An uncountable countable set
in [Math]
|
Prev: integral problem
Next: Prime numbers
From: Virgil on 9 Oct 2006 13:45 In article <452a6912(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: I'm not much into answering the same > question multiple times per person. > > TOny TO's not much into answering hard questions at all.
From: Virgil on 9 Oct 2006 13:54 In article <452a6a68(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <e3ab0$4529ffd9$82a1e228$22026(a)news2.tudelft.nl>, > > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > > >> Virgil wrote: > >> > >>> In article <1160332251.241188.301420(a)i3g2000cwc.googlegroups.com>, > >>> Han.deBruijn(a)DTO.TUDelft.NL wrote: > >>>> But ah, a picture says more than a thousand words: > >>>> > >>>> http://hdebruijn.soo.dto.tudelft.nl/jaar2006/ballen.jpg > >>> But the function one should be graphing is not continuous at any of the > >>> time of entry or exit of any ball, nor at noon (except it is > >>> right-continuous at noon). > >> Correct. I have "continuized" the function. The true (discrete) function > >> should be like a staircase with a stair at each red ball. But a discrete > >> version likewise explodes (i.e. not implodes) at noon. > > > > Actually not precisely "at" noon. > > And not before and not after. Like, you mean, never, right? Not to anyone who can read how I defined the function B(t). One can choose a sequence of times, t_n, before noon which converge to noon such that the sequence B(t_n) is defined at all t_n, strictly increasing and unbounded. But those sequences do nor affect the value of B(t) AT t = noon. > > > > > Let A_n(t) be equal to > > 0 at all times, t, when the nth ball is out of the vase, > > 1 at all times, t, when the nth ball is in the vase, and > > undefined at all times, t, when the nth ball is in transition. > > > > Note that noon is not a time of transition for any ball, though it is a > > cluster point of such times. > > Thent he vase does not empty at noon, nor does it empty before, or > after. Got an option 4? It IS empty at noon, since A_N(noon) = 0 for all n in N so B(noon) = Sum_{n in N} A_n(noon) = Sum_{n in N} 0 = 0 > > > > > let B(t) = Sum_{n in N} A_n(t) represent the number of balls in the vase > > at any non-transition time t. > > > > B(t) is clearly defined and finite at every non-transition point, as > > being, essentially, a finite sum at every such non-transition point. > > > > Further, A_n(noon) = 0 for every n, so B(noon) = 0. > > Similarly when t > noon, every A_n(t) = 0, so B(t) = 0
From: Virgil on 9 Oct 2006 13:56 In article <452a6b11(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > I like mine better: > > > > Let A_n(t) be equal to > > 0 at all times, t, when the nth ball is out of the vase, > > 1 at all times, t, when the nth ball is in the vase, and > > undefined at all times, t, when the nth ball is in transition > > (times at which a ball changes location). > > > > Note that noon is not a time of transition for any ball, though it is a > > cluster point of such times. > > > > let B(t) = Sum_{n in N} A_n(t) represent the number of balls in the vase > > at any non-transition time t. > > > > B(t) is clearly defined and finite at every non-transition point, as > > being, essentially, a finite sum at every such non-transition point, > > and is undefined at each transition point. > > > > Further, A_n(noon) = 0 for every n, so B(noon) = 0. > > Similarly when t > noon, every A_n(t) = 0, so B(t) = 0 > > It's nice that you like yours. I'd hate to see you treating it badly. I > don't mean to be mean to it. It's just that it kind of smells and keeps > trying to crawl on my lap and drool. If you could keep it off the > furniture, I'd appreciate it. > > Tony When TO has nothing relevant to say, he tries to be cute. But he doesn't do very well at that, either.
From: David R Tribble on 9 Oct 2006 14:04 Tony Orlow wrote: >> That doesn't seem "real", and the axiom of choice aside, I don't see >> there being any well ordering of the reals. The closest one can come is >> the H-riffic numbers. :) > David R Tribble wrote: >> Hardly. The H-riffics are a simple countable subset of the reals. >> Anyone mathematically inclined can come up with such a set. > Tony Orlow wrote: > You never paid enough attention to understand them. They cover the reals. They omit an uncountable number of reals. Any power of 3, for example, which you never showed as being a member of them. Show us how 3 fits into the set, then we'll talk about "covering the reals".
