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
historicHistory & 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.
Connected problems
- related The Continuum Hypothesis
References
- primaryRead the primary source →
- sourceWikipedia — abelian group
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.
Verified by curator — see the API for full claim provenance.