Published online by Cambridge University Press: 12 March 2014
Let S be the group of all permutations of the set of natural numbers. The cofinality spectrum CF(S) of S is the set of all regular cardinals λ such that S can be expressed as the union of a chain of λ proper subgroups. This paper investigates which sets C of regular uncountable cardinals can be the cofinality spectrum of S. The following theorem.is the main result of this paper.
Theorem. Suppose that V ⊨ GCH. Let C be a set of regular uncountable cardinals which satisfies the following conditions.
(a) C contains a maximum element.
(b) If μ is an inaccessible cardinal such thatμ = sup(C ∩ μ), thenμ ∈ C.
(c) if μ is a singular cardinal such thatμ = sup(C ∩ μ), thenμ+ ∈ C.
Then there exists a c.c.c. notion of forcing ℙ such that Vℙ ⊨ CF(S) = C.
We shall also investigate the connections between the cofinality spectrum and pcf theory; and show that CF(S) cannot be an arbitrarily prescribed set of regular uncountable cardinals.