[FOM] finite choice question
Stephen Fenner
fenner at cse.sc.edu
Wed Nov 16 12:38:02 EST 2005
This is a basic question about ZF set theory.
It's not hard to see that the following is true:
METATHEOREM: For any fixed natural number n, the sentence, "For any
sequence <X1,...,Xn> of pairwise disjoint, nonempty sets, there is a set C
such that (C intersect Xi) is a singleton for each i in {1,...,n}" is a
theorem of ZF.
But is the following a theorem of ZF?
"For any natural number n and any sequence <X1,...,Xn> of pairwise
disjoint, nonempty sets, there is a set C such that (C intersect Xi) is a
singleton for each i in {1,...,n}."
Note that, unlike other finite versions of AC, the cardinalities of the Xi
are unrestricted.
Steve Fenner
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