An uncountable countable set
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From: Tony Orlow on 27 Oct 2006 10:59 cbrown(a)cbrownsystems.com wrote: > Tony Orlow wrote: >> MoeBlee wrote: >>> Ross A. Finlayson wrote: >>>> Please identify something you see as incorrect or don't understand. >>> What's the point? People have been pointing out your incorrect >>> statements for years. You just sail right past every time. >>> >>> So really think you are correct and that you make sense? >>> >>> It must be nice. >>> >>> MoeBlee >>> >> For what it's worth, and I know this doesn't add a lot of credibility to >> Ross in your eyes, coming from me, but I think Ross has a genuine >> intuition that isn't far off with respect to what's controversial in >> modern math. > > I think this "controversy" is in fact the difference between relying > utterly on inutition (as Ross seems to do) and relying utterly on > logical conclusions of some explicit set of assumptions (which is the > domain of what I would call "mathematics") which is at issue here. > > Just as logical conclusions from some set of assumptions can at times > be in conflict with our intuitions; so it is also true that we can hold > intuitions which are not logically compatible with each other. > > In my use of the word, mathematics is within that domain of discussion > which eschews the latter in favor of the former. It is a specialisation > of the domain of logic, rather than of the domain of physics. > >> Sure, he gets repetitive and I don't agree with everything >> he says, but his cryptic "Well order the reals", which I actually >> haven't seen too much of lately, is a direct reference to his EF >> (Equivalence Function, yes?) between the naturals and the reals in >> [0,1). The reals viewed as discrete infinitesimals map to the >> hypernaturals, anyway, and his EF is a special case of my IFR. So, to >> answer your question, I think Ross makes some sense. > > Of course; there is nothing he says that is completely without /some/ > sort of sense. But I would say he is speaking /poetically/, not > mathematically; so in the context of sci.math, I can't respond to his > remarks. > > (So if you read this Ross; it's not that I don't respond to you because > I don't like you; I don't respond to you becuase I have no common > conceptual gound with you. You actually strike me as a pretty nice guy, > overalll. You're always polite and well meaning; that's all one can > ask!) > > A poet would say that "A rose is still a rose by any other name"; a > mathematician would say that "By 'a rose' we mean a repesentative of an > equivalence class of those herbacious plants having the following > properties: thorns, leaves found on alternating sides of the stem; > flowers having a a sweet smell, vaselike growth pattern, ... From this, > we can deduce that the assertion of the heavy metal ballad, 'Every rose > has its thorn', logically follows." > > Sometimes these different modes of thinking overlap; but more often, > they lead to different conclusions about what is or isn't the state of > affairs. Very true, but like the Zen archer, we have to train our intuitions, and when they are in harmony with the universe, the arrow hits its mark.:) > >> But, of course, >> coming from me, that probably doesn't mean much. :) >> > > On the contrary; I think you have accurately identified the nub of > whatever "controversy" has arisen regarding arguments you have asserted > in this and any other threads. > > Cheers - Chas > Gee, thanks, Chas. We've certainly had some interesting debates, where I've ended up screwing my brow trying to figure out what the crux of the disagreement was. It's not always easy to dig down to the logical roots and pinpoint the exact problem that leads to differing conclusions. Of course, I rather doubt that means you agree with all my points, but that okay. Agreeing all the time gets a little boring, eh? Have a nice day. Tony
From: Tony Orlow on 27 Oct 2006 11:05 cbrown(a)cbrownsystems.com wrote: > Tony Orlow wrote: >> cbrown(a)cbrownsystems.com wrote: >>> Tony Orlow wrote: >>>> Mike Kelly wrote: >>> <snip> >>> >>>>> My question : what do you think is in the vase at noon? >>>>> >>>> A countable infinity of balls. >>>> >>>> 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. >>> No one disagrees with the above statements. >>> >>>> The vase can only become empty through the removal of balls, >>> Note that this is not identical to saying "the vase can only become >>> empty /at time t/, if there are balls removed /at time t/"; which is >>> what it seems you actually mean. >>> >>> This doesn't follow from (1)..(8), which lack any explicit mention of >>> what "becomes empty" means. However, we can easily make it an >>> assumption: >>> >>> (T1) If, for some time t1 < t0, it is the case that the number of balls >>> in the vase at any time t with t1 <= t < t0 is different than the >>> number of balls at time t0, then balls are removed at time t0, or balls >>> are added at time t0. >>> >>>> 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. >>> Now your logical argument is complete, assuming we also accept >>> (1)..