Forcing over Models of Determinacy

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.

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