an:05563900
Zbl 1178.03066
Busche, Daniel; Schindler, Ralf
The strength of choiceless patterns of singular and weakly compact cardinals
EN
Ann. Pure Appl. Logic 159, No. 1-2, 198-248 (2009).
00250042
2009
j
03E35 03E45 03E55 03E60
large cardinals; determinacy; inner models; premice; core models; core model induction
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.
Martin Weese (Potsdam)