Varieties of Infinity
Brendan Frost (Brendan_Frost@cch.com)
Tue, 30 Jun 1998 11:36:06 -0400
JR>One aside: it's tricky to apply things like "minority" and
>"majority" to infinite subsets of infinite sets of universes [if each set
>is of the same cardinality] -- one needs some non-counting-based
>way to define such. E.g., it seems natural to say that only a minority
>of the natural numbers are powers of 1000, but since there are
>infinitely many of each [and the same "order" of infinity], one can't
>say there are more of one than the other, merely quantitatively!
>similarly: are there more points in a 5x5 square of Euclidean space
>than a 1x1 square? No, even though the area of the 5x5 square is 25
>times larger. But that isn't determined by counting points.
SEJ>This type of thing is why I don't believe that an actual infinity of
>physical things is possible.
If anyone is interested in this type of question, it is my understanding
that
the advanced mathematics of Georg Cantor, his work with "transfinite"
functions
in the late 1800's, deals with precisely this type of question, i.e.
weighing
different values of different generations of infinity. I also understand
there's
a moralistic, polemical element to his work as well. Wish I knew
more---does
anyone out there have a better grasp they would like to impart?
Brendan Frost
Washington, D.C.