In group theory, a branch of abstract algebra, the Whitehead problem is the following question: Is every abelian group A with Ext1(A, Z) = 0 a free abelian group?Saharon Shelah proved that Whitehead's problem is independent of ZFC, the standard axioms of set theory.
It should be mentioned that if this condition is strengthened by requiring that the exact sequence must split for any abelian group C, then it is well known that this is equivalent to A being free.
[2] Progress for larger groups was slow, and the problem was considered an important one in algebra for some years.
Shelah later showed that the Whitehead problem remains undecidable even if one assumes the continuum hypothesis.
[3][4] In fact, it remains undecidable even under the generalised continuum hypothesis.