Galileo's Paradox
in [Math]
|
From: Bob Kolker on 28 Nov 2006 23:25 Tony Orlow wrote: > > That is correct, and that is where cardinality fails as a measure of > such sets. Raw bijection determines cardinality, but measure involves a > consideration of the actual mapping function which establishes the > bijection. The two are not incompatible, Bob. Cardinality was never intended as a measure. It was intended as a count. Compare and contrast the following questions: How much vs. How many. Bob Kolker
From: Tony Orlow on 28 Nov 2006 23:36 Bob Kolker wrote: > Tony Orlow wrote: > >> >> That is correct, and that is where cardinality fails as a measure of >> such sets. Raw bijection determines cardinality, but measure involves >> a consideration of the actual mapping function which establishes the >> bijection. The two are not incompatible, Bob. > > Cardinality was never intended as a measure. It was intended as a count. > > Compare and contrast the following questions: How much vs. How many. > > Bob Kolker > Hi Bob - 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? :) Tony
From: Virgil on 29 Nov 2006 00:33 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: > > > >> 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? 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.
From: Virgil on 29 Nov 2006 00:41 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. 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.
From: Virgil on 29 Nov 2006 00:42
In article <456cfe2b(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <456c9c4f(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> I certainly consider 1 inch to be twice as > >> infinitely many points in a row as 2 inches, nonstandard as that may > >> sound. > > > > As well as being totally nonstandard, it sounds excessively idiotic. > > > > More points in less space is a bit much even for TO. > > Yeah, oops, you're right. I'm not myself these days. My woman is driving > me nuts. Sorry, I meant half as infinitely many. Thanks for the correction. Even Homer nods. |