Proper Forcing pp 394-409 | Cite as

Notes on Improper Forcing

  • Saharon Shelah
Part of the Lecture Notes in Mathematics book series (LNM, volume 940)

Abstract

In X we prove general theorems on semi-proper forcing notions, and iterations. We use them by iterating several forcing, one of them, and an important one, is Namba forcing. But to show Namba forcing is semi-proper, we need essentially that 2 is in a large cardinal which was collapsed to 2 (more exactly — a consequence of this on Galvin games). In XI we take great trouble to use a notion considerably more complicated than semi-properness which is satisfied by Namba forcing. However it was not clear whether all this is necessary as we do not exclude the possibility that Namba forcing is always semi-proper, or at least some other forcing, fulfilling the main function of Namba forcing (i.e., making the cofinality of 2 to ω without collapsing 1). But we prove in 1.2 here, that: there is such semi-proper forcing, iff Namba forcing is semi-proper, iff player II wins Gm({ 1},ω, 2) (a game similar to Galvin games) and, in 1.3, that this implies Chang conjecture. Now it is well known that Chang conjecture implies 0# exists, so e.g., in ZFC we cannot prove the existence of such semi-proper forcing.

Keywords

Hull 

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Copyright information

© Springer-Verlag Berlin Heidelberg 1982

Authors and Affiliations

  • Saharon Shelah
    • 1
    • 2
    • 3
    • 4
  1. 1.Institute of MathematicsThe Hebrew UniversityJerusalemIsrael
  2. 2.Department of MathematicsUniversity of CaliforniaBerkeleyUSA
  3. 3.Department of MathematicsOhio State UniversityColumbusUSA
  4. 4.Institute of Advanced StudiesThe Hebrew UniversityJerusalemIsrael

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