Galileo's Paradox
in [Math]
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From: Tony Orlow on 5 Dec 2006 13:20 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? That description by Spinoza doesn't lead to mathematics, does it? 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. 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.
From: Tony Orlow on 5 Dec 2006 13:26 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. >> >> Bob Kolker > > > 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
From: Eckard Blumschein on 5 Dec 2006 13:29 I wonder who educated Bob. Didn't he intend to kill the maybe million of muslims who live in Germany by means of atomic bombs? On 12/5/2006 4:14 PM, Bob Kolker wrote: >> >> Cantor has won his psycho-battle against Kronecker who eventually got >> ill and gave up when Cantor got admired for his masterly >> misinterpretation. Kronecker died already in 1891. It was perhaps >> Cantor's own feeling to be possibly wrong which prompted his mental >> breakdowns for the first time in 1884 after Cantor believed to have a > > Depression is a purely physical/chemical condition. It is all about > seritonin re-uptake. There is strong evidence that depression is > hereditary. There is no such thing as a mental disease since there is no > such thing as a mind. However the brain and nervous system, like any > other subsystem of the physical body is subject to disease and disfunction. > > Bob Kolker
From: Eckard Blumschein on 5 Dec 2006 13:30 On 12/5/2006 4:46 PM, Tonico wrote: > Eckard Blumschein ha escrito: > >> On 12/4/2006 11:52 PM, Virgil wrote: >> >> >> I am an electrical engineer >> > >> > Shocking! >> >> Why? We love mathematics. > *************************************************** > Oh, I bet elec. eng. love maths; the problem seems to be that maths > does not correspond AT ALL that love, at least in the case of several > engineers...**sigh**...tough. > Tonio What is maths?
From: Bob Kolker on 5 Dec 2006 15:54
Eckard Blumschein wrote: > I wonder who educated Bob. Didn't he intend to kill the maybe million of > muslims who live in Germany by means of atomic bombs? Not in Germany. In the Middle East. The Germans can take care of their own Muslims. They have a lot of practice at dealing with undersirable populations. Bob Kolker |