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 are answers which, in turning away wrath, only send it to the other end of the room, and to have a discussion coolly waived when you feel that justice is all on your own side is even more exasperating in marriage than in philosophy.”
—George Eliot [Mary Ann (or Marian)
“My companion and I, having a minute’s discussion on some point of ancient history, were amused by the attitude which the Indian, who could not tell what we were talking about, assumed. He constituted himself umpire, and, judging by our air and gesture, he very seriously remarked from time to time, “you beat,” or “he beat.””
—Henry David Thoreau (1817–1862)
“If we had had more time for discussion we should probably have made a great many more mistakes.”
—Leon Trotsky (1879–1940)