# [FOM] finite choice question

Thomas Forster T.Forster at dpmms.cam.ac.uk
Thu Nov 17 02:30:23 EST 2005

There is an article by Conway on this.  He gives some interesting
nontrivial relations between the AC_n.  The broad picture is that AC_n,
for n finite, is known to be weaker than AC_{< \aleph_0} (where the Xi
must all be finite but there is no bound stipulated.  That in turn in
know to be weaker than the ordering principle.   See the article by Levy
in the Skolem memorial NH volume 1965(?)

On Wed, 16 Nov 2005, Stephen
Fenner wrote:

> 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
> _______________________________________________
> FOM mailing list
> FOM at cs.nyu.edu
> http://www.cs.nyu.edu/mailman/listinfo/fom
>

--

URL: www.dpmms.cam.ac.uk/~tf   Tel: +44-1223-337981
(U Cambridge); +44-20-7882-3659 (QMW); +32-2-650-5853 (ULB)