Galileo's Paradox
in [Math]
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From: cbrown on 3 Dec 2006 14:22 Tony Orlow wrote: > cbrown(a)cbrownsystems.com wrote: > > Tony Orlow wrote: > > > > <snip> > > > >> Anyway, ala Leibniz, each object IS the set of properties which it > >> possesses, so any two objects with the exact same set of properties are > >> the same object. > > > > But this begs the question: what do we mean, exactly, by a SET of > > properties? What exactly are we trying to say when we say "This set of > > properties is the same as this other set of properties"? > > > > Cheers - Chas > > > > Well, what we really mean is that there is a set of universal > properties, each of which is a set of values, and that each object is > defined by a set of values, one from each set of property values, such > that any two distinct objects differ in at least one property value. Was > that specific enough? Well, you used the term "set" four times in your above definition of what we mean by a "set". That's why I said "this begs the question, what do we mean, exactly, by a set of properties?". There's something that we intuitively seem to think of as a "set"; but unless such a thing is carefully defined, we end up with the contradictions of naive set theory: http://en.wikipedia.org/wiki/Naive_set_theory Cheers - Chas
From: Lester Zick on 3 Dec 2006 15:29 On Sat, 02 Dec 2006 23:26:48 -0500, Tony Orlow <tony(a)lightlink.com> wrote: >Virgil wrote: >> In article <45705ad3(a)news2.lightlink.com>, >> Tony Orlow <tony(a)lightlink.com> wrote: >> >>> 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? >> >> As long as it is only TO playing his silly games, nobody much cares. >> >> If TO were ever to produce anything like a coherent system with actual >> proofs, we might actually have to pay some attention. > >Of course, that would be nice, to get some attention around here, >especially from Virgil. But, he never pays me no mind. He hardly >responds to anything I say. Very good, Tony. ~v~~
From: cbrown on 3 Dec 2006 15:35 Tony Orlow wrote: > cbrown(a)cbrownsystems.com wrote: > > Tony Orlow wrote: > > <snip> > Your other question about 3 needs to be thought about a bit. You could > ask Dave Rusin, He confirmed a couple of years ago it was a simple > repeating bit pattern. I suppose I need to do my own legowrk in that > area now. > Usually what we mean when we say "(some set of operations on a base set) generates (some other set)" is "by a finite number of applications of the rules on the base set". But no /finite/ number of applications of your rules will generate "3". So you need to define a limit, in addition to your two generating rules. (Unless you want to define your own personal meaning to "generates"; in which case you will just be confusing people :) ). Note that your generator certainly will not produce -1, whether we allow limits or not; so really the question is does it generate (in the limit) all non-negative reals (R+)? To prove this, you need to prove that the closure under finite applications of your rules is dense in R+; otherwise, there will be gaps. Without limits, there are only a countable set of values generated; so this would clearly fail to generate all reals. Cheers - Chas
From: Lester Zick on 3 Dec 2006 19:47 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. 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. >> 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. >> 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. >> 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.) >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. >_______________________________________________________________________ > >> 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~~
From: Lester Zick on 3 Dec 2006 20:05
On Sat, 02 Dec 2006 13:38:47 -0500, Tony Orlow <tony(a)lightlink.com> wrote: >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. Oh I agree. But if you add them as pieces of fruit for example that requires a conversion of metrics between apples, oranges, and pieces of fruit or anything else they have in common according to a common metric. With a finite r and infinitesimal dr you don't have that without some common basis of metric integration. >>>> 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. Well you can say you have a common metric but that doesn't mean you can show one. In my opinion taking the derivative of a straight line such as r with respect to itself produces 0dr and shows definitively that there is no metric in common between r and dr. >> 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? Well that seems to be pretty easy. At least taking the derivative of a cube like rrr with respect to r indicates a ratio of infinitesimals is 3rr dr for its square and with respect to r would be 6r dr. >> >> 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. But I just said above that points are not units of measure any more than zero is a metric, Tony. Do you intend to say how you avoid that issue? Points cannot be integrated into lines despite Bob's contention a year or two ago. >>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. So exactly how do you expect to show any infinite number of points apart from the finite number of points which define the figure itself? I mean you can have any number of points defined by intersection within a line segment but they have nothing to do with definition of the segment and certainly nothing to do with any kind of mechanically integratable dr. >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. How can you tell? There could be any number. ~v~~ |