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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

Boolean Algebra Winning Strategy Regular Cardinal Force Notion Stationary Subset 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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