[FOM] Simple and difficult

[Timothy Y. Chow]

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