One of the central topics of set theory since Cantor has been the study of the power function κ→2 κ . The basic problem is to determine all the possible values of 2 κ 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→2 κ for regular κ can behave in any prescribed way consistent with the Zermelo-König inequality, which entails cf (2 κ )>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→2 κ 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.
KeywordsDirect Extension Generic Subset Regular Cardinal Measurable Cardinal Force Notion
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