Whitehead Problem - Shelah's Proof

Shelah's Proof

Saharon Shelah (1974) showed that, given the canonical ZFC axiom system, the problem is independent of the usual axioms of set theory. More precisely, he showed that:

• If 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 either 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.