(8): If the number of balls at time t = 0, then by (7), (5) and >>> (6), the number of balls changes at time 0; and therefore by (T1), >>> balls are either placed or removed at time 0, implying by (5) and (6) >>> that there is a natural number n such that -1/n = 0; which is absurd. >>> Therefore, by reductio ad absurdum, the number of balls at time 0 >>> cannot be 0. >>> >>> However, it does not follow that the number of balls in the vase is >>> therefore any other natural number n, or even infinite, at time 0; >>> because that would /equally/ require that the number of balls changes >>> at time 0, and that in turn requires by (T1) that balls are either >>> added or removed at time 0; and again by (5) or (6) this implies that >>> there is a natural number n with -1/n = 0; which is absurd. So again, >>> we get that any statement of the form "the number of balls at time 0 is >>> (anything") must be false by reductio absurdum. >>> >>> So if we include (T1) as an assumption as well as (1)..(8), it follows >>> logically that the number of balls in the vase at time 0 is not >>> well-defined. >>> >>> Of course, we also find that by (1)..(8) and (T1), it /still/ follows >>> logically that the number of balls in the vase at time t is 0; and this >>> is a problem: we can prove two different and incompatible statements >>> from the same set of assumptions >>> >>> So at least one of the assumptions (1)..(8) and (T1) must be discarded >>> if we are to resolve this. What do you suggest? Which of (1)..(8) do >>> you want discard to maintain (T1)? >>> >>> Cheers - Chas >>> >> This is a very good question, Chas. Thanks. I'll have to think about it, >> and I'm rather tired right now, but at first glance it seems like it >> could be a sound analysis. I've cut and pasted for perusal when I'm >> sharper tomorrow. >> > > Here's some of my thoughts: > > When you say "noon doesn't occur"; I think "he doesn't accept (1): by a > time t, we mean a real number t" That doesn't mean t has to be able to assume ALL real numbers. The times in [-1,0) are all real numbers. > > When you say "if we always add more balls than we remove, the number of > balls in the vase at time 0 is not 0", I think "he doesn't accept (8): > if the numbers of balls in the vase is not 0, then there is a ball in > the vase." No, I accept that. There is no time after t=-1 where there is no ball in the vase. > > When you say "an infinite number of balls are removed at time 0", I > think "he does not agree with (6) if balls are removed at some time t, > they are removed in accordance with the problem statement: i.e. there > exists some natural number n s.t. n = -1/t and (some other stuff)". I didn't say that exactly. If 0 occurs, then all finite balls are gone, but infinite balls have been inserted such that 1/n=0 for those balls. So, at noon the vase is not empty, even if it occurs in the problem, which it doesn't. > > All these assertions follow a simgle theme: "If I require that my > statemnents be /logically/ consistent, does the given problem make > sense; and if so, what is a reasonable resonse?". > > Cheers - Chas > That there is a contradiction in your conclusion if you assume that all events must occur at some time, and that becoming empty is the result of events that happen in the vase. It cannot become empty until noon, when nothing happens to cause it.
From: Tony Orlow on 27 Oct 2006 11:15 Ross A. Finlayson wrote: > Tony Orlow wrote: > >> For what it's worth, and I know this doesn't add a lot of credibility to >> Ross in your eyes, coming from me, but I think Ross has a genuine >> intuition that isn't far off with respect to what's controversial in >> modern math. Sure, he gets repetitive and I don't agree with everything >> he says, but his cryptic "Well order the reals", which I actually >> haven't seen too much of lately, is a direct reference to his EF >> (Equivalence Function, yes?) between the naturals and the reals in >> [0,1). The reals viewed as discrete infinitesimals map to the >> hypernaturals, anyway, and his EF is a special case of my IFR. So, to >> answer your question, I think Ross makes some sense. But, of course, >> coming from me, that probably doesn't mean much. :) >> >> TOE-Knee > > Hi, > > What is this IFR, "inverse function rule"? I've heard you mention it. > Is it just general EF? > > Ross > Hey Ross! The Inverse Function Rule uses infinite-case induction to finely order infinite sets of reals mapped from a standard set, N. Where there is a bijection between N and a set S using f(n)=s, there is a mapping from S to N using g(s)=n, where g(f(x))=f(g(x)) (inverse functions for the bijection). The size of the set S over the interval [a,b] is given by floor(g(b)-g(a)+1). (I think I wrote that correctly). This works for all finite sets of reals. The number of square roots, for instance, between 1 and 100 is floor(100^2-1^2+1), 10000 square roots, from sqrt(1) to sqrt(10000). IFR can easily be used to show that the evens are half as numerous as the naturals, and other interesting "facts". EF is the special case of IFR mapping the naturals in [0,oo) to the reals in [0,1), using the mapping function f(n)=n/oo. Isn't that how you define the equivalency function? Given this mapping, we can say g(s)=s*oo, so that over the entire real line, we have oo^2 reals, oo in each unit interval, over oo unit intervals. Does that sound about right? Tony
