Galileo's Paradox
in [Math]
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From: David Marcus on 5 Dec 2006 23:58 Tony Orlow wrote: > Eckard Blumschein wrote: > > On 11/24/2006 1:20 PM, Richard Tobin wrote: > >>> * If you have two sets of infinite size, is the union of these sets > >>> than larger than infinity? What would larger than infinity mean? > >> That this is not a stumbling block should be clear of you replace > >> "infinite" with "big". The union of two big sets can be bigger than > >> either of the original big sets. Like "big", "infinity" is not a > >> number; it's a description of certain numbers. > > > > According to Spinoza, infinity is something that cannot be enlarged > > (and also not exhausted). It is a quality, not a quantity. > > > > There is only one such ideal concept, not different levels of infinity. > > Isn't the purpose of math be to quantify? No. > That description by Spinoza > doesn't lead to mathematics, does it? Nothing EB has quoted leads to mathematics. > But, infinite is a term often > applied to quantities, whether of space, or time, or knowledge, or power... > > We have a unit interval, consisting of some infinity of distinct points > (more than any finite number). We have the unit square, consisting of an > equally infinite number of distinct parallel unit line segments. The > unit cube, again, has this infinite number of parallel unit squares > within it. Now, does it stand to reason that each added dimension to > this figure multiplies the number of points by this infinite number? > When we divide the line by this number, we get one point, the 0D square. > There's really no way you can convince me that the cube does not have > infinitely more points than the square, or the square than the segment, > or the segment than the point. I think we can all agree that there is really no way to convince you of anything. > These are different levels of infinity. > > As far as sequences go, we can also distinguish between different > infinities, certainly where one is a subset of the other, but also where > quantitative elements are mapped by formulas referencing their values. > Those mapping formulas describe the relative sizes of the sets, > parametrically. > > So, Spinoza might be reconsidering a little, were he still around. -- David Marcus
From: David Marcus on 6 Dec 2006 00:00 Eckard Blumschein wrote: > On 11/29/2006 3:26 PM, Six wrote: > > For me, anyway. I am just trying to work things out in > > my own mind. > > Continue! > > > If it's a matter of comparative success or fruitfulness, I have no > > complaint. I do not have the mathematics to judge. > > The less biased the better. You have an odd notion of what the word "biased" means. -- David Marcus
From: David Marcus on 6 Dec 2006 00:03 Tonico wrote: > Tony Orlow ha escrito: > > Eckard Blumschein wrote: > > > On 12/4/2006 9:56 PM, Bob Kolker wrote: > > >> Eckard Blumschein wrote: > > >> > > >>> 2*oo is not larger than oo. Infinity is not a quantum but a quality. > > >> But aleph-0 is a quantity. > > > > > > To those who belive in the usefulness of that illusion. > > > > > Aleph_0 is a phantom. The aleph_0th natural starting from 1 would be > > aleph_0. It's not a count of the naturals. There is no smallest infinity > > but, sorry to have to tell you, Eckard, a whole spectrum of infinities > > that extend above and below any given infinite expression. Sure, > > transfinitology is quasi-religious. Actual infinity can be quite > > sensible, though. :) > > > > Tony > *********************************************************** > Just like good'ol Mad Journal with Spy vs Spy, but here it is "Troll vs > Troll"...fascinating. A good analogy. -- David Marcus
From: David Marcus on 6 Dec 2006 00:14 Tony Orlow wrote: > Virgil wrote: > > In article <456f334d$1(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: > >>> In article <456e4621(a)news2.lightlink.com>, > >>> Tony Orlow <tony(a)lightlink.com> wrote: > > > >>>> Where standard measure is the same, there still may be an infinitesimal > >>>> difference, such as between (0,1) and [0,1], if that's what you mean. > >>> The outer measure of those two sets is exactly the same. > >> Right, and yet, the second is missing two elements, and is therefore > >> infinitesimally smaller in measure. > > > > Except that in outer measure there are no infinitesimals, and the outer > > measure of the difference set, {0,1} is precisely and exactly zero. > > So, you're saying infinitesimals cannot be considered? You're saying one > is not ALLOWED to consider the removal of a finite set from an ifninite > set to make any difference in measure? I say you're wrong. I guess you still haven't figured out that in mathematics we make precise statements and then make deductions. We don't just decide the theorems based on what we wish. If a certain concept (e.g., cardinality, measure, outer measure, Hausdorff measure, continuity, absolute continuity, differentiability) doesn't do what we want, then we come up with a new concept, state it precisely, and see if we can prove that it does what we want. The term "outer measure" has a precise meaning that is given in courses and books on Real Analysis. -- David Marcus
From: David Marcus on 6 Dec 2006 00:30
Bob Kolker wrote: > You obviously have no knowledge of fractal dimension or Hausdorf > dimension. For example the Peano space filling curves have a dimension > between 1 and 2. The image of a space filling curve is a square, so has dimension 2. -- David Marcus |