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Core models with more Woodin cardinals. (English) Zbl 1012.03055
The author proves two theorems about the construction of core models with infinitely many Woodin cardinals. A new wrinkle is the iterability of \(K^c\) with respect to uncountable iteration trees. Confirming a conjecture of Feng, Magidor, and Woodin, he proves that for measurable \(\Omega\) the following are equivalent: (a) for all posets \(\mathbb{P,Q}\in V_{\Omega}, L({\mathbb R})^{V^{\mathbb P}}\equiv L({\mathbb R})^{V^{\mathbb Q}}\); (b) for every \({\mathbb P}\in V_{\Omega}, V^{{\mathbb P}}\vDash AD^{L({\mathbb R})}\); (c) for every \({\mathbb P}\in V^{\Omega}, V^{{\mathbb P}}\vDash\) there is no uncountable sequence of distinct reals in \(L({\mathbb R})\); (d) there is an \(\Omega\)-iterable premouse of height \(\Omega\) which satisfies “there are infinitely many Woodin cardinals”. The author notes that Woodin had (slightly) earlier proved this theorem under the weaker hypothesis that \(\Omega\) is inaccessible. He also notes that as a corollary one obtains that if every set of reals in \(L({\mathbb R})\) is weakly homogeneous, then \(\text{AD}^{L({\mathbb R})}\) holds.
The author’s second result is an improved lower bound for the consistency strength of the failure of the unique branches hypothesis. He proves: if there is a non-overlapping iteration tree \(\mathcal T\) on \(V\) having distinct cofinal branches \(b\) and \(c\), then there is an inner model with infinitely many Wooden cardinals, and if in addition, \(\delta(\mathcal T)\in \operatorname {ran}(i_b)\cap\operatorname {ran}(i^c)\), then there is an inner model with a strong cardinal which is a limit of Woodin cardinals.

MSC:
03E45 Inner models, including constructibility, ordinal definability, and core models
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