1. Iterated Forcing and Elementary Embeddings

In this chapter we present a survey of the area of set theory in which iterated forcing interacts with elementary embeddings. The original plan was to concentrate on forcing constructions which preserve large cardinal axioms, particularly Reverse Easton iterations. However this plan proved rather restrictive, so we have also treated constructions such as Baumgartner’s consistency proof for the Proper Forcing Axiom. The common theme of the constructions which we present is that they involve extending elementary embeddings. We have not treated the preservation of large cardinal axioms by “Prikrytype” forcing, for example by Radin forcing or iterated Prikry forcing. For this we refer the reader to Gitik’s chapter in this Handbook [22]. After some preliminaries, the bulk of this chapter consists of fairly short sections, in each of which we introduce one or two technical ideas and give one or more examples of the ideas in action. The constructions are generally of increasing complexity as we proceed and have more techniques at our disposal. Especially at the beginning, we have adopted a fairly leisurely and discursive approach to the material. The impatient reader is encouraged to jump ahead and refer back as necessary. At the end of this introduction there is a brief description of the contents of each section. Here is a brief review of our notation and conventions. We defer the discussion of forcing to Section 5.