Galileo's Paradox
in [Math]
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From: Lester Zick on 2 Dec 2006 12:44 On Fri, 01 Dec 2006 14:02:54 -0700, Virgil <virgil(a)comcast.net> wrote: >In article <457059e6(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Lester Zick wrote: >> > On Thu, 30 Nov 2006 09:52:49 +0100, Eckard Blumschein >> > <blumschein(a)et.uni-magdeburg.de> wrote: >> > > >TO verus LZ versus EB. It only wants WM to become the ultimate battle of >the pigmies As are Nam, Brian, Stephen, and Virgil the gang of four horses asses of the apocalypse and thoroughly uninquiring minds. ~v~~
From: Tony Orlow on 2 Dec 2006 13:13 MoeBlee wrote: > Tony Orlow wrote: >> Bob Kolker wrote: >>> Tony Orlow wrote: >>> >>>> Are x and y ordered? The Cartesian plane is ordered in two dimensions, >>>> not a linear order, but a 2D ordered plane with origin. >>> >>> The plane is not a linearly ordered set of points. >>> >>> Bob Kolker >> That's what I just said. > > So what kind of ordering do you contend it is? > > MoeBlee > It's the superposition of two continuous linear orders, such that each point obeys trichotomy with each other point, along each of the two dimensions, or linear orderings of each element of the n-tuple describing each point. Surely, you don't consider the Cartesian plane to be completely unordered??? ToeKnee
From: Tony Orlow on 2 Dec 2006 13:26 Lester Zick wrote: > On Fri, 01 Dec 2006 11:39:57 -0500, Tony Orlow <tony(a)lightlink.com> > wrote: > >> Lester Zick wrote: >>> On Thu, 30 Nov 2006 12:08:03 +0100, Eckard Blumschein >>> <blumschein(a)et.uni-magdeburg.de> wrote: >>> >>>> On 11/29/2006 8:13 PM, Bob Kolker wrote: >>>>> Tony Orlow wrote: >>>>>> Uncountable simply means requiring infinite strings to index the >>>>>> elements of the set. That doesn't mean the set is not linearly ordered, >>>>>> or that there exist any such strings which do not have a successor. >>>>> Uncountable means infinite but not of the same cardinality as the >>>>> integers. For example the set of real numbers. It is an infinite set, >>>>> but it cannot be put into one to one correspondence with the set of >>>>> integers. >>>> Uncountable means: Counting is impossible. This property obviously >>>> belongs to fictitious elements of continuum. There is simply too much of >>>> them. So counting is not feasible. As long as one looks at a finite, >>>> just potentially infinite heap of single integers, one has to do with >>>> individuals. The set of all integers is something else. It is a fiction. >>>> It is to be thought constituted of an uncountable amount of >>>> non-elementary elements. Well this looks nonsensical. There is indeed a >>>> selfcontradiction within the notion of an infinite set. >>>> Non-elementary means not having a distinct numerical address. Element >>>> means "exactly defined by an impossible task". >>> You make the same mistake of assuming "infinite" means "larger than" >>> when it only means numerically undefined. Infinites are neither large >>> nor small; they're only undefined. Consequently there are no numerical >>> relations or operations possible between them and finites. The reason >>> counting is not possible is not because infinites are huge or because >>> they form a continuum but because there is no numeric metric defined >>> for them and counting as well as every other arithmetic relation and >>> operation requires some kind of numeric definitional metric. >>> >>> ~v~~ >> Huh! So, what happens if I declare a number, Big'un, and say that that >> is the number of reals in (0,1]? What if I say the real line is >> homogeneous, so every unit interval contains the same number of points? > > If you do, Tony, then what you've defined as "points" are in point of > fact infinitesimals not points. > As you should know by now, I don't disagree, but you must remember that in standard mathematics, a line segment of zero length is considered to be the same object as a point. So, it's more a matter of terminology that you respond to, which is a relative and arbitrary set of words, depending on context. 0.999...<1 or 0.999...=1? Depends. >> And then, what if I say the positive number line is going to include >> Big'un such unit intervals, so it has Big'un^2 reals up to Big'un? Does >> the universe collapse, or all tautologies suddenly become false? > > You could do this but the result wouldn't be finite in numerical terms Meaning? If I can represent uncountable values in finite strings, then... ? > and anything you might try to do between them and finites would cause > the universe to collapse because they just aren't there on the same > line and have different properties because they are finitely infinite. > No, dear Lester, the line is infinite, and straightness is about as approachable as absolute 0 or c. > This is the price to be paid for absurdities like the real number line > and putting infinities on them. "Infinite" means "not finite" and you > just can't do finite arithmetic with "not finites". This includes even > simple