Editing Whitehead problem (section)

==Discussion==
[[J. H. C. Whitehead]], motivated by the [[second Cousin problem]], first posed the problem in the 1950s. Stein answered the question in the affirmative for [[countable]] groups.{{r|Stein1951}} Progress for larger groups was slow, and the problem was considered an important one in [[abstract algebra|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 later showed that the Whitehead problem remains undecidable even if one assumes the continuum hypothesis.{{r|Shelah1977|Shelah1980}} In fact, it remains undecidable even under the [[generalised continuum hypothesis]].<ref>{{cite web |url=https://www.karlin.mff.cuni.cz/~trlifaj/ANK_5.pdf |title=The Whitehead Problem and Beyond (Lecture notes for NMAG565) |last=Triflaj |first=Jan |date=16 February 2023 |website= |publisher=[[Charles University]] |access-date=26 September 2024 |quote=}}</ref> The Whitehead conjecture is true if all sets are [[constructible universe|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]].