The aim of this chapter is to develop the theory of proper forcings and their iteration and to provide interesting examples of its usefulness and range of applications. Our presentation is detailed and should be accessible to any reader who is familiar with the Solovay and Tennebaum technique of finite support iteration and the proof that the c.c.c. property is preserved. We present the basic preservation theorem of the properness property under countable support iteration, and continue with additional preservation theorems—all due to Shelah. Each abstract preservation theorem is followed by an application which shows its relevance. A longer section deals with posets that add no new countable sets and their iteration.
KeywordsChromatic Number Continuum Hypothesis Countable Support Elementary Substructure Supercompact Cardinal
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