An uncountable countable set
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From: David R Tribble on 30 Oct 2006 13:54 David R Tribble wrote: >> Each ball n is placed into the vase at time 2^int(n/10), and then later >> removed at time n. This happens for every ball before noon. So every >> ball is inserted and then later removed from the vase before noon. >> >> At any given time n before noon, ten balls are added to the vase and >> then ball n (which was added to the vase in a previous step) is >> removed. Your entire confusion results from assuming a "last" time >> prior to noon, but there is no such time. > Tony Orlow wrote: > At no time prior to noon are all balls removed. Nor are any removed at > noon. It cannot be empty, then. The problem states that every ball (every ball) is added to the vase and then later removed from the vase. We conclude from this that every ball is removed (eventually). You conclude that at no time are all balls removed. Obviously you think that there are balls left in the vase that never got removed. In fact, you say that there are an infinitude of balls left in the vase. Yet somehow you cannot name a single one of them.
From: Lester Zick on 30 Oct 2006 14:16 On Mon, 30 Oct 2006 11:18:19 -0500, Tony Orlow <tony(a)lightlink.com> wrote: >Ross A. Finlayson wrote: >> Lester Zick wrote: >>> Oops. My bad. Hit the wrong button on the duplicate reply. - LZ >>> >> >> De nada. > >Hi Ross, Lester. Nice to see the two of you conversing. Just haven't had much to say previously on a technical level. > Mind if I >sprinkle some thoughts about? Not at all. Just keep it clean. >>>> ... >> >>>> Various considerations of the natural integers have there being a point >>>> at infinity. That can be useful in a form of nonstandard analysis, to >>>> say that, for example, an infinite sum with a limit actually equals in >>>> evaluation that limit. Consider for example something along the lines >>>> of >>>> >>>> 1 >>>> s = ----- >>>> 2^n >>>> >>>> for n from zero to infinity. The limit exists and is two and then >>>> consider replacing the 2 in the denominator with s. When you can >>>> actually say that the sum equals that limit, you get the same >>>> expression for s. >>>> >>>> Otherwise, s could never equal 2. >>> Okay. I take the point, Ross. But what rule is there requiring 00 to >>> be part of the same set as finites raised to a power of infinity? I >>> think the power of infinity could be defined using the "number of >>> infinitesimals" which is reciprocally defined with differentiation. > >I think you both derive your concept of infinitesimals largely from >differentiation as dx->0 and 1/dx->oo. Would that be a fair assessment? Probably although I explicitly rely on the Newtonian notion and haven't tried to reason through the actual processes involved apart from noting definite integration and differentiation are reversible and mutually reciprocal processes. >I think Ross is also deriving the same dx through subdivision, but I >would caution that A n>=0 1/n>1/2^n. In my book, for a set of size N, >using an alphabet of size S, we require strings of length L so as to >have enough strings to enumerate the set of size N, such that N=S^L. A >two-symbol alphabet such as binary mirrors the structure given by power >set, where there are two logical values, producing 2^L strings of length >L. Complete languages like digital systems, with alphabets of size S, >mirror the power set in structure also, where there are S possible >logical values, rather than just two, in the logical system. So, what am >I rambling about? Oh yeah, it's true, dividing the unit into n segments >is not equivalent to dividing it in half n times. They're two different, >not not incompatible, notions. Well here, Tony, you're frankly losing me. As noted above I just have very little interest in technical issues of this sort. I can't even make out what you're trying to say with "A n>=0 1/n>1/2^n" and at this stage of my life I'm really not inclined to try. Furthermore if as it seems you're trying to draw some computer analogy to general mathematics and science I'd consider the effort totally misdirected. Show me any infinite or infinitesimal inside a machine and I might reconsider. >>>> There are other reasons to consider the infinite naturals as containing >>>> an infinite element, that N E N. >>> N E N? > >The nth element is n. If there are n elements, n is a member of the set. >The number of elements up to and including n, starting at 1, is always >n. The set of the first n naturals starting at 1 always has a maximum >element of n. So, N is not just the size of the set, it's also a member. >That sounded too much like Sy Sterling, from the Hair Club for Men..... ;) Okay. I agree with what you say here certainly with respect to the naturals. >>>> In these arguments here, if there is no infinite value for n, then the >>>> process never completes. >>> Well infinitesimal subdivision certainly never completes. >>> >> >> Les, Lester, there is some consideration that it does. Is not the >> differential intuitively the atomic subdivision of one? >> >> In an interesting way, variously on what you consider interesting, the >> notion of subatomic particles in physics is a similar one as the >> consideration of sub-iota reals in mathematics. >> > >Lester hates when I talk mathematical TOE. Only because it's not true, Tony. People call things "theories" when they're at best nothing but undemonstrable rank speculation and at worst when they're nothing but analytical academic formalisms and what they're really after is just the cachet of scientific truth. >>>> Re Zeno, the arrow goes half and then half >>>> and then half again ad infinitum to reach the mark, that it does is >>>> well-known. That's similar to the above equation with the domain over >>>> (1, 2, ... (infinitely many times). The distance is one, the arrow >>>> travels the distance, the distance can be decomposed in that manner to >>>> partial distances, thus the sum is the distance. >>> But the problem with Zeno's paradoxes is that what is established >>> primarily is that the arrow or whatever goes the distance first. In >>> other words the paradox establishes unity and attempts to subdivide >>> the unity infinitesimally then claims the unity cannot exist unless >>> the process of infinitesimal subdivision can complete and be finite. >>> Infinitesimal subdivision is not a finite process. It's only used as a >>> method of drawing tangents for the purpose of comparing otherwise >>> infinites. >>> >> >> Right, it's not a "finite" process in the sense that there are finitely >> many integers, but I think you would agree that it is a "finite" >> process to the extent that the above description is self-contained. >> >> > >It's bounded, but infinitely. It's internally infinite, as opposed to >bounded but externally infinite (with a limit). > >Axiom of internal infinity: >(E xeR ^ E zeR ^ x<z -> E yeR ^ x<y ^y<z) - R is internally infinite. >(That means dense, basically) Yeah, Tony, I'll have to pass on this stuff. Doesn't make any sense to me. >>>> Similarly for the real numbers there are considerations of points at >>>> infinity, in the "projectively extended" real numbers for example. >>>> Similarly as to how division by zero in the "complete" ordered field is >>>> undefined, ie that every other number than zero has a multiplicative >>>> inverse, there i
From: David R Tribble on 30 Oct 2006 15:29 Virgil wrote: >> Are the properties of "Finlayson Numbers" known to anyone except >> Ross himself? > Tony Orlow wrote: > 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? Only a countable infinity of them? Then the number of infinitesimals in [0,1] is exactly the same as the reciprocals 1/n for every natural n>0, right? But there are c reals in [0,1], so are there more reals than infinitesimals?
