Prikry-Type Forcings

Chapter

Abstract

One of the central topics of set theory since Cantor has been the study of the power function κ→2κ. The basic problem is to determine all the possible values of 2κ for a cardinal κ. Paul Cohen proved the independence of CH and invented the method of forcing. Easton building on Cohen’s results showed that the function κ→2κ for regular κ can behave in any prescribed way consistent with the Zermelo-König inequality, which entails cf (2κ)>κ. This reduces the study to singular cardinals.

It turned out that the situation with powers of singular cardinals is much more involved. Thus, for example, a remarkable theorem of Silver states that a singular cardinal of uncountable cofinality cannot be the first to violate GCH. The Singular Cardinal Problem is the problem of finding a complete set of rules describing the behavior of the function κ→2κ for singular κ’s.

There are three main tools for dealing with the problem: pcf theory, inner model theory and forcing involving large cardinals. The purpose of this chapter is to present the main forcing tools for dealing with powers of singular cardinals.

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