Galileo's Paradox
in [Math]
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From: MoeBlee on 4 Dec 2006 13:17 Tony Orlow wrote: > 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. I was hoping you might answer the question rather than spew a bunch more of your doubletalk. > Surely, you don't consider the Cartesian plane to > be completely unordered??? If you give me a precise set theoretic definition of 'completely unordered', then I might address the question. Meanwhile, I'm just asking you to specify what ordering you have in mind (what set of ordered pairs; RxR is a set of ordered pairs, but is not itself an ordering) and what kind of ordering it is - partial, linear, well, or other. MoeBlee
From: Tony Orlow on 4 Dec 2006 12:43 Lester Zick wrote: > Tony, I'm making some far ranging speculations here which I think are > correct. But if I make some mistakes I hope you'll be patient. > Er, okay. > On Sun, 03 Dec 2006 00:28:04 -0500, Tony Orlow <tony(a)lightlink.com> > wrote: > >> Lester Zick wrote: >>> Tony, let me see if I can provide an alternative line of reasoning to >>> support my analysis. >>> >> Okely Dokums! >> >>> 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. >>> >> Yes, I thought it was encouraging to see the notion of a mixture there, >> though it didn't seem like it followed necessarily. It was more like, >> there could be an infinitesimal change, that wouldn't be detected. In >> any case, do go on... ;) > > It's more than a matter of detection. There are probably finite > changes which couldn't be detected either. The problem is that r is > finite and the changes infinitesimal. Different metrics entirely. > But, isn't the change infinitesimal because the slice of time in the rotation is infinitesimal? Does the radius really change, or is the point simply accelerating in the diection of the center, without changing velocity toward that point? >>> Now the problem for you and your idea of combining finites and >>> infinitesimals arithmetically is that you can't combine finites and >>> infinitesimals directly. >> Oh, then no dr/dt for you tonight, young man. And after you brushed your >> teeth already... >> >> In other words there is no way to say r+dr>r >> >> Except you just did it. > > Okay. You can say it but you can't demonstrate it. > Then there is no way to say dr>0. x+y=y -> x=0 >>> as you're trying to suggest because finites and infinitesimals don't >>> lie together on a common line with the same metric. >> Same line, different scale. > > Not quite, Tony. The issue is how you arrive at dr. The derivative of > a straight line r with respect to itself is 0dr. You only find nonzero > derivatives with respect to other metrics such as other lines or time. > Well, the derivative of a line is 0 with respect to x where it's horizontal. Do you really take derivatives of obhjects with respect to themselves, or as one of their independent variables changes? >>> 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. >> Uh, what? A finite dr of infinitesimal magnitude? What makes it finite? > > The definite integral between 0 and dt. Just because dt is an > infinitesimal doesn't mean the interval between 0 and dt is an > infinitesimal (although I'd like some more time to consider this > aspect of the problem as I'm not entirely happy with what I'm saying > here.) > Yeah, I think you need to think about that statement. It doesn't fill me with glee either. Sorry. >> 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. >>> >> Yeah. That's what IFR's about. The line, Man. That's the common metric. > > The problem though, Tony, is what I pointed out above, the production > of dr. If you take the derivative of a straight line r with respect to > r you get 0dr not ndr or 00dr. > I don't understand. You take a derivative of a formula with respect to one of the variables in the formula. You don't take the derivative of the formula with respect to itself. Do you? Do you take the derivative of a square with respect to x^3? You can take the third derivative, with the right unit, and get 8, the number of points. But.... ? >> _______________________________________________________________________ >> >>> 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. >> Actually I got lost at the end there. Infinitesimals are things that, if >> you multiply them together, they disappear. *poof* > > Not sure exactly what you're getting at here, Tony. If you want to > combine finites and infinitesimals you have to first produce the > infinitesimals then you have to produce some finite interval by > definite integration between limits. Only then is some combination > possible. When I say the definite integral of dr/dt between 0 and dt > in the context of angular mechanics is finite but of infinitesimal > magnitude it's only because I'm integrating dr/dt with respect to an > infinitesimal dt not dr. (In any event this seems to be the way it > works out at the moment.) > > ~v~~ Alright. Hopefully all will work out in the end, eh? 01oo
From: MoeBlee on 4 Dec 2006 13:44 Tony Orlow wrote: > Lexicographic ordering corresponds to some multidimesnional ordering, > such as is obvious here. :) Hmm, I don't recall what Kolker said, but if I'm not mistaken, the lexicographic ordering (taken from the standard linear ordering of R) of RxR is a linear ordering of RxR. I've been giving you a hard time about the question of ordering RxR, though I now recognize that, if I'm not mistaken, RxR does have a linear ordering. However, you provided no help in this except to reel off a bunch of your personal mumbo jumbo. In this discussion, the idea of using the lexicographic ordering from the standard ordering comes from Kolker and reporting that this is a linear ordering comes from me. I believe this is a special case of a general theorem: If m is a linear ordering of S and n is a linear ordering of T, then the lexicographic ordering made from m and n is a linear ordering of SxT. In the present instance, both m and n are the standard linear ordering of R and both S and T are R, so we get that the lexicographic ordering is a linear ordering of RxR. MoeBlee
From: Eckard Blumschein on 4 Dec 2006 13:52 On 12/1/2006 7:32 PM, Bob Kolker wrote: > Eckard Blumschein wrote: > >> >> Any rational number has a numerical address. Just the entity of all >> rational numbers is a fiction. > > The set of rational numbers exists in exactly the same sense that the > set of integers exists. > > Bob Kolker You are right if you consider each time the complete set. Neither the integers not the rationals are actually complete. So both complete sets are just irreal fictions. They are uncountable, and the belonging power sets are also uncountable.
From: Eckard Blumschein on 4 Dec 2006 14:00
On 12/1/2006 7:04 PM, MoeBlee wrote: > > The Cartesian plane is the set of ordered pairs of real numbers. Doesn't coordinate transform e.g. into circular coordinates require fictitious real numbers? Here engineers might have a better feeling for the categorical difference between number and continuum. |