[FOM] Error in statement of AC
Matt Insall
montez at fidnet.com
Wed Nov 5 08:06:48 EST 2003
I want to thank Michael Kremer for correctly pointing out to me that the
statements I sent previously are not correct reformulations of the axiom of
choice. I neglected to include the requiirement that the intersection of
the class in question with every nonempty class is a singleton. When this
correction is made, the number of quantifiers is increased, so that the
corrected statements do not comprise an answer to Harvey's question. The
corrected form of GC is of course
(GC) There is a class whose intersectoin with every nonempty set is a
singleton.
The formal version of this is
(\exists A)(\forall x)[(\exists y)(y\in x) implies {(\exists y)(y\in A and
y\in x) and (\forall y)(\forall z)[(y\in A and y\in x and z\in A and z\in x)
implies y=z]}.
As you can see, this statement has 6 quantifiers. A similar correction of
my ill-formulated AC also yields a statement with more than 6 quantifiers.
My apologies to the list for this error.
Dr. Matt Insall
Associate Professor of Mathematics
Department of Mathematics and Statistics
University of Missouri - Rolla
Rolla MO 65409-0020
insall at umr.edu
(573)341-4901
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