An uncountable countable set
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From: Ross A. Finlayson on 29 Oct 2006 21:58 Lester Zick wrote: > > On 29 Oct 2006 12:27:33 -0800, "Ross A. Finlayson" > ... > > 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. > .... > > I'm not quite sure yet what the mechanical implications of zero are. > But basically if you're going to integrate something that something > has been infinitesimally differentiated and if not you aren't doing > definite integration. Either way the zero and the infinity defined > refer to the limits of infinitesimal integration and not to natural > numbers. But I certainly appreciate your considered comments. > > ~v~~ The differential was the atomic infinitesimal for hundreds of years before Gauss invented the notation and use of limit that was later adopted under the Cauchy/Weierstrass movement towards formalization, because, that was the simplest way to formalize the differential. As you can see here, wide-ranging discussions continue to this day about the differential, as an infinitesimal. Consider what's sitting there under the integral bar, which is an S for summation, of differentials, per Leibni(t)z' convention: functions, variables, constants, and differentials. It's there to help signify the infinitesimal, patch, element, Riemann or Lebesgue segment, often in analogous to physical terms of length, distance, volume, or position, velocity, acceleration, and etc. The summation is built right into the notation. Robinso(h)n's hyperreals are similar to Newton's rules. Newton was of the mind that there were infinitesimals, called fluxions, where Newton's notation of the derivative is f, f', f'', .... These were interval subdivisions of the unit, unity, one. There, unity is the fluent to that fluxion, perhaps in reference to continuum and flow, fluid, fluide. Then, he figured the fluxion was a fluent to further fluxion, ad infinitum, and a very parallel, although probably not interchangeable, construction is seen in the hyperreals today. So, you can see the progression of f, f', f'', ..., as fluent to next fluxion. Start with the unit interval, divide it in half then half again and so on. Multiply each least value by the positive integers less than its reciprocal, eg 1/2^200, 2/2^200, ..., 2^200-1 / 200, 2^after dividing the unit interval into 2^200 segments. In summing the areas of the rectangles in those segments under the Riemann integrable function's curve, a closer and closer approximation to actual area under the curve is "derived" through integration. For no non-constant curve is the value correct. So, the differential can not be finite, else it would be a plain Sigma for summation and delta x for the differential of x. Where the sum of zeros is zero the differential can not be zero. So, the differential is not zero, and, it is less than 1/n for positive finite integer n. So, for values n > 2 of 1/n, it is closer to zero than one. For values n-1 / n, it is closer to one than zero, and of 1/n, 2/n, ..., n-1/n, they are naturally ordered by the natural ordering of the natural integers, they're trichotomous, and as n->oo, 1/n -> 0 and n-1 /n -> 1. That would appear to be a partition of the real number line, or rather the unit interval segment of the real number line, into degenerate "intervals". That due to formalisms of a sort it is not, can not deny that it is. It is what is said to be "not a real function", where that is also used to describe the unit impulse function. As well, it's argued that is not a function because it would be a bijection between the naturals and unit interval of reals. Thus, it has been carefully compared against those arguments, EF, the natural/unit equivalency function. Ross
From: Tony Orlow on 30 Oct 2006 11:18 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. Mind if I sprinkle some thoughts about? >>> ... > >>> 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? 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. >> >>> 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..... ;) >> >>> 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. > > >>> 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) >>> 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 is some consideration and ready application of there >>> being points at infinity, and meaningfully that their multiplicative >>> inverses are defined in similar ways as zero's. >> In other words people are just trying to make zero a natural number. >> It isn't. It has very specialized restrictions when it comes to use in >> arithmetic contexts. >> > > When I say "naturals" it includes zero and its successors. Zero and > its successors is the naturals. Notice the verb is conjugated "is", > not "are." Do you see how that subtle play on words, representing > mathematical concepts, leads to N E N? > Actually, Ross, your argument works much better starting from 1, so the set size equals the max value, inductively, at every step. Then it's impossible to imagine how two values incrementing in tandem can ever become any different, much less one remaining finite while the other is infinite. I have to agree with Lester on this one. The naturals are positive, and while 0 is often considered to be positive, it's not, because it's also negative. It's the origin, compared to which all measures are made. It's an arbitrary point which exists before any counting has begun. So, I see Lester's point on that. > > >>> There are applications in integral transforms and so on for points at >>> infinity, for which the "standard" real numbers are insufficient, and >>> they could only be real numbers. >> Well the so called "points at infinity" don't have to be defined
From: Tony Orlow on 30 Oct 2006 11:22 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.
From: Tony Orlow on 30 Oct 2006 11:24 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.
From: Tony Orlow on 30 Oct 2006 11:31
Virgil wrote: > In article <454364ae(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Apparently you are not aware of my >> position on the subject. Bijections alone do not prove equinumerosity >> for infinite sets. > > If they do not then nothing does. > > > >> Cardinality is a rough measure of equivalence class, >> not a precise measure of the size of a set. > > There is no better measure. > > >> In order to precisely >> compare such infinite sets of values, one must measure over a common >> infinite value range formulaically. > > Except that TO has never proved that his "formulaic measures" form a > proper partial order relation on sets the way cardinality does. > > For cardinality, one can easily show that if |A| >= |B| then it is false > that |B| > |A|. > > Can TO prove a similar result for his "formulaic measures"? > > At any rate he has never done so. > > And absent such a proof, and other proofs necessary for a partial > ordering, his "formulaic measures" are, at best, dubious. Since the ordering on the set sizes is done using inductively proven inequalities between formulas describing them, such that the inequality holds for all values greater than some finite n, without having a limit of 0 as x->oo, that fact goes without saying. If I can prove E n e N f(x)>g(x) for all x>n, then I cannot also prove g(x)>f(x) over that range. So, doesn't that result follow naturally, that this partial ordering holds? |