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.
Read more about this topic: Whitehead Problem
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