From: Tony Orlow on 27 Oct 2006 11:18 cbrown(a)cbrownsystems.com wrote: > Ross A. Finlayson wrote: >> Tony Orlow wrote: >> >>> For what it's worth, and I know this doesn't add a lot of credibility to >>> Ross in your eyes, coming from me, but I think Ross has a genuine >>> intuition that isn't far off with respect to what's controversial in >>> modern math. Sure, he gets repetitive and I don't agree with everything >>> he says, but his cryptic "Well order the reals", which I actually >>> haven't seen too much of lately, is a direct reference to his EF >>> (Equivalence Function, yes?) between the naturals and the reals in >>> [0,1). The reals viewed as discrete infinitesimals map to the >>> hypernaturals, anyway, and his EF is a special case of my IFR. So, to >>> answer your question, I think Ross makes some sense. But, of course, >>> coming from me, that probably doesn't mean much. :) >>> >>> TOE-Knee >> Hi, >> >> What is this IFR, "inverse function rule"? I've heard you mention it. > > That which can be done; can be undone. > > Cheers - Chas > The inverse mapping gives the number of elements over a given value range, which can be applied to the entire set of reals, providing intuitive results (like half as many events as naturals) for infinite sets when that real range is considered constant. Tony
From: Tony Orlow on 27 Oct 2006 11:26
David Marcus wrote: > Tony Orlow wrote: >> MoeBlee wrote: >>> Tony Orlow wrote: >>>> Uh, if Robinson's thesis is built upon transfinite set theory, then that >>>> is evidence right there that it's inconsistent, since you have a >>>> smallest infinity, omega, but Robinson has no smallest infinity. >>> We JUST agreed that 'smallest infinity' means two different things when >>> referring to ordinals and when referring to certain kinds of other >>> orderings! It is AMAZING to me that even though I took special care to >>> make sure this was clear, and then you agreeed, you NOW come back to >>> conflate the two ANYWAY! >> Ahem. I said that Robinson's analysis seems to have nothing to do with >> transfinitology. They appear to be unrelated. However, they cme to two >> very different conclusions regarding a basic question: is there a >> smallest infinite number? It seems clear to me there is not, for the >> very same reason that Robinson uses: if there is an infinite number, you >> can subtract 1 and get a different, smaller infinite number. It's the >> same logic y'all use to argue that there's no largest finite. It's >> correct. The Twilight Zone between finite and infinite CANNOT really be >> pinpointed that way. >> >> So, that's a verrry basic discrepancy. There is clearly a contradiction >> between the two theories. They can't both be right about that, can they? > >> Is there, and at the same time is not, a smallest infinite number? >> >> Tach me more about mathematical logic and consistency. > > We've been trying, but you don't seem to want to learn. I guess you missed the sarcastic tone there. I have a hard time taking lessons in logic from people who think things can happen in some kind of time without being anywhere in time. The contradiction here is glaring. > > 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. > > The two theories can both be right about the "numbers" that they are > talking about, since they are talking about different things. To avoid > confusion, the simplest solution is to be specific. We could say that > there is a smallest infinite ordinal, but there is no smallest infinite > non-standard real number. When phrased this way, no contradiction is > apparent. The apparent contradiction is due to your using "infinite > number" to mean both "infinite ordinal" and "infinite non-standard real > number". > > A similar problem would arise if you used the word "cat" to refer to > both domestic cats and lions and were to say that "Cats make good pets". > Thanks for the logic lesson, but if transfinitologists are going to claim to have a "correct" answer, and that any other interpretation is, as Virgil would say, WRONG!!!, well then, they should feel obligated to explain what's wrong with non-standard analysis, infinitesimals, infinite series, limits, infinite-case induction, and other approaches. If set theory is just a theory that's interesting in itself, fine, but the pompous attitude of set theorists towards anyone disagreeing with their nonsense only invites controversy. So, I don't really apologize for putting down transfinitology. What goes around comes around. :) TOny |