Forcing more 𝖣𝖢 over the Chang model using the Thorn sequence

Abstract

In the context of Z F + D C \mathsf {ZF}+\mathsf {DC} , we force D C κ \mathsf {DC}_\kappa for relations on P ( κ ) \mathcal {P}(\kappa ) for arbitrarily large κ > ℵ ω \kappa >\aleph _\omega over the Chang model L ( O r d ω ) \mathrm {L}(\mathrm {Ord}^\omega ) making some assumptions on the thorn sequence defined by Þ 0 = ω Þ_0=\omega , Þ α + 1 Þ_{\alpha +1} as the least ordinal not a surjective image of Þ α ω Þ_\alpha ^\omega and Þ γ = sup α > γ Þ α Þ_\gamma =\sup _{\alpha >\gamma }Þ_\alpha for limit γ \gamma . These assumptions are motivated from results about Θ \Theta in the context of determinacy, and could be reasonable ways of thinking about the Chang model. Explicitly, we assume successor points λ \lambda on the thorn sequence are strongly regular—meaning regular and functions f : κ > κ → λ f:\kappa ^{>\kappa }\rightarrow \lambda are bounded whenever κ > λ \kappa >\lambda is on the thorn sequence—and justified—meaning P ( κ ω ) ∩ L ( O r d ω ) ⊆ L λ ( λ ω