Worked solution: Shelah's proof that the Whitehead problem is independent of ZFC (1974)

Working under $V = L$, hence assuming $\diamondsuit_{\aleph_1}$, Shelah (1974) showed: given an $\aleph_1$-free Whitehead group $W$ with filtration $\langle W_\alpha : \alpha < \omega_1 \rangle$, the diamond sequence can be used to predict, stationarily often, exactly the kind of "bad" quotient $W_{\alpha+1}/W_\alpha$ that would obstruct Pontryagin's freeness criterion; whenever such a prediction is confirmed, the vanishing of $\mathrm{Ext}^1(W, \mathbb{Z})$ is used to correct course at that stage, ensuring the obstruction never actually manifests.

Because $\diamondsuit$ guarantees stationarily many correct predictions, every possible obstruction gets caught and defeated, so the filtration ends up well-behaved on a club of stages, and Pontryagin's criterion certifies $W$ is free. Shelah extended this argument, by an additional induction on cardinality (a technique later systematized as "singular compactness"), to Whitehead groups of every infinite cardinality, not just $\aleph_1$.

Hence $V = L$ implies every Whitehead group is free: combined with Stein's countable case, this settles Whitehead's question with the answer "yes" throughout the entire constructible universe. Since $L$ satisfies $\mathrm{ZFC}$ (as in the companion proof of the continuum hypothesis), this shows $\mathrm{Con}(\mathrm{ZFC}) \implies \mathrm{Con}(\mathrm{ZFC} + \mathrm{Free}(\mathrm{Wh}))$, where $\mathrm{Free}(\mathrm{Wh})$ abbreviates "every Whitehead group is free" — one half of the independence result.