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Iterated Forcing and Elementary Embeddings

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Handbook of Set Theory

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.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    James Cummings

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  1. James Cummings
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Correspondence to James Cummings .

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Editors and Affiliations

  1. School of Physical Sciences, University of California, Irvine, Irvine, 92697-3875, U.S.A.

    Matthew Foreman

  2. Dept. Manufacturing Engineering, Boston University, St. Mary's St. 15, Boston, 02215, U.S.A.

    Akihiro Kanamori

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Cummings, J. (2010). Iterated Forcing and Elementary Embeddings. In: Foreman, M., Kanamori, A. (eds) Handbook of Set Theory. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-5764-9_13

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