On Tue, 30 Jun 1998 11:36:06 -0400, Brendan Frost wrote:
>JR>...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.
>SJ>This type of thing is why I don't believe that an actual infinity
>>of physical things is possible.
BF>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?
Thanks. Cantor did indeed show that there were various levels of
infinity among *numbers*:
"In 1873 Cantor demonstrated that the rational numbers, though
infinite, are countable (or denumerable) because they may be placed
in a one-to-one correspondence with the natural numbers (i.e., the
integers, as 1, 2, 3, ...). He showed that the set (or aggregate) of real
numbers (composed of irrational and rational numbers) was infinite
and uncountable. Even more paradoxically, he proved that the set of
all algebraic numbers (the solutions of simple algebraic equations, as,
sqrt 2 and 5) contains as many components as the set of all integers
and that transcendental numbers (those that are not algebraic, as pi),
which are a subset of the irrational are uncountable and are therefore
more numerous than integers, which must be conceived as infinite."
("Cantor, Georg", Encyclopaedia Britannica, Benton: Chicago, 15th
edition, 1984, Vol. 3, p784)
But my points was whether "an actual infinity of *physical things* is
possible."
While mathematicians might be able to imagine an infinite series of
*numbers*, it does not necessarily follow that those numbers
correspond to any actual infinite series of *physical things* in the real
universe.
For example, while the numbers in a series of integers, as 1, 2, 3,...
are infinite, the actual maximum number of physical things (ie.
elementary particles) is finite:
"There are estimated to be no more than 10^80 elementary particles
in the universe." (Dembski W.A., in Moreland J.P. ed., "The Creation
Hypothesis," 1994, p124)
Therefore if one tried to correspond the infinite number series 1, 2,
3... against all the particles in the universe, ie. count them, one
would have to stop when one reached 10^80. This would still be
an infinite distance short of infinity!
Steve
"Evolution is the greatest engine of atheism ever invented."
--- Dr. William Provine, Professor of History and Biology, Cornell University.
http://fp.bio.utk.edu/darwin/1998/slides_view/Slide_7.html
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3 Hawker Avenue / Oz \ Steve.Jones@health.wa.gov.au
Warwick 6024 ->*_,--\_/ Phone +61 8 9448 7439
Perth, West Australia v "Test everything." (1Thess 5:21)
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