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.
Read more about this topic: Whitehead Problem
Famous quotes containing the word discussion:
“There exist few things more tedious than a discussion of general ideas inflicted by author or reader upon a work of fiction.”
—Vladimir Nabokov (1899–1977)
“We should seek by all means in our power to avoid war, by analysing possible causes, by trying to remove them, by discussion in a spirit of collaboration and good will. I cannot believe that such a programme would be rejected by the people of this country, even if it does mean the establishment of personal contact with the dictators.”
—Neville Chamberlain (1869–1940)
“What chiefly distinguishes the daily press of the United States from the press of all other countries is not its lack of truthfulness or even its lack of dignity and honor, for these deficiencies are common to the newspapers everywhere, but its incurable fear of ideas, its constant effort to evade the discussion of fundamentals by translating all issues into a few elemental fears, its incessant reduction of all reflection to mere emotion. It is, in the true sense, never well-informed.”
—H.L. (Henry Lewis)