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:
“It was heady stuff, recognizing ourselves as an oppressed class, but the level of discussion was poor. We explained systemic discrimination, and men looked prettily confused and said: But, I like women.”
—Jane OReilly, U.S. feminist and humorist. The Girl I Left Behind, ch. 2 (1980)
“Bigotry is the disease of ignorance, of morbid minds; enthusiasm of the free and buoyant. Education and free discussion are the antidotes of both.”
—Thomas Jefferson (17431826)
“If the abstract rights of man will bear discussion and explanation, those of women, by a parity of reasoning, will not shrink from the same test: though a different opinion prevails in this country.”
—Mary Wollstonecraft (17591797)