Advertisement

Archive for Mathematical Logic

, Volume 55, Issue 3–4, pp 493–504 | Cite as

Mathias–Prikry and Laver type forcing; summable ideals, coideals, and +-selective filters

  • David Chodounský
  • Osvaldo Guzmán González
  • Michael Hrušák
Article

Abstract

We study the Mathias–Prikry and the Laver type forcings associated with filters and coideals. We isolate a crucial combinatorial property of Mathias reals, and prove that Mathias–Prikry forcings with summable ideals are all mutually bi-embeddable. We show that Mathias forcing associated with the complement of an analytic ideal always adds a dominating real. We also characterize filters for which the associated Mathias–Prikry forcing does not add eventually different reals, and show that they are countably generated provided they are Borel. We give a characterization of \({\omega}\)-hitting and \({\omega}\)-splitting families which retain their property in the extension by a Laver type forcing associated with a coideal.

Keywords

Mathias–Prikry forcing Laver type forcing Mathias like real \({+}\)-Selective filter Dominating real Eventually different real \({\omega}\)-Hitting 

Mathematics Subject Classification

Primary 03E05 03E17 03E35 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Brendle J., Hrušák M.: Countable Fréchet Boolean groups: an independence result. J. Symb. Logic 74(3), 1061–1068 (2009)CrossRefzbMATHGoogle Scholar
  2. 2.
    Bartoszyński, T., Judah, H.: Set theory: on the structure of the real line. A K Peters, Ltd., Wellesley (1995)Google Scholar
  3. 3.
    Brendle J., Raghavan D.: Bounding, splitting, and almost disjointness. Ann. Pure Appl. Logic 165(2), 631–651 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Canjar R. M.: Mathias forcing which does not add dominating reals. Proc. Am. Math. Soc. 104(4), 1239–1248 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chodounský, D., Repovš, D., Zdomskyy, L.: Mathias forcing and combinatorial covering properties of filters. J. Symbolic Logic 80(4), 1398–1410 (2015). doi: 10.1017/jsl.2014.73
  6. 6.
    Dow A.: Two classes of Fréchet–Urysohn spaces. Proc. Amer. Math. Soc. 108(1), 241–247 (1990)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Farah I.: Analytic quotients: theory of liftings for quotients over analytic ideals on the integers. Mem. Am. Math. Soc. 148(702), xvi–177 (2000)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Farah I.: How many Boolean algebras \({\mathscr{P}( {\mathbb{N}})/\mathscr{I}}\) are there?. Illinois J. Math. 46(4), 999–1033 (2002)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Farah, I., Solecki, S.: Two \({F_{\sigma\delta}}\) ideals. Proc. Am. Math. Soc. 131(6), 1971–1975 (electronic) (2003)Google Scholar
  10. 10.
    Guzmán, O., Hrušák, M., Martínez-Celis, A.: Canjar filters. Notre Dame J. Form. Log. (2014)Google Scholar
  11. 11.
    Groszek M. J.: Combinatorics on ideals and forcing with trees. J. Symb. Logic 52(3), 582–593 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hrušák M., Minami H.: Mathias-Prikry and Laver-Prikry type forcing. Ann. Pure Appl. Logic 165(3), 880–894 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hrušák M., Verner J. L.: Adding ultrafilters by definable quotients. Rend. Circ. Mat. Palermo (2) 60(3), 445–454 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hrušák M., Zapletal J.: Forcing with quotients. Arch. Math. Logic 47(7-8), 719–739 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Just W., Krawczyk A.: On certain Boolean algebras \({\mathscr{P}(\omega )/I}\). Trans. Am. Math. Soc. 285(1), 411–429 (1984)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Laflamme, C.: Filter games and combinatorial properties of strategies. In: Set theory (Boise, ID, 1992–1994). Contemporary mathematics, vol. 192, pp. 51–67. American Mathematical Society, Providence, RI (1996). doi: 10.1090/conm/192/02348
  17. 17.
    Mathias A.R.D.: Happy families. Ann. Math. Logic 12(1), 59–111 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Oliver M. R.: Continuum-many Boolean algebras of the form \({{\mathscr{P}}(\omega)/\mathscr{I} \mathscr{I}}\) Borel. J. Symb. Logic 69(3), 799–816 (2004)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Pawlikowski J.: Undetermined sets of point-open games. Fund. Math. 144(3), 279–285 (1994)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Shelah S.: Properness without elementaricity. J. Appl. Anal. 10(2), 169–289 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Steprāns, J.: Many quotient algebras of the integers modulo co-analytic ideals. Logic Its Appl. Contemp. Math., vol. 380, Am. Math. Soc., Providence, pp. 271–281 (2005)Google Scholar
  22. 22.
    Vopěnka, P., Hájek, P.: The theory of semisets. Academia (Publishing House of the Czechoslovak Academy of Sciences), Prague, p. 332 (1972)Google Scholar
  23. 23.
    Zapletal J.: Forcing Idealized, Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (2008)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • David Chodounský
    • 1
  • Osvaldo Guzmán González
    • 2
  • Michael Hrušák
    • 3
  1. 1.Institute of Mathematics, Czech Academy of SciencesŽitná 25Praha 1Czech Republic
  2. 2.Instituto de MatemáticasUniversidad Nacional Autónoma de MéxicoMoreliaMéxico
  3. 3.Instituto de MatemáticasUniversidad Nacional Autónoma de México, Área de la Investigación Científica, Circuito Exterior, Ciudad UniversitariaMéxicoMéxico

Personalised recommendations