Worked solution: Shelah's proof that the Whitehead problem is independent of ZFC (1974)
is pure homological algebra, computed the same way whether or not anyone worries about uncountable cardinals or forcing. Yet whether its vanishing forces to be free turned out to hinge entirely on which extra set-theoretic axiom, beyond , one assumes — says yes, says no.
This stunned algebraists in the 1970s: an "ordinary" question about abelian groups, phrased with no reference to set theory whatsoever, turned out to be exactly as undecidable from as the continuum hypothesis itself.
Combining the two halves: relative to , both (Step 5, via , meaning "every Whitehead group is free") and (Step 6, via , meaning "some Whitehead group of cardinality is not free") are consistent theories. So Whitehead's problem — is every Whitehead group free? — is formally undecidable from alone, exactly as defined for the continuum hypothesis in the companion proof.
This is widely regarded as the first major example of an independent statement from mainstream algebra, outside of set theory or logic themselves. Why does set theory enter here at all? Because the statement quantifies over all abelian groups of arbitrary infinite cardinality, and any such statement about uncountable structures can encode enough combinatorics about how countable pieces fit together to become sensitive to exactly the kind of axioms — , Martin's Axiom — that govern that combinatorics. Most algebra avoids this simply because it deals with countable or finitely generated structures, where Stein's greedy argument (Step 3) always succeeds and no such sensitivity can arise.
Shelah's 1974 result, for this and later related work, contributed to his reputation as one of the most prolific and influential logicians of the twentieth century; the next step surveys how this discovery reshaped the study of infinite abelian groups as a field.