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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.

03E45 Inner models, including constructibility, ordinal definability, and core models
Full Text: DOI
[1] DOI: 10.1016/S0168-0072(96)00032-2 · Zbl 0868.03021
[2] DOI: 10.1090/S0894-0347-1994-1224594-7
[3] Universally Baire sets · Zbl 0781.03034
[4] Israel Journal of Mathematics
[5] Handbook of set theory
[6] Fine structure and iteration trees (1994) · Zbl 0805.03042
[7] The wellfoundedness of the Mitchell order 58 pp 931– (1993)
[8] DOI: 10.1016/0168-0072(93)90037-E · Zbl 0805.03043
[9] DOI: 10.1090/S0002-9947-99-02411-3 · Zbl 0928.03059
[10] Fine structure for tame inner models 61 pp 621– (1996)
[11] A weak Dodd-Jensen lemma 64 pp 1285– (1999)
[12] The core model iterability problem (1996) · Zbl 0864.03035
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