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.
Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.