Every now and then, somebody will tell me about a question. When I start thinking about it, they say, "actually, it's undecidable in ZFC."

For example, suppose A is an abelian group such that every short exact sequence of abelian groups 0→ℤ→B→A→0 splits. Does it follow that A is free? This is known as Whitehead's Problem, and it's undecidable in ZFC.

What are some other statements that aren't directly set-theoretic, and you'd think that playing with them for a week would produce a proof or counterexample, but they turn out to be undecidable? One answer per post, please, and include a reference if possible.

Every now and then, somebody will tell me about a question. When I start thinking about it, they say, "actually, it's undecidable in ZFC."

For example, suppose $A$ is an abelian group such that every short exact sequence of abelian groups $0\to\mathbb Z\to B\to A\to0$ splits. Does it follow that $A$ is free? This is known as Whitehead's Problem, and it's undecidable in ZFC.

What are some other statements that aren't directly set-theoretic, and you'd think that playing with them for a week would produce a proof or counterexample, but they turn out to be undecidable? One answer per post, please, and include a reference if possible.