The problem
Question: must every abelian group A satisfying Ext¹(A, ℤ) = 0 (a 'Whitehead group') be free? Resolution: the statement is INDEPENDENT of ZFC — true in Gödel's constructible universe L, false under Martin's Axiom + ¬CH.
Whitehead's Problem
historicReferences
History & significance
Whitehead asked circa 1952 while studying homotopy. Eklof and Sabbagh connected it to set-theoretic axioms by 1971; Saharon Shelah proved independence in 1974 (under V=L every Whitehead group is free; forcing yields non-free ones). Unlike CH — a statement ABOUT sets — this is an ordinary-looking algebraic classification question whose answer depends on which universe you inhabit.
The resolution (human proof)
Independence established by Saheloh Shelah, 1974, refining the Jensen cover/ladder machinery from L. The problem became the textbook demonstration that everyday algebra can probe the set-theoretic sky: no ZFC verdict exists, none is expected, and 'which axiom system?' is now simply part of the answer. Companion piece to our Continuum Hypothesis entry on the historic shelf.