Editing Whitehead problem (section)

==Shelah's proof==
Saharon Shelah showed that,  given the canonical [[ZFC]] axiom system, the problem is [[Independence (mathematical logic)|independent of the usual axioms of set theory]].{{r|Shelah1974}} More precisely, he showed that:
* If [[Axiom of constructibility|every set is constructible]], then every Whitehead group is free;
* If [[Martin's axiom]] and the negation of the [[continuum hypothesis]] both hold, then there is a non-free Whitehead group.
Since the [[consistency]] of ZFC implies the consistency of both of the following:
*The axiom of constructibility (which asserts that all sets are constructible);
*Martin's axiom plus the negation of the continuum hypothesis,
Whitehead's problem cannot be resolved in ZFC.