Is every abelian group A with Ext¹⁽A, Z) = 0 a free abelian group?

Shelah (1974) proved that Whitehead's problem is undecidable within standard ZFC set theory.

Refinement

The condition Ext¹⁽(A, Z) = 0 can be equivalently formulated as follows: whenever B is an abelian group and f : B → A is a surjective group homomorphism whose kernel is isomorphic to the group of integers Z, then there exists a group homomorphism g : A → B with fg = id_A. Abelian groups satisfying this condition are sometimes called Whitehead groups, so Whitehead's problem asks: is every Whitehead group free?

Caution: The converse of Whitehead's problem, namely that every free abelian group is Whitehead, is a well known group-theoretical fact. Some authors call Whitehead group only a non-free group A satisfying Ext¹⁽(A, Z) = 0. Whitehead's problem then asks: do Whitehead groups exist?
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.
Discussion

J. H. C. Whitehead, motivated by the second Cousin problem, first posed the problem in the 1950s. Stein (1951) answered the question in the affirmative for countable groups. Progress for larger groups was slow, and the problem was considered an important one in algebra for some years.

Shelah's result was completely unexpected. While the existence of undecidable statements had been known since Gödel's incompleteness theorem of 1931, previous examples of undecidable statements (such as the continuum hypothesis) had all been in pure set theory. The Whitehead problem was the first purely algebraic problem to be proved undecidable.

Shelah (1977, 1980) later showed that the Whitehead problem remains undecidable even if one assumes the Continuum hypothesis. The Whitehead conjecture is true if all sets are constructible. That this and other statements about uncountable abelian groups are provably independent of ZFC shows that the theory of such groups is very sensitive to the assumed underlying set theory.
See also

Free abelian group
Whitehead group
List of statements undecidable in ZFC
Statements true if all sets are constructible

References

Eklof, Paul C. (1976), "Whitehead's Problem is Undecidable", The American Mathematical Monthly (The American Mathematical Monthly, Vol. 83, No. 10) 83 (10): 775–788, doi:10.2307/2318684, JSTOR 2318684 An expository account of Shelah's proof.
Eklof, P.C. (2001), "Whitehead problem", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1556080104
Shelah, S. (1974), "Infinite Abelian groups, Whitehead problem and some constructions", Israel Journal of Mathematics 18 (3): 243–256, doi:10.1007/BF02757281, MR0357114
Shelah, S. (1977), "Whitehead groups may not be free, even assuming CH. I", Israel Journal of Mathematics 28 (3): 193–203, doi:10.1007/BF02759809, MR0469757
Shelah, S. (1980), "Whitehead groups may not be free, even assuming CH. II", Israel Journal of Mathematics 35 (4): 257–285, doi:10.1007/BF02760652, MR0594332
Stein, Karl (1951), "Analytische Funktionen mehrerer komplexer Veränderlichen zu vorgegebenen Periodizitätsmoduln und das zweite Cousinsche Problem", Math. Ann. 123: 201–222, doi:10.1007/BF02054949, MR0043219

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