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From: markp@tekmdp.UUCP (Mark Paulin)
Newsgroups: net.math
Subject: Axiom of Choice
Message-ID: <2286@tekmdp.UUCP>
Date: Fri, 7-Oct-83 14:18:50 EDT
Article-I.D.: tekmdp.2286
Posted: Fri Oct  7 14:18:50 1983
Date-Received: Sun, 9-Oct-83 20:13:01 EDT
Lines: 22


My favorite formulation of the "Axiom of Choice" is:

"The cartesian product of a nonempty family of nonempty sets is nonempty."


The counter-intuitive result mentioned is the so-called "Banach-Tarski Paradox"
which is sometimes stated as, "a sphere the size of a pea may be cut into
finitely many pieces which may be assembled into a life-size statue of Banach."

It has been shown that the axiom of choice is required to prove the existence
of a non-lebesgue-measureable set.

To me, acceptance/rejection of the axiom hinges on the difference between what
*can* be done and what *may* be done.  There may be *no way* to actually find
the cartesian product in the formulation above (or the "choice set" in the
other formulation) but that does not mean (to me) that these sets do not exist,
so I accept the axiom.


Mark Paulin
...tektronix!tekmdp!markp