[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
>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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