Galileo's Paradox
in [Math]
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From: Virgil on 29 Nov 2006 03:22 In article <456d31a0(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <456cfcac$1(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Virgil wrote: > >>> In article <456C5361.40706(a)et.uni-magdeburg.de>, > >>> Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >>>> The relations smaller, equally large, and larger are invalid for > >>>> infinite quantities. > >>> For the lengths of line segments, longer, equally long, and shorter, are > >>> essential to Euclidean geometry. To deny that is to "throw out the baby > >>> with the bath water". And I doubt that Galileo did so. > >>> > >>> For the intersections of lines determining points, any two line segments > >>> can be shown to have a one to one correspondence of points. > >>> > >>> All one needs do is divorce the "length" from the "number of points", > >>> which is probably what Galileo did, as being different sorts of measures > >>> (like weight versus volume), and the problem disappears. > >> Does one "need" to do any such thing, or rather, does one need to > >> integrate the two concepts into a coherent theory including both? To tie > >> measure with count in the infinite is the task here, regarding such sets. > > Unless one can equate mass with volume, and many other similar measures, > > one must allow that there are different measures of "size" for a single > > object. Why should it be different for sets? > > > > In fact since finite intervals can contain infinitely many real or > > rational numbers, we have already admitted multiple measures of a single > > set. > > Of course, my dear Virgil. Set sizes are counts of integral entities > called members or elements. Some such sets, especially most infinite > ones, have elements that denote some sort of measure. Where there is > some sort of order, usually a correspondence exists between the value > range of a set's members, and the size of the set within that range. :) The "value ranges", i.e., lenghts, of rational intervals will be equal to those of the corresponding real intervals though the sets of equal value ranges will have different cardinalities. So that lengths and cardinalities are measures of different qualities of subsets of the reals, and their quantities need not correlate.
From: Virgil on 29 Nov 2006 03:32 In article <456d3270(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <456cfd7e(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Bob Kolker wrote: > >>> Tony Orlow wrote: > >>>> It also seems reasonable to use measures of set density, and more > >>>> sophisticated methods of comparison, such as are employed in the > >>>> converse situation, with infinite series. It seems natural to say > >>>> that, if half the elements of A are in B, and all elements in B are in > >>>> A, then B is half the size of A, as is the case where A=N and B=E. The > >>>> proper subset as a smaller set should not be a notion violated by set > >>>> theory, in my opinion. > >>> Do you know the difference between cardinality and measure? > >> I know that cardinality is a purported method of measure of a set. > >> Otherwise it is is not a quantity of any sort relating to anything. > >> > >>> A straight line segment unit length and a straight line segment twice > >>> unit length have the same cardinality (taken as sets of points). But one > >>> has twice the measure of the other. > >> That is correct, and that is where cardinality fails as a measure of > >> such sets. > > > > That would only hold if one insists that there is only one way to > > measure a set. In analysis, for example, there are many ways to measure > > sets, and restricting things to any single measurement of sets would > > require throwing out a great deal of analysis. > It is a direct consequence of the notion that a proper subset is always > smaller, in some sense, than the base set. But not in every sense, and those other senses may be quite valid measures of size. There are many order relations that may be applied to sets within such set theories aS ZFC and NBG, and not all of them require subsets to be smaller. For example, the length of a bounded subset of the reals is the least upper bound on distances between pairs of points of the set, and the lengths of the set {0,1} and the set (0,1) and the set [0,1] are all the same, even though {0,1} is a proper subset of (0,1) which is in turn a proper subset of [0,1]. > > > > > Raw bijection determines cardinality, but measure involves a > >> consideration of the actual mapping function which establishes the > >> bijection. The two are not incompatible, Bob. > > > > "Measure", in the sense of measure theory, is not preserved merely by > > bijection, but neither is it preserved by any of TO's methods. > > Sure it is, Virgle! If TO wishes to convince me he must show me more than a mere claim unsupported by any evidence.
