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Ultrapowers as sheaves on a category of ultrafilters

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Abstract.

In the paper we investigate the topos of sheaves on a category of ultrafilters. The category is described with the help of the Rudin-Keisler ordering of ultrafilters. It is shown that the topos is Boolean and two-valued and that the axiom of choice does not hold in it. We prove that the internal logic in the topos does not coincide with that in any of the ultrapowers. We also show that internal set theory, an axiomatic nonstandard set theory, can be modeled in the topos.

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References

  1. Barr, M., Diaconescu, R.: Atomic Toposes. J. Pure and Appl. Algebra 17, 1–24 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  2. Blass, A.: Two closed categories of filters. Fund. Math. 94 (2), 129–143 (1977)

    MATH  Google Scholar 

  3. Blass, A.: The Rudin-Keisler ordering of P-points. Trans. Amer. Math. Soc. 179, 145–166 (1973)

    MATH  Google Scholar 

  4. Chang, C.C., Keisler, H.J.: Model theory third ed. North-Holland 1990

  5. Comfort, W.W., Negrepontis, S.: The theory of ultrafilters Springer-Verlag 1974

  6. Ellerman, D.P.: Sheaves of structures and generalized ultraproducts. Ann. Math. Logic 7, 163–195 (1974)

    Article  MATH  Google Scholar 

  7. Johnstone, P.T.: Topos theory. Academic Press 1977

  8. Koubek, V., Reiterman, J.: On the category of filters. Comment. Math. Univ. Carolinae 11, 19–29 (1970)

    MATH  Google Scholar 

  9. Mac Lane, S., Moerdijk, I.: Sheaves in geometry and logic, Springer-Verlag 1994

  10. Moerdijk, I.: A model for intuitionistic non-standard arithmetic. Ann. Pure Appl. Logic 73 (1), 37–51 (1995)

    Article  MATH  Google Scholar 

  11. Nelson, E.: Internal set theory: a new approach to nonstandard analysis. Bull. Amer. Math. Soc. 83 (6), 1165–1198 (1977)

    MATH  Google Scholar 

  12. Nelson, E.: The syntax of nonstandard analysis. Ann. Pure Appl. Logic 38, 123–134 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  13. Palmgren, E.: A sheaf-theoretic foundation of nonstandard analysis. Ann. Pure Appl. Logic 85 (1), 68–86 (1997)

    Google Scholar 

  14. Palmgren, E.: Developments in constructive nonstandard analysis. Bull. Symbolic Logic 4 (3), 233–272 (1998)

    MATH  Google Scholar 

  15. Palmgren, E.: Real numbers in the topos of sheaves over the category of filters. J. Pure and Appl. Algebra 160 (2-3), 275–284 (2001)

    Google Scholar 

  16. Palmgren, E.: Unifying Constructive and Nonstandard Analysis. In: U. Berger, H. Osswald, and P. Schuster, editors, Reuniting the Antipodes–Constructive and Nonstandard Views of the Continuum Proceedings of the Symposion in San Servolo/Venice, Italy, May 17-22, 1999

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Authors and Affiliations

  1. Department of Mathematics, Uppsala University, P.O. Box 480, S-751 06, Uppsala, Sweden

    Jonas Eliasson

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  1. Jonas Eliasson
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Correspondence to Jonas Eliasson.

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Mathematics Subject Classification (2000): Primary 03G30, 03C20, Secondary 03E05, 03E70, 03H05

The author would like to thank the Mittag-Leffler Institute for partial suport.

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Eliasson, J. Ultrapowers as sheaves on a category of ultrafilters. Arch. Math. Logic 43, 825–843 (2004). https://doi.org/10.1007/s00153-004-0228-0

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