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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:
 3e+45 Inner models, including constructibility, ordinal definability, and core models
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##### References:
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