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Constructibility and Class Forcing

In: Handbook of Set Theory

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  • Sy D. Friedman

    (Universität Wien, Kurt Gödel Research Center for Mathematical Logic)

Abstract

In this article we discuss the basic theory and applications of class forcing, with an emphasis on three problems posed by Solovay which can be resolved using it. As class forcing, unlike traditional set forcing, does not in general preserve $\mathop {\rm ZFC}$ , we first isolate the first-order property of tameness, necessary and sufficient for this preservation. After mentioning four basic examples, we discuss the question of the generic existence for class forcing before turning to the most important technique in the subject, the technique of Jensen coding. Armed with these ideas we then proceed to describe the solutions to the Solovay problems. We next discuss generic saturation, a concept which helps to explain the special role of 0# in this theory. We end by briefly describing some other applications.

Suggested Citation

  • Sy D. Friedman, 2010. "Constructibility and Class Forcing," Springer Books, in: Matthew Foreman & Akihiro Kanamori (ed.), Handbook of Set Theory, chapter 8, pages 557-604, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4020-5764-9_9
    DOI: 10.1007/978-1-4020-5764-9_9
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