An uncountable countable set
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From: Randy Poe on 26 Oct 2006 23:39 Tony Orlow wrote: > Randy Poe wrote: > > Tony Orlow wrote: > >> Randy Poe wrote: > >>> Tony Orlow wrote: > >>>> t=0 is precluded by n e N and t(n) = -1/n. > >>> Really? > >>> > >>> I hope you will accept as true that noon occurred yesterday. > >>> > >>> Let's define noon yesterday as t=0. Now let's define a set of values > >>> t_n = -1/n seconds for n=1, 2, 3, ... , that is, for all FINITE > >>> natural numbers n. > >>> > >>> Has my giving these names to those times somehow > >>> precluded noon yesterday from occurring? Retroactively? > >>> > >> Do you live in the gedanken? Oy. Nothing happens at noon. > > > > Did noon occur? > > Not within the constraints of the experiment. Nothing is allowed to > happen at noon. > > > > >> Your desired result does not happen before noon. > > > > What desired result? I didn't have an experiment, I > > just named a bunch of times. Is noon "precluded" by > > my defining that countable set of variables? > > > > Can anything happen at noon? What can change, in the vase, at noon? What vase? I'm just naming a bunch of times that occurred before noon yesterday. Infinitely many other times also occurred, at noon and after, but they happen not to be named. > What is the state before noon? Of what? I've just defined a set {t_n = -1/n, n=1, 2, ...}. No states, no vase, no balls. - Randy
From: Virgil on 26 Oct 2006 23:42 In article <45417528$1(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> 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. 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. :) Coming from TO it damns Ross.
From: Tony Orlow on 26 Oct 2006 23:52 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. Cheers - Tony
From: Virgil on 26 Oct 2006 23:53 In article <45417714(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> MoeBlee wrote: > >>> Tony Orlow wrote: > >>>>> But none of Robinson's non-standard numbers are cardinalities. > >>>> No kidding. They actually make sense. > >>> You said you have not properly studied chapter II in the book - the one > >>> that includes mathematical logic, model theory, and set theory (does it > >>> not? I'll stand corrected if it doesn't). What are you going to say > >>> when you find out that what you say makes sense rests on a foundation > >>> of set theory that you say doesn't make sense? Or, if I'm incorrect > >>> that Robinson's work in non-standard analysis doesn't presuppose basic > >>> mathematical logic, model theory, and set theory, then I'll benefit by > >>> being corrected in my admittedly cursory understanding of the matter. > >>> > >>> MoeBlee > >>> > >> Uh, if Robinson's thesis is built upon transfinite set theory, > > > > Nonstandard analysis is built on mathematical logic as is 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. > >> Robinson doesn't use ordinals or cardinals that I've seen. He basically > >> defines what a well-formed formula is in his system, which is a little > >> more restrictive that some others, it seems, and uses the language to > >> extend what can be said about finite n in N to include infinite n in *N. > > > > That doesn't mean that mathematical logic is being properly applied in > the transfinite case. Since Robinson's ultrafilters are highly transfinite, if the infinite is being improperly applied anywhere, that would be where. > There is yet some debate as to which logical > constructions are valid and which are not in that situation, as I see > it. That something conf0rms to the way "TO sees it" is enough to condemn any theory. >It's hardly cut and dried. If Euclidean geometry as universal truth > can be overturned after more than 2,000 years, what makes one so sure > that transfinitology is "correct" after less than 200, especially when > it's at odds with all the intuitions we've developed over those > thousands of years.? What was "overturned"? The parallel postulate, which has been viewed with some suspicion by mathematicians since Euclid's time, finally turned out not to be necessary for geometry. > I think the problem is solved most immediately by adopting the method of > inductive proof in the infinite case, provided that one is proving > either an equality between formulas, or an inequality where the > difference establishing it does not have a limit of 0 as n->oo. The possibility of TO thinking straight about anything like this, after all the evidence he has presented that he cannot, is less than miniscule and not greater than infinitesimal.
From: cbrown on 26 Oct 2006 23:56
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. > 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 |