LOCAL DEFINABILITY OF HOD $\mathsf {HOD}$ sans serif upper H upper O upper D IN L ( R ) $L(\mathbb {R})$ upper L left parenthesis double struck upper R right parenthesis

Abstract We show that in L ( R ) $L(\mathbb {R})$ upper L left parenthesis double struck upper R right parenthesis , assuming large cardinals in V , HOD ∥ η + HOD $\mathsf {HOD}{\parallel }\eta ^{+\mathsf {HOD}}$ sans serif upper H upper O upper D parallel to eta Superscript plus sans serif upper H upper O upper D is locally definable from HOD ∥ η $\mathsf {HOD}{\parallel }\eta $ sans serif upper H upper O upper D parallel to eta for all HOD $\mathsf {HOD}$ sans serif upper H upper O upper D -cardinals η ∈ [ δ 1 2 , Θ ) $\eta \in [\boldsymbol {\delta }^2_1,\Theta )$ eta element of left bracket bold italic delta 1 squared comma normal upper Theta right parenthesis . This is a further elaboration of the statement “ HOD L ( R ) $\mathsf {HOD}^{L(\mathbb {R})}$ sans serif upper H upper O upper D Superscript upper L left parenthesis double struck upper R right parenthesis is a core model below Θ