it or not. As > you say, Abraham Robinson was an original thinker, but I don't think I > would be insulting him to say no more original in particular than > thousands of other mathematicians all brighter than me. > That sounds like some kind of compliment, though I would say he's probably more original than "thousands" of known mathematicians. > Do you think you are an "original thinker"? Do you? It seems like almost everything that comes out of my mouth (via my fingers and this computer) is objectionable to someone here, and yet, it all fits together. I get the feeling Robinson would tire of my lack of rigor, but probably agree with most of my conclusions, if he were here. I try to think on my own, without excessive guidance. The road less traveled, or the bushwhack, yields the seldom-seen. :) What conclusion of his did I question? Oh yeah, that there was a prime number greater than any finite number. That's the only one, so I'll take it as a possibility for now...what do you think? > > I've more or less said this before, but I think one reason I'm just > light-years (in the popular sense) ahead of you on all this is that I > have sat in undergraduate algebra lectures when Simon Norton asked > questions. I have some concept of how much dimmer I am than a bright > mathematician. You have only the proud memory of being top all the time > in the village school (as I was, of course). > Oh. Did you get any 800's on your college entrance exams? How long does it take to traverse a light-year, anyway? >> 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? > > Did Boole or Cantor what? > Agree with anything I'm talking about? Never mind. >> 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. > > Could be. > >>> 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? >>> >> A function in poetry? > > Yes, you're writing poetry, we gather. What, poetically speaking, do > you see a function as being? > > Brian Chandler > http://imaginatorium.org > A function is a relation between one set X and another Y, such that each xeX has one yeY associated with it, over the range allowed for x. If the function is invertible over a given range of y, then each yeY has a unique xeX associated with it, and here is an inverse function, and a bijection, at least over the stated ranges of x and y. The little flowers reach for the sunny skies, while the butterflies flit joyfully, smelling the sweet flowers, without a care. How was that? Pretty lame, eh? Tony
From: David Marcus on 28 Oct 2006 15:23 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: > >>>>>>>> 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? I'm afraid I don't see. As for what I mean by "exist", you would have to give me a complete sentence that I've used that contains the word "exist". I've never said "noon exists" or "noon doesn't exist". You were the one who said "noon does not exist". I was trying to clarify in which problems you would say "noon exists" and in which problems you would say "noon does not exist". I already know that in the original problem you say that "noon does not exist". For Mike's problem (add ball n at time -1/n, don't remove any balls), does "noon exist" or does "noon not exist"? Please answer "yes" or "no". -- David Marcus
From: stephen on 28 Oct 2006 15:22
Tony Orlow <tony(a)lightlink.com> wrote: > stephen(a)nomail.com wrote: <snip> >> >> What does that have to do with the sets IN and OUT? IN and OUT are >> the same set. You claimed I was losing the "formulaic relationship" >> between the sets. So I still do not know what you meant by that >> statement. Once again >> IN = { n | -1/(2^(floor(n/10))) < 0 } >> OUT = { n | -1/(2^n) < 0 } >> > I mean the formula relating the number In to the number OUT for any n. > That is given by out(in) = in/10. What number IN? There is one set named IN, and one set named OUT. There is no number IN. I have no idea what you think out(in) is supposed to be. OUT and IN are sets, not functions. >> Given that for every positive integer -1/(2^(floor(n/10))) < 0 >> and -1/(2^n) < 0, both sets are in fact the same set, namely N. >> >> Do you agree, or not? Or is it the case that the >> "formulaic between the sets is lost." >> ? >> >> Stephen > The formulaic relationship is lost in that statement. When you state the > relationship given any n, then the answer is obvious. What relationship? For a given n, -1/(2^(floor(n/10))) < 0 if and only if -1/(2^n) < 0. The two conditions are logically equivalent for positive integers. If n is a member of IN, n is a member of OUT, and vice versa. What other relationship do you think there is between -1/(2^(floor(n/10))) < 0 and -1/(2^n) < 0 ?? Do you think there exists a positive integer n such that -1/(2^(floor(n/10))) < 0 and -1/(2^n) >= 0 Stephen |