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In group theory, the Whitehead problem is the following question:
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 = idA.
The question was asked by J. H. C. Whitehead in the 1950s, motivated by the second Cousin problem. The affirmative answer for countable groups was already found in the 1950s. Progress for larger groups was slow, and the problem was considered one of the most important ones in algebra for many years.
In 1973, Saharon Shelah showed that from the standard ZFC axiom system, the statement can be neither proven nor disproven.
This result was completely unexpected. While the existence of undecidable statements had been known since Gödel's incompleteness theoremIn mathematical logic, Godel's incompleteness theorems are two celebrated theorems proved by Kurt Godel in 1930. Somewhat simplified, the first theorem states: In any consistent formalization of mathematics that is sufficiently strong to define the concep of 1931, previous examples of undecidable statements (such as the Continuum hypothesisIn mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. Georg Cantor introduced the concept of cardinality to compare the sizes of infinite sets, and he showed that the set of integers is strictly smaller than t) had been confined to the realm of set theory. The Whitehead problem was the first purely algebraic problem that was shown to be undecidable.
The Whitehead problem remains undecidable even if one assumes the Continuum hypothesis, as shown by Shelah in 1980. Various similar independence statements were proved and it was realized more and more that the theory of abelian groups depends very sensitively on the underlying set theory.