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The strength of choiceless patterns of singular and weakly compact cardinals. (English) Zbl 1178.03066
The authors start with models of ZF and want to show that the axiom of determinacy is consistent relative to the hypothesis “each uncountable successor cardinal is weakly compact and each uncountable limit cardinal is singular” and “each uncountable cardinal is singular”, respectively.
They show that each one of the following two hypotheses individually implies that AD holds in the $$L(\mathbb R)$$ of a generic extension of HOD: (a) ZF + every uncountable cardinal is singular and (b) ZF + every infinite successor cardinal is weakly compact and every uncountable limit cardinal is singular.
The authors use the so-called core model induction, which was originally developed by W. Hugh Woodin and John R. Steel. They introduce two special kinds of premice and define a mouse closure operation.

##### MSC:
 3e+35 Consistency and independence results 3e+45 Inner models, including constructibility, ordinal definability, and core models 3e+55 Large cardinals 3e+60 Determinacy principles
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##### References:
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