Galileo's Paradox
in [Math]
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From: Bob Kolker on 28 Nov 2006 17:19 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? 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. Bob Kolker
From: Virgil on 28 Nov 2006 19:06 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.
From: Tony Orlow on 28 Nov 2006 22:21 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.
From: Tony Orlow on 28 Nov 2006 22:24 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. Raw bijection determines cardinality, but measure involves a consideration of the actual mapping function which establishes the bijection. The two are not incompatible, Bob. > > Bob Kolker >
From: Tony Orlow on 28 Nov 2006 22:27
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. |