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Matheology § 176
Posted:
Dec 10, 2012 4:23 AM
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Matheology § 176
Here's a paradox of infinity noticed by Galileo in 1638. It seems that
the even numbers are as numerous as the evens and the odds put
together. Why? Because they can be put into one-to-one correspondence.
The evens and odds put together are called the natural numbers. The
first even number and the first natural number can be paired; the
second even and the second natural can be paired, and so on. When two
finite sets can be put into one-to-one correspondence in this way,
they always have the same number of members.
Supporting this conclusion from another direction is our intuition
that "infinity is infinity", or that all infinite sets are the same
size. If we can speak of infinite sets as having some number of
members, then this intuition tells us that all infinite sets have the
same number of members.
Galileo's paradox is paradoxical because this intuitive view that the
two sets are the same size violates another intuition which is just as
strong {{and as justified! If it is possible to put two sets A and B
in bijection but also to put A in bijection with a proper subset of B
and to put B in bijection with a proper subset of A, then it is insane
to judge the first bijection as more valid than the others and to talk
about equinumerousity of A and B.}}
[Peter Suber: "Infinite Reflections", St. John's Review, XLIV, 2
(1998) 1-59]
http://www.earlham.edu/~peters/writing/infinity.htm#galileo
Regards, WM
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