From: Lester Zick on 9 Oct 2006 14:08
On Sun, 08 Oct 2006 22:07:23 -0400, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: >> On Sun, 08 Oct 2006 15:12:28 -0400, Tony Orlow <tony(a)lightlink.com> >> wrote: >> >>> Lester Zick wrote: >>>> On Sat, 07 Oct 2006 23:45:23 -0400, Tony Orlow <tony(a)lightlink.com> >>>> wrote: >>>> >>>>> mueckenh(a)rz.fh-augsburg.de wrote: >>>>>> Tony Orlow schrieb: >>>>>> >>>>>>> mueckenh(a)rz.fh-augsburg.de wrote: >>>>>>>> Tony Orlow schrieb: >>>>>>>> >>>>>>>> >>>>>>>>>>> Why not? Each and every number of the list terminates. That one is a number >>>>>>>>>>> that does *not* terminate. >>>>>>>>>>> >>>>>>>>>>> > If you think that 0.111... is a number, but not in the list, >>>>>>>>>> It is me who insists that it is not a representation of a number. >>>>>>>>> Well, Wolfgang, that sets us apart, though I agree it's not a "specific" >>>>>>>>> number. It's still some kind of quantitative expression, even if it's >>>>>>>>> unbounded. Would you agree that ...333>...111, given a digital number >>>>>>>>> system where 3>1? >>>>>>>> That is the similar to 0.333... > 0.111.... But all these >>>>>>>> representations exist only potentially, in my opinion. The difference >>>>>>>> is, that 0.333... can be shown to lie between two existing numbers, so >>>>>>>> we can calculate with it, while for ...333 this cannot be shown. >>>>>>> I think it can be shown to lie between ...111 and ...555, given that >>>>>>> each digit is greater than the corresponding digit in the first, and >>>>>>> less than the corresponding digit in the second. >>>>>> Yes, but only if we define, for instance, >>>>>> >>>>>> A n eps |N : 111...1 < 333...3 where n digits are symbolized in both >>>>>> cases. >>>>>> >>>>>> This approach would be comparable with the "measure" which gives >>>>>> >>>>>> A n eps |N : |{1,2,3,...,2n}| = 2*|{2,4,6,...,2n}|. >>>>>> >>>>>> I don't know whether these definitions are of any use, but I am sure >>>>>> that they are not less useful than Cantor's cardinality. >>>>>> >>>>>> Regards, WM >>>>>> >>>>>> . >>>>>> >>>>> My opinion about that is, if one wants to talk about what happens "at >>>>> infinity", that's the way that makes sense, not the measureless way of >>>>> abstract set theory. I trust limit concepts, but not limit ordinals. >>>> Tony, would it be fair to characterize what you're trying to say as >>>> that there is some kind of positive/negative crossover at infinity >>>> such that {-00, . . .,-1, 0, +1, . . . +00}? I haven't really been >>>> following this thread too closely so I'm trying to understand what >>>> you're after here in basic terms instead of the exact arguments >>>> involved. >>>> >>>> ~v~~ >>> Hi Lester, how's thangs? >> >> Hey, Tony, pretty much as usual. >> >>> I wasn't saying that right here, but agreeing with Wolfgang that limit >>> concepts make sense, while the transfinitological approach doesn't. What >>> I did say was that there are two ways to view the number line, one where >>> oo and -oo are polar opposites, and a number circle where they are the >>> same. >> >> But there is no single real number line. There's a single rational/ >> irrational line but not even a single transcendental line, Tony. So on >> the surface I don't see what this speculation has to recommend it. And >> more to the point I don't see any way to effect a crossover in >> mechanical terms. > >I've certainly heard you discuss your views on the number line, and how >pi lies on a curve and rationals lie on straight lines, etc. To me, it >sounds like a matter of construction, or meaning of the number, but not >one of raw quantity. In terms of raw quantity on the real number line, >they all obey the law of trichotomy, for any a and b, either a=b, a>b, >or a<b. So, it's a linear order. As far as transcendentals are concerned, Tony, the only thing that can lie on a real number line in common with rationals/irrationals are straight line segment approximations. That's the only linear order possible. So either you give up transcendentals or a real number line. >>> When it comes to reality, pretty much everything is in circles, >>> including finite number systems so this model makes some sense, even if >>> it doesn't in terms of limits of, say, powers. lim(x->-oo)=0 and >>> lim(x->oo)=oo for n^x where n>1, so there the two are different. >> >> Well in a way I'm not sure I disagree. I'm interested in this aspect >> of the problem mainly because I think that perhaps the issue both of >> you may be trying to address is the closing of otherwise open sets. >> Remember x->0 or plus or minus 00 doesn't mean x ever gets there. > >Perhaps not, but if, say, y=1/x is both oo and -oo at x=0, and oo=-oo, >then the function is continuous in every respect, which is what we might >desire in such a fundamental algebraic relation. But for the division operation x never becomes zero. Which indicates that there can be no plus or minus infinity and no continuity. >>> When it comes down to this argument, Wolfgang's argument, I agree with >>> his logic concerning the naturals and the identity function between >>> element count and value. >> >> To me "element count" and the number of commas are the same. > >Sure, that sounds okay to me. In the naturals, the first is 1, the >second 2, etc. What is the aleph_0th? I don't know what you're asking here, Tony. If there is no real number line aleph there are no aleph ordinals either. There can be aleph infinitesimals but that represents a continual process of subdivision and not one of division in which case the ordinality would be one of relation between various curvatures where straight lines would be first or minimal and the ordinality of others judged in relation to it. I think the question you really need to be asking in this context is whether there can be such a thing as an open set. If not then the question becomes how to close open sets and whether there can be anything besides finites in closed sets. >>> He chooses then to reject infinities, while I >>> choose to retool them. I think the problem he and I both see clearly is >>> that using finiteness as a "bound" on the set simply doesn't tell you >>> anything, but rather clouds the entire subject. >>> >>> Did that answer your question? >> >> Kinda, Tony, although I really don't get all the arguing over vases >> a |