Is there a set of reals not in \(K(\mathbb{R})\)?
The real core model, \(K(\mathbb R)\), was introduced by \textit{D. W. Cunningham} [Ann. Pure Appl. Logic 72, No. 3, 213-289 (1995; Zbl 0828.03025)], who has shown that \(K(\mathbb R)\) is an inner model containing reals and definable scales beyond those in \(L(\mathbb R)\). In the paper under review, using the fine structure of \(K(\mathbb R)\), the author proves that the theory ZF\({}+{}\)AD\({}+(\exists X\subseteq\mathbb R)\allowbreak [X\notin K(\mathbb R)]\) implies the existence of an inner model of ZF\({}+{}\)AD\({}+{}\)DC with a measurable cardinal greater than \(\Theta\), the supremum of all ordinals which are a surjective image of \(\mathbb R\). This generalizes a result of Woodin which states that \(\text{HOD} ^{L(\mathbb R)}=L(P)\) for some \(P\subseteq\Theta^{L(\mathbb R)}\). To get this result, the author proves a modification of the generalized continuum hypothesis in \(K(\mathbb R)\), i.e., for every ordinal \(\lambda\), \(|\mathcal P(\lambda\times\mathbb R)|=\lambda^+\) holds in \(K(\mathbb R)\). Using this together with a theorem of Vopěnka, he computes \(\text{HOD} ^{K(\mathbb R)}\) and shows that \(\text{HOD} ^{K(\mathbb R)}=K(P)\) for some \(P\subseteq(\Theta^+)^{K(\mathbb R)}\) where \(K(P)\) is the Dodd-Jensen core model relative to \(P\). As he works without the axiom of choice, he has to prove Loś's theorem for the ultrapowers of these inner models without choice and he develops some tools for weak iterability of \(K(\mathbb R)\) without DC. Assuming AD he proves that \(\neg\)DC\(_{\mathbb R}\) implies the existence of a set \(A\) of reals such that \(X\notin K(\mathbb R)\) and the existence of \(\mathbb R^\dagger\) (dagger). Many results of the paper are obtained in the theory ZF\({}+{}\)AD\({}+(\exists X\subseteq\mathbb R)\allowbreak [X\notin K(\mathbb R)]\).
scientific article; zbMATH DE number 1308060
scientific article