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Iterated Forcing and Elementary Embeddings

In: Handbook of Set Theory

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  • James Cummings

    (Carnegie Mellon University, Department of Mathematics)

Abstract

I give a survey of some forcing techniques which are useful in the study of large cardinals and elementary embeddings. The main theme is the problem of extending a (possibly generic) elementary embedding of the universe to a larger domain, which is typically a generic extension of the ground model by some iterated forcing construction. Topics covered include (a) building, transfer and alteration of generic objects (b) strong and weak master conditions (c) the use of guessing principles (d) term forcing (e) the Kunen, Levy, Mitchell and Silver collapses The techniques which I discuss are illustrated with many examples. These include the failure of GCH at a measurable cardinal, the consistency of PFA, the laver indestructibility theorem, and the existence of a saturated ideal on the least uncountable cardinal.

Suggested Citation

  • James Cummings, 2010. "Iterated Forcing and Elementary Embeddings," Springer Books, in: Matthew Foreman & Akihiro Kanamori (ed.), Handbook of Set Theory, chapter 12, pages 775-883, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4020-5764-9_13
    DOI: 10.1007/978-1-4020-5764-9_13
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