From: Virgil on 29 Nov 2006 03:39 In article <456d3361(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <456d0e2e(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> I understand the difference between count and measure. I also understand > >> that the count of a set of real measures may be calculated, from the > >> formula defining the set, and its range of measure. Where count and > >> measure are related, it behooves us to cite the connection, does it not? > >> > > If TO "understands" what he says he understands, he is using systems of > > counting and measuring as yet unknown in civilized society. > > Well, I'm a little feral, to be honest, but I've scampered myself this > far. Indeed, bijective functions have inverse bijective functions which > allow us to gauge the infinities relatively, formulaically, and produce > an ordering of infinite sets unavailable using cardinality. Grrrr... > Hope that's not tooo scary. ;) I have always insisted that cardinality is not the only measure of set size, but that there are many measures. That there are sets that are of the same size by cardinality measure which have different sizes by other measures is hardly surprising. It would be much more surprising if it were not so. It is only those who claim only one measure can be applied to sets who are in a position to claim that cardinality cannot be that measure.
From: Virgil on 29 Nov 2006 03:48 In article <456D3D5B.4000909(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 11/29/2006 6:33 AM, Virgil wrote: > > > Unless one can equate mass with volume, and many other similar measures, > > one must allow that there are different measures of "size" for a single > > object. Why should it be different for sets? > > Mathematical sets are ideal objects built with just the notion "unity" > and reasonable rules agreed on. Their size is countable in general, > provided the notion set corresponds to common sense. When one considers the rationals, some sets of rationals, the bounded sets, have a measure called length, which is the least upper bound of the absolute differences between pairs of members. This is the case for all finite non-empty sets of rationals, and for bounded but infinite sets of rationals. > > Objects that exhibit more than one size are not just physical ones like > mass but already such continuous mathematical objects like area (value > and circumference) and volume (value and surface). Then by EB's criteria the rationals must be continuous. > > > > In fact since finite intervals can contain infinitely many real or > > rational numbers, we have already admitted multiple measures of a single > > set. > > No. They are thought to contain infinitely many rational numbers, yes. > However, they do not contain infinitely many but uncountably much real > "numbers". Countable or uncountable is an alternative but no measure. > Incidentally, do not equate intervals and sets. Continuously filled > intervals are merely fictitious sets. It is EB's entire version of mathematics which is fiction. At least everywhere but in his own mind. What is in EB's mind fictitious is elsewhere no more fictitious than anything else in mathematics.
From: Mike Kelly on 29 Nov 2006 06:06
Tony Orlow wrote: > Virgil wrote: > > In article <456C5361.40706(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > > >> On 11/28/2006 3:48 AM, Virgil wrote: > >>> In article <456AF6F8.5020307(a)et.uni-magdeburg.de>, > >>> Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >>> > >>>> On 11/27/2006 3:21 AM, stephen(a)nomail.com wrote: > >>>> > >>>>> There is no need to resolve the paradox. There exists a > >>>>> one-to-correspondence between the natural numbers and the > >>>>> perfect squares. The perfect squares are also a proper > >>>>> subset of the natural numbers. This is not a contradiction. > >>>> What is better? Being simply correct as was Galilei or being more than > >>>> wrong? (Ueberfalsch) > >>> Galileo was both right and wrong. He applied two standards to one > >>> question and was confused when they gave different answers. > >> Initially he was confused, yes. However, he found the correct answer: > >> The relations smaller, equally large, and larger are invalid for > >> infinite quantities. > > > > For the lengths of line segments, longer, equally long, and shorter, are > > essential to Euclidean geometry. To deny that is to "throw out the baby > > with the bath water". And I doubt that Galileo did so. > > > > For the intersections of lines determining points, any two line segments > > can be shown to have a one to one correspondence of points. > > > > All one needs do is divorce the "length" from the "number of points", > > which is probably what Galileo did, as being different sorts of measures > > (like weight versus volume), and the problem disappears. > > Does one "need" to do any such thing, or rather, does one need to > integrate the two concepts into a coherent theory including both? No, one doesn't need to integrate count and measure. Cardinality is a notion of count(bijectability) that applies to all sets. It is quite a useful idea in some mathematical proofs. Various types of measure can also be applied to sets, these can also be useful in mathematical proofs. Nobody has provided an integrated notion of count and measure that applies to all sets, least of all you. Nobody ever will; one does not exist. Furthermore, it is not even clear why you consider it so important that one exist. What new and useful mathematics could be derived from such a thing? >To tie measure with count in the infinite is the task here, regarding such sets. Why? What is the utility of performing this "task"? What new mathematics does it allow for? -- mike. |