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Forcing over Models of Determinacy

In: Handbook of Set Theory

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  • Paul B. Larson

    (University of Florida, Department of Mathematics)

Abstract

A theorem of Woodin states that the existence of a proper class of Woodin cardinals implies that the theory of the inner model L(ℝ) cannot be changed by set forcing. The Axiom of Determinacy is part of this fixed theory for L(ℝ). The partial order ℙmax is a forcing construction in L(ℝ) which lifts the absoluteness properties of L(ℝ) to models of the Axiom of Choice. The structure H(ω 2) in the ℙmax extension of L(ℝ) (assuming AD L(ℝ)) satisfies every Π2 sentence φ for H(ω 2) which is forceable from a proper class of Woodin cardinals. Furthermore, the partial order ℙmax can be easily varied to produce other consistency results and canonical models. We attempt to give a complete account of the basic analysis of the ℙmax extension of L(ℝ), relative to published results. We then briefly survey some of the issues surrounding ℙmax, in particular ℙmax variations and forcing over larger models of determinacy. We also briefly introduce Woodin’s Ω-logic, in order to properly state the maximality properties of the ℙmax extension.

Suggested Citation

  • Paul B. Larson, 2010. "Forcing over Models of Determinacy," Springer Books, in: Matthew Foreman & Akihiro Kanamori (ed.), Handbook of Set Theory, chapter 24, pages 2121-2177, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4020-5764-9_25
    DOI: 10.1007/978-1-4020-5764-9_25
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