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The cofinality spectrum of the infinite symmetric group

Published online by Cambridge University Press:  12 March 2014

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

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 62 , Issue 3 , September 1997 , pp. 902 - 916
Copyright
Copyright © Association for Symbolic Logic 1997

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References

REFERENCES

[1]Burke, M. R. and Magidor, M., Shelah's pcf theory and its applications, Annals of Pure and Applied Logic, vol. 50 (1990), pp. 207–254.CrossRefGoogle Scholar
[2]Gitik, M. and Magidor, M., The singular cardinal hypothesis revisited, Set theory of the continuum (Judah, H., Just, W., and Woodin, H., editors), Mathematical Sciences Research Institute Publications, vol. 26, Springer-Verlag, 1992, pp. 243–279.CrossRefGoogle Scholar
[3]Hodges, W., Hodkinson, I., Lascar, D., and Shelah, S., The small index property for ω-stable ω-categorical structures and for the random graph, Journal of the London Mathematical Society, vol. 48 (1993), no. 2, pp. 204–218.CrossRefGoogle Scholar
[4]Kunen, K., Set theory, an introduction to independence proofs, North Holland, Amsterdam, 1980.Google Scholar
[5]Macpherson, H. D. and Neumann, P. M., Subgroups of infinite symmetric groups, Journal of the London Mathematical Society, vol. 42 (1990), no. 2, pp. 64–84.CrossRefGoogle Scholar
[6]Sharp, J. D. and Thomas, Simon, Uniformisation problems and the cofinality of the infinite symmetric group, Notre Dame Journal of Formal Logic, vol. 35 (1994), pp. 328–345.CrossRefGoogle Scholar
[7]Sharp, J. D. and Thomas, Simon, Unbounded families and the cofinality of the infinite symmetric group, Arch. Math. Logic, vol. 34 (1995), pp. 33–45.CrossRefGoogle Scholar
[8]Shelah, S., Further cardinal arithmetic, to appear in Israel Journal of Mathematics.Google Scholar
[9]Shelah, S., PCF and infinite free subsets, submitted to Archive for Mathematical Logic.Google Scholar
[10]Shelah, S., Strong partition relations below the power set: Consistency. Was Sierpinski right? II, in Proceedings of the Conference on Set Theory and its Applications in honor of A, Hajnal and V. T. Sos, Budapest, Sets, Graphs and Numbers, vol. 60 of Colloquia Mathematica Societatis Janos Bolyai (1991), pp. 637–668.Google Scholar
[11]Shelah, S., Cardinal arithmetic for skeptics, American Mathematical Society Bulletin, New Series, vol. 26 (1992), pp. 197–210.CrossRefGoogle Scholar
[12]Shelah, S., Advances in cardinal arithmetic, Finite and infinite combinatorics in sets and logic (Sauer, N. W., Woodrow, R. E., and Sands, B., editors), Kluwer Academic Publishers, 1993, pp. 355–383.CrossRefGoogle Scholar
[13]Shelah, S., Cardinal arithmetic, Oxford Logic Guides, vol. 29, Oxford University Press, 1994.CrossRefGoogle Scholar