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from Part II - What is New in Set Theory
Published online by Cambridge University Press: 28 September 2020
The last sections of Chapter 17 show why the forcing techniques that we have at N1 cannot possibly work at singular cardinals. Similar concerns apply to the successors of singular cardinals. For example, if we wish to study values of various cardinal invariants at κ, then we had better make sure that they are not trivially equal to κ+. Therefore we wish to work in the context of 2κ > κ+. If, in addition, we have that κ is a strong limit cardinal, then we are automatically dealing with the failure of SCH and so with large cardinals. In this situation, the cardinal invariants at κ+ are also affected. This situation presents many challenges and at this moment there is no unique technique or an axiom that deals with it. However, some techniques have emerged in recent years, two of which will be described below.
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