LOCAL DEFINABILITY OF
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DOI: 10.1017/jsl.2026.10261 ISSN: 0022-4812

LOCAL DEFINABILITY OF

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

$\Theta $

normal upper Theta

” made by John Steel.