[FOM] Error in statement of AC
Michael Kremer
kremer at midway.uchicago.edu
Wed Nov 5 15:57:10 EST 2003
What is given below is still not a correct version of global choice. It is
provably not the case that there is a class whose intersection with every
non-empty set is a singleton.
Suppose that A were such a class; then the intersection of A and {0} would
be {0}; the intersection of A and {1} would be {1}; consequently both 0
and 1 would be in A; consequently the intersection of A and {0,1} is
{0,1}; by assumption this is a singleton, so 0=1.
--Michael Kremer
At 07:06 AM 11/5/2003 -0600, Matt Insall wrote:
>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]}.
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