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Precipitous Ideals

DOI: 10.1016/s0049-237x(09)70443-4
  • Mathematics


Publisher Summary This chapter describes precipitous ideals and presents an example of application of generic ultrapowers. The chapter presents the theorem of a typical application of precipitous ideals and generic ultrapowers, along with an observation based on Mitchell's proof. Ultrapowers have been used, among others, to obtain results on cardinal exponentiation. While small cardinals do not carry κ-complete prime ideals, they do carry κ-complete ideals that are not prime. The chapter introduces Magidor's model that is a generic extension of a model, which has a supercompact cardinal. It is reasonable to conjecture that the assumption that the closed unbounded ideal is precipitous is stronger than measurability (consistency wise); for example, one might expect that the assumption implies existence of inner models with many measurable cardinals.

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