processes like numeric comparison of smaller and larger. You > can't have "non finites" and "finites" on the same line in conceptual > terms because "finites" have some numeric metric and "non finites" > don't. Incorrect, old chap. An infinite unit of measure is every bit as viable as a finite or infinitesimal one. Why can't you say 2*oo>oo? I can. Repeat after me.... > > In other words even in the elementary case of arithmetic infinities > infintesimals can't be added, subtracted, or compared in size to > finites because their metric is completely different and is described > in finitely unrelatable terms unless some metric can be established > between finites and non finites through a mechanism like calculus and > comparison through L'Hospital's rule. Actually they have a very real relationship to finites in the infinitesimal calculus. The parts that drop out in the final calculation using infinitesimal units are terms of higher than the first power. Those are infinitesimal, compared to infinitesimals, and can be considered insignificant in the final analysis. Them's "nilpotent" infinitesimals. > > In other words just because you say "bigun" doesn't indicate if it is > a finite "bigun" or not and just because you use the phrase "number > of" doesn't indicate you have any finite number subject to arithmetic > in finite terms. > > ~v~~ Big'un is uncountably large. It's the number of points, or infinitesimal if you prefer, in the unit interval. It's also the measure of a basic dimension of the universe. In other words, within every "one" is a universe of "infinity", and this even pertains, in the non-nilpotent model, to infinitesimal "ones", or as Ross would call them, "iotas". "Epsilons" works as well, or if you must, "deltas". Depends what Greek your arguing with, I imagine. :) 01oo (btw, do you like my Zickesque signature? It's like when I sign responses to MoeBlee as "ToeKnee", though I'm starting to like TOEknee, better, what with the Theory Of Everything reference and all. Mathematical truth, tautologically determined or otherwise, IS the basis for all reality. You'll see that eventually, Herr Lester. :))
From: Tony Orlow on 2 Dec 2006 13:38 Lester Zick wrote: > On Fri, 01 Dec 2006 11:36:00 -0500, Tony Orlow <tony(a)lightlink.com> > wrote: > >> Lester Zick wrote: >>> On Thu, 30 Nov 2006 09:52:49 +0100, Eckard Blumschein >>> <blumschein(a)et.uni-magdeburg.de> wrote: >>> >>>> On 11/29/2006 6:37 PM, Bob Kolker wrote: >>>>> Tony Orlow wrote: >>>>>> It has the same cardinality perhaps, but where one set contains all the >>>>>> elements of another, plus more, it can rightfully be considered a larger >>>>>> set. >>> Tony, you know we've been over this previously. All "infinite" means >>> is lack of definition for a particular predicate such as numerical >>> size. And when you add numerical finites to numerical infinites the >>> result is still infinite. >> When you add anything to anything, you have more than you had, eh? >> That's pretty basic. Let's try to keep that in mind. > > Provided they have the same metric and you can just "add anything to > anything". Next you'll be trying to add apples and oranges. > I can do that, by the pound, or apiece. That's a different problem. >>> This problem mainly arises I suspect because mathematikers insist on >>> portraying infinites as larger than naturals and somehow coming beyond >>> the range of naturals such as George Gamow's famous 1, 2, 3, . . . 00. >>> Then mathematkers try to establish certain numerical properties for >>> infinities by comparative numerical analysis and mapping with >>> numerically defined finites. However one cannot do comparative >>> numerical analysis and numerical analysis with numerically undefined >>> infinites anymore than one can do arithmetic. Infinites are neither >>> large nor small; they're just numerically undefined. >> Uh, what if you define them, and even work out a language for expressing >> them, and arithmetic that be performed on them, and they produce >> intuitive results that include measure, as well as count? Why do you >> claim that's impossible, because you don't like the idea? > > I don't like the idea because you can't establish any metric for them > not because you work out all kinds of things you claim are intuitive. > You can establish a common metric, even if you can't describe one in terms of the other in a finite formula. > Points are no more units of measure than zero is a metric. Someone > wrote the other day that Cantor was surprized that cubes have the same > number of points as squares and I was tempted to reply that if he was > he really didn't understand what he was talking about because cubes > and squares certainly have different numbers of infinitesimals. They most certainly do, as well as having infinitesimals of different dimensions. How many infinitesimals would you say a cube has, compared to a cube? Can you express that relationship? > > How many points are there in a finite interval? Big'un, times the length in number of units of measure, plus or minus some finite number. Technically when it > comes to arithmetic and comparison there are only the two points > defining the interval metric. Yes, two "defining" it, meaning "marking the ends of" it. Notice the "fin" in "define"? There are two endpoints, and in between, and infinite number of intermediate points. Similarly for cubes and squares. Yes, 2^2 and 2^3 "endpoints". And > presumably there are as many points in a point as zeroes in a zero. One? Think again. > > ~v~~ 01oo
