Galileo's Paradox
in [Math]
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From: Bob Kolker on 3 Dec 2006 20:24 Lester Zick wrote: > 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. dr/dr = 1. in general d(k*r)/dr = k, where k is a constant. > > > 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. d(r^3)/dr = 3*r^2 That is calculus. What you are babbling about is Zickulus. Bob Kolker
From: Lester Zick on 3 Dec 2006 20:30 On Sat, 02 Dec 2006 13:26:13 -0500, Tony Orlow <tony(a)lightlink.com> wrote: >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. And I wouldn't disagree, Tony. But infinitesimals are not of zero length. They're of infinitesimal length. I don't argue that line segments of zero length are points. But I do argue that points are not units of measure any more than zero is a metric becausee points cannot be integrated into lines. > 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. It's not a matter of terminology at all because infinitesimals can be integrated into lines but points cannot. >>> 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... ? I'm not sure what this means exactly, Tony. But if it means you think infinite numbers of points can be integrated into lines you're flatout wrong. There's a huge difference between trying to integrate zero and trying to integrate 0dr. That's what I mean by different metrics. Zero is not a metric whereas 0dr is. >> 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. Well even that's not at all true if you're talking about conventional set "theory" definitions of lines because nothing I'm aware of in the Peano axioms shows whether or how consecutive finites fall on any succession of colinear elements. Everybody thinks they do and claims they do but no one seems too anxious to show how the trick is done. >> 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.... Well my problem with infinite units of measure comes when people refer to a specific raidus of rotation they say it's finite not infinite. >> 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. Sure. But as far as I know that's the only relationship they have and without going through that connection I see no common metric. > 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. Not sure what all this means, Tony. >> 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 can't be both, Tony. Points are not units of measure.Infinitesimals are because they can be integrated. Calling infinitesimals points doesn't make them integratable. > 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. :) Well, Tony, I don't really think any of this matters. These just seem to be word strings having nothing to do with the matters at hand. ~v~~
From: Tonico on 4 Dec 2006 01:41 Bob Kolker wrote: >> dr/dr = 1. > > in general d(k*r)/dr = k, where k is a constant. ****************************************************** Since k^r = e^(r*lnk), d(k^r)/dr = (lnk)k^r.... Tonio
From: Eckard Blumschein on 4 Dec 2006 02:57 On 12/3/2006 8:22 PM, cbrown(a)cbrownsystems.com wrote: > Tony Orlow wrote: > > 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 Is it really justified to blame an allegedly insufficient definition of the term set for obvious antinomies of set theory? Notice, there is not even a valid definition of a set which includes infinite sets. Cantor's definition has been declared untennable for decades.
From: Virgil on 4 Dec 2006 03:47
In article <4573D4DA.4040709(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/3/2006 8:22 PM, cbrown(a)cbrownsystems.com wrote: > > Tony Orlow wrote: > > > > 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 > > Is it really justified to blame an allegedly insufficient definition of > the term set for obvious antinomies of set theory? As "set" and "is a member of" are primitives in axiomatic set theory, any "definition" of them is outside of set theory and irrelevant to it. |