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Hindman's theorem: an ultrafilter argument in second order arithmetic

Published online by Cambridge University Press:  12 March 2014

Henry Towsner
Henry Towsner*
Affiliation:
Department Of Mathematics, University of California Los Angeles, Los Angeles, Ca 90095-1555, USA, E-mail: htowsner@gmail.com
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Abstract

Hindman's Theorem is a prototypical example of a combinatorial theorem with a proof that uses the topology of the ultrafilters. We show how the methods of this proof, including topological arguments about ultrafilters, can be translated into second order arithmetic.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 76 , Issue 1 , March 2011 , pp. 353 - 360
Copyright
Copyright © Association for Symbolic Logic 2011

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References

REFERENCES

[1]Avigad, Jeremy, An effective proof that open sets are Ramsey, Archive for Mathematical Logic, vol. 37 (1998), no. 4, pp. 235–240.CrossRefGoogle Scholar
[2]Baumgartner, James E., A short proof of Hindman's theorem, Journal of Combinatorial Theory, Series A, vol. 17 (1974), pp. 384–386.CrossRefGoogle Scholar
[3]Blass, Andreas R., Hirst, Jeffry L., and Simpson, Stephen G., Logical analysis of some theorems of combinatorics and topological dynamics, Logic and combinatorics (Areata, Calif, 1985) (Providence, RI), Contemp. Math., vol. 65, Amer. Math. Soc., 1987, pp. 125–156.CrossRefGoogle Scholar
[4]Comfort, W. W., Ultrafilters: some old and some new results, Bulletin of the American Mathematical Society, vol. 83 (1977), no. 4, pp. 417–455.CrossRefGoogle Scholar
[5]Ellis, Robert, Lectures on topological dynamics, W. A. Benjamin, Inc., New York, 1969.Google Scholar
[6]Shore, Richardet al., Computability, reverse mathematics and combinatorics: Open problems, 12 2008, http://robson.Mrs.ca/~08w5019/problems.pdf.Google Scholar
[7]Hindman, Neil, Finite sums from sequences within cells of a partition of N, Journal of Combinatorial Theory, Series A, vol. 17 (1974), pp. 1–11.CrossRefGoogle Scholar
[8]Hindman, Neil and Strauss, Dona, Algebra in the Stone-Čech compactification, de Gruyter Expositions in Mathematics, vol. 27, Walter de Gruyter & Co., Berlin, 1998.CrossRefGoogle Scholar
[9]Hirst, Jeffry L., Hindmari's theorem, ultrafilters, and reverse mathematics, this Journal, vol. 69 (2004), no. 1, pp. 65–72.Google Scholar
[10]Simpson, Stephen G., Subsystems of second order arithmetic, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1999.CrossRefGoogle Scholar
[11]Towsner, Henry, A combinatorial proof of the dense Hindman's theorem, http://arxiv.org/abs/1002.0347.Google Scholar
[12]Towsner, Henry, A simple proof of Hindman's theorem, http://arxiv.org/abs/0906.3885.Google Scholar