Analysis of HOD for Admissible Structures

Abstract

Let 𝑛 ≥ 1 and assume that there is a Woodin cardinal. For 𝑥 ∈ ℝ let 𝛼ₓ be the least 𝛽 such that L_β[𝑥] models Σ_n-KP + ∃κ (“κ is inaccessible and κ^+ exists”). We adapt the analysis of HOD^{L[x,G]} as a strategy mouse to L_{α_x}[x, G] for a cone of reals x. That is, we identify a mouse M^{n-ad} and define a class H ⊆ L_{α_x}[x, G] as a natural analogue of HOD^{L[x,G]} ⊆ L[x, G], and show that H = M_∞[Σ_0], where M_∞ is an iterate of M^{n-ad} and Σ_0 a fragment of its iteration strategy.