From: Lester Zick on 2 Dec 2006 13:40
Tony, let me see if I can provide an alternative line of reasoning to support my analysis. Over the past couple of years in addition to tautological analysis I've also provided analysis of angular mechanics in corrected form. And in that analysis I do make use of arithmetic combinations of finites and infinitesimals. In particular I take finites such as the radius of circles, r, and combine them with infinitesimal changes in radius, dr, showing that for any finite multiple of dr, such as ndr, the combination can change only infinitesimally such that r+ndr=r and r remains finitely constant. I don't know if you followed that discusion but the mechanics involved are identical to what you suggest. Now the problem for you and your idea of combining finites and infinitesimals arithmetically is that you can't combine finites and infinitesimals directly. In other words there is no way to say r+dr>r as you're trying to suggest because finites and infinitesimals don't lie together on a common line with the same metric. In the case of angular mechanics this is also true. However I provide a common metric for them by definitely integrating a finite velocity, dr/dt, between 0 and dt which provides a finite dr of infinitesimal magnitude. In other words you can't provide an arithmetic sum for finites and infintesimals directly without first providing a common finite metric for them through definite integration of some kind. This is how we can know arithmetic combinations of finites of finite magnitude and finites of infinitesimal magnitude. Mathematically modern mathematikers incorrectly analyze the same problem in the reciprocal terms of n/dr instead of ndr and wind up with various kinds of 00 they like to pretend follow the finites on a common real number line. However this makes the proper analysis of angular mechanics impossible unless one takes r to be an infinite and ndr to be finite. In any event I hope this clears up my perspective on analysis of the arithmetic combination of finites and infinitesimals. On Fri, 01 Dec 2006 11:39:57 -0500, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: >> On Thu, 30 Nov 2006 12:08:03 +0100, Eckard Blumschein >> <blumschein(a)et.uni-magdeburg.de> wrote: >> >>> On 11/29/2006 8:13 PM, Bob Kolker wrote: >>>> Tony Orlow wrote: >>>>> Uncountable simply means requiring infinite strings to index the >>>>> elements of the set. That doesn't mean the set is not linearly ordered, >>>>> or that there exist any such strings which do not have a successor. >>>> Uncountable means infinite but not of the same cardinality as the >>>> integers. For example the set of real numbers. It is an infinite set, >>>> but it cannot be put into one to one correspondence with the set of >>>> integers. >>> Uncountable means: Counting is impossible. This property obviously >>> belongs to fictitious elements of continuum. There is simply too much of >>> them. So counting is not feasible. As long as one looks at a finite, >>> just potentially infinite heap of single integers, one has to do with >>> individuals. The set of all integers is something else. It is a fiction. >>> It is to be thought constituted of an uncountable amount of >>> non-elementary elements. Well this looks nonsensical. There is indeed a >>> selfcontradiction within the notion of an infinite set. >>> Non-elementary means not having a distinct numerical address. Element >>> means "exactly defined by an impossible task". >> >> You make the same mistake of assuming "infinite" means "larger than" >> when it only means numerically undefined. Infinites are neither large >> nor small; they're only undefined. Consequently there are no numerical >> relations or operations possible between them and finites. The reason >> counting is not possible is not because infinites are huge or because >> they form a continuum but because there is no numeric metric defined >> for them and counting as well as every other arithmetic relation and >> operation requires some kind of numeric definitional metric. >> >> ~v~~ > >Huh! So, what happens if I declare a number, Big'un, and say that that >is the number of reals in (0,1]? What if I say the real line is >homogeneous, so every unit interval contains the same number of points? >And then, what if I say the positive number line is going to include >Big'un such unit intervals, so it has Big'un^2 reals up to Big'un? Does >the universe collapse, or all tautologies suddenly become false? > >01oo ~v~~ |