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 

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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

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