Iterated hyper-extensions and an idempotent ultrafilter proof of Rado’s Theorem

Di Nasso, Mauro

doi:10.1090/S0002-9939-2014-12342-2

Iterated hyper-extensions and an idempotent ultrafilter proof of Rado’s Theorem
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by Mauro Di Nasso

Proc. Amer. Math. Soc. 143 (2015), 1749-1761

DOI: https://doi.org/10.1090/S0002-9939-2014-12342-2

Published electronically: December 8, 2014

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Abstract:

By using nonstandard analysis, and in particular iterated hyper-extensions, we give foundations to a peculiar way of manipulating ultrafilters on the natural numbers and their pseudo-sums. The resulting formalism is suitable for applications in Ramsey theory of numbers. To illustrate the use of our technique, we give a (rather) short proof of Milliken-Taylor’s Theorem and a ultrafilter version of Rado’s Theorem about partition regularity of diophantine equations.

References

David Ballard and Karel Hrbáček, Standard foundations for nonstandard analysis, J. Symbolic Logic 57 (1992), no. 2, 741–748. MR 1169206, DOI 10.2307/2275304
Vieri Benci, A construction of a nonstandard universe, Advances in dynamical systems and quantum physics (Capri, 1993) World Sci. Publ., River Edge, NJ, 1995, pp. 11–21. MR 1414687
Ben Barber, Neil Hindman, and Imre Leader, Partition regularity in the rationals, J. Combin. Theory Ser. A 120 (2013), no. 7, 1590–1599. MR 3092686, DOI 10.1016/j.jcta.2013.05.011
Vitaly Bergelson and Neil Hindman, Nonmetrizable topological dynamics and Ramsey theory, Trans. Amer. Math. Soc. 320 (1990), no. 1, 293–320. MR 982232, DOI 10.1090/S0002-9947-1990-0982232-5
Greg Cherlin and Joram Hirschfeld, Ultrafilters and ultraproducts in non-standard analysis, Contributions to non-standard analysis (Sympos., Oberwolfach, 1970), Studies in Logic and the Foundations of Mathematics, Vol. 69, North-Holland, Amsterdam, 1972, pp. 261–279. MR 0485344
C. C. Chang and H. J. Keisler, Model theory, 3rd ed., Studies in Logic and the Foundations of Mathematics, vol. 73, North-Holland Publishing Co., Amsterdam, 1990. MR 1059055
Mauro Di Nasso, ${}^\ast \textrm {ZFC}$: an axiomatic ${}^\ast$approach to nonstandard methods, C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), no. 9, 963–967 (English, with English and French summaries). MR 1451233, DOI 10.1016/S0764-4442(97)87868-8
Mauro Di Nasso, An axiomatic presentation of the nonstandard methods in mathematics, J. Symbolic Logic 67 (2002), no. 1, 315–325. MR 1889553, DOI 10.2178/jsl/1190150046
Vieri Benci and Mauro Di Nasso, Alpha-theory: an elementary axiomatics for nonstandard analysis, Expo. Math. 21 (2003), no. 4, 355–386. MR 2022004, DOI 10.1016/S0723-0869(03)80038-5
Mauro Di Nasso, $\text {ZFC}[\Omega ]$: A nonstandard set theory where all sets have nonstandard extensions (abstract), Bull. Symb. Logic, vol. 7, 2001, p. 138.
Mauro Di Nasso, Hyper-natural numbers as ultrafilters, unpublished manuscript.
Robert Goldblatt, Lectures on the hyperreals, Graduate Texts in Mathematics, vol. 188, Springer-Verlag, New York, 1998. An introduction to nonstandard analysis. MR 1643950, DOI 10.1007/978-1-4612-0615-6
Neil Hindman, Imre Leader, and Dona Strauss, Open problems in partition regularity, Combin. Probab. Comput. 12 (2003), no. 5-6, 571–583. Special issue on Ramsey theory. MR 2037071, DOI 10.1017/S0963548303005716
Neil Hindman, Imre Leader, and Dona Strauss, Infinite partition regular matrices: solutions in central sets, Trans. Amer. Math. Soc. 355 (2003), no. 3, 1213–1235. MR 1938754, DOI 10.1090/S0002-9947-02-03191-4
N. Hindman and D. Strauss, Algebra in the Stone-Čech Compactification, Theory and Applications (2nd edition), W. de Gruyter, 2011.
Vladimir Kanovei and Michael Reeken, Nonstandard analysis, axiomatically, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2004. MR 2093998, DOI 10.1007/978-3-662-08998-9
L. Luperi Baglini, Hyperintegers and Nonstandard Techniques in Combinatorics of Numbers, Ph.D. dissertation, 2012, University of Siena, arXiv:1212.2049.
Amir Maleki, Solving equations in $\beta \Bbb N$, Semigroup Forum 61 (2000), no. 3, 373–384. MR 1832314, DOI 10.1007/PL00006036
Siu-Ah Ng and Hermann Render, The Puritz order and its relationship to the Rudin-Keisler order, Reuniting the antipodes—constructive and nonstandard views of the continuum (Venice, 1999) Synthese Lib., vol. 306, Kluwer Acad. Publ., Dordrecht, 2001, pp. 157–166. MR 1895391