A diamond-plus principle consistent with AD

Abstract

After showing that $$AD +DC $$ refutes $$\lozenge ^+_\kappa $$ for all regular cardinals $$\kappa \ge \omega _1$$ , we present a diamond-plus principle $$\lozenge _{{{\mathbb {R}}} }^+$$ concerning all subsets of $$\varTheta $$ . Using a forcing argument, we prove that $$\lozenge _{{{\mathbb {R}}} }^+$$ holds in Steel’s core model $${{{{\mathbf {K}}}({{{{\mathbb {R}}} }})}}$$ , an inner model in which the axiom of determinacy can hold. The combinatorial principle $$\lozenge _{{{\mathbb {R}}} }^+$$ is then extended, in $${{{{\mathbf {K}}}({{{{\mathbb {R}}} }})}}$$ , to successor cardinals $$\kappa >\varTheta $$ and to certain cardinals $$\kappa >\varTheta $$ that are not ineffable. Here $$\varTheta $$ is the supremum of the ordinals that are the surjective image of the set of reals $${{{\mathbb {R}}} }$$ .