The cofinality spectrum of the infinite symmetric group

Journal of Symbolic Logic 62 (3):902-916 (1997)
Abstract
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 $V \models GCH$ . 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 that $\mu = \sup(C \cap \mu)$ , then μ ∈ C. (c) If μ is a singular cardinal such that $\mu = \sup(C \cap \mu)$ , then μ + ∈ C. Then there exists a c.c.c. notion of forcing P such that $V^\mathbb{P} \models CF(S) = C$ . 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
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