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The cofinality spectrum of the infinite symmetric group

Published online by Cambridge University Press:  12 March 2014

Saharon Shelah
Mathematics Department, Bilkent University, Ankara, Turkey
Simon Thomas
Mathematics Department, The Hebrew University, Jerusalem, Israel Mathematics Department, Rutgers University, New Brunswick, New Jersey 08903, USA


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 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(Cμ), thenμC.

(c) if μ is a singular cardinal such thatμ = sup(Cμ), thenμ+C.

Then there exists a c.c.c. notion of forcing ℙ such that V ⊨ 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.

Research Article
Copyright © Association for Symbolic Logic 1997

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