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18 - Forcing at Singular Cardinals and Their Successors

from Part II - What is New in Set Theory

Published online by Cambridge University Press:  28 September 2020

Mirna Džamonja
Mirna Džamonja
Affiliation:
Institute for the History and Philosophy of Science and Technology, Paris
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Summary

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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Fast Track to Forcing , pp. 133 - 136
Publisher: Cambridge University Press
Print publication year: 2020

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