[FOM] Simple and difficult
Timothy Y. Chow
tchow at alum.mit.edu
Fri Apr 5 13:20:54 EDT 2013
Joe Shipman wrote:
> What's the shortest or simplest sentence you can come up with in the
> language of set theory that is either (1) not settled (2) provably not a
> theorem of ZFC if ZFC is consistent?
There's Frankl's union-closed sets conjecture.
http://en.wikipedia.org/wiki/Union-closed_sets_conjecture
One catch is that this conjecture involves the notion of a finite set, and
expressing finiteness is a bit of a nuisance. Maybe there's some way to
get around this?
Once you can express finiteness, the conjecture is that if
1. S is finite;
2. if x is in S then x is finite;
3. S is different from {{}};
4. x in S and y in S implies x U y in S;
then there exists z and a surjection from A := {x in S : z in x} onto S\A.
Tim
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