From: Virgil on 30 Oct 2006 15:56 In article <454626e1$1(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Randy Poe wrote: > > Tony Orlow wrote: > >> Randy Poe wrote: > >>> Tony Orlow wrote: > >>>> Virgil wrote: > >>>>> In article <4542201a(a)news2.lightlink.com>, > >>>>> Tony Orlow <tony(a)lightlink.com> wrote: > >>>>> > >>>>>> cbrown(a)cbrownsystems.com wrote: > >>>>>>> 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. > >>>>> By what mechanism does TO propose to stop time? > >>>> By the mechanism of unfinishablility. > >>> But that's why I asked you a question about variables labelling > >>> times yesterday, when noon clearly occurred. > >> > >> The experiment occurred in [-1,0). Talk of time outside that range is > >> irrelevant. Times before that are imaginary, and times after that are > >> infinite. Only finite times change anything, so if something changes, > >> it's at a finite, negative time. > >> > >>> I can define a list of times t_n = noon yesterday - 1/n seconds, > >>> for all n=1, 2, 3, ... > >> Are there balls in the vase for t<-1? No. > > > > What balls? What vase? > > > > I'm naming times. They're just numbers. > > > >>> Clearly this list of times has no end. But didn't noon happen? > >> Nothing happened at noon to empty the vase, \ > > > > What vase? Why are you obsessed with vases? > > > > Do you deny me the ability to create a set of variables > > t_n, n = 1, 2, ...? Why do vases have to come into it? > > > > - Randy > > > > I thought we were trying to formulate the problem. TO is only trying to confuse the problem.
From: Virgil on 30 Oct 2006 15:59
In article <45462736(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <454360a5(a)news2.lightlink.com>, > > 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: > >>>>>>>> David Marcus wrote: > >>>>>>>>> You are mentioning balls and time and a vase. But, what I'm asking > >>>>>>>>> is > >>>>>>>>> completely separate from that. I'm just asking about a math > >>>>>>>>> problem. > >>>>>>>>> Please just consider the following mathematical definitions and > >>>>>>>>> completely ignore that they may or may not be > >>>>>>>>> relevant/related/similar > >>>>>>>>> to the vase and balls problem: > >>>>>>>>> > >>>>>>>>> -------------------------- > >>>>>>>>> For n = 1,2,..., let > >>>>>>>>> > >>>>>>>>> A_n = -1/floor((n+9)/10), > >>>>>>>>> R_n = -1/n. > >>>>>>>>> > >>>>>>>>> For n = 1,2,..., define a function B_n: R -> R by > >>>>>>>>> > >>>>>>>>> B_n(x) = 1 if A_n <= x < R_n, > >>>>>>>>> 0 if x < A_n or x >= R_n. > >>>>>>>>> > >>>>>>>>> Let V(x) = sum_n B_n(x). > >>>>>>>>> -------------------------- > > > >>> I thought we agreed above to not use the word "time" in discussing this > >>> mathematics problem? > >> If that's what you want, then why don't you remove 't' from all of your > >> equations? > > > > I have taken the liberty of replacing the 't' with 'x' in those > > equations. It does not change the conclusions. > >>> As for your question, let's look at B_2 (the argument is similar for the > >>> other B_n). > >>> > >>> B_2(x) = 1 if A_2 <= x < R_2, > >>> 0 if x < A_2 or x >= R_2. > >>> > >>> Now, A_2 = -1 and R_2 = -1/2. So, > >>> > >>> B_2(x) = 1 if -1 <= x< -1/2, > >>> 0 if x < -1 or x >= -1/2. > >>> > >>> In particular, B_2(x) = 0 for x >= -1/2. So, the value of B_2 at zero is > >>> zero and the limit as we approach zero is zero. So, B_2 is continuous at > >>> zero. > >>> > >> Oh. For each ball, nothing is happening at 0 and B_n(0)=0. That's for > >> each finite ball that one can specify. > > > > As there are no other balls, what is your point? > > > > The only relevant question is "According to the rules set up in the > > problem, is each ball which is inserted into the vase before noon also > > removed from the vase before noon?" > > > > An affirmative answer confirms that the vase is empty at noon. > > A negative answer directly violates the conditions of the problem. > > > > How does TO answer? > > That doesn't occur at any time before noon. Does TO wish to claim some balls are not removed before noon? Which one(s), TO? |