Forcing More DC Over the Chang Model Using the Thorn Sequence
James Holland, Grigor Sargsyan
Abstract
In the context of $\mathsf{ZF}+\mathsf{DC}$, we force $\mathsf{DC}_κ$ for relations on $\mathcal{P}(κ)$ for $κ<\aleph_ω$ over the Chang model $\mathrm{L}(\mathrm{Ord}^ω)$ making some assumptions on the thorn sequence defined by ${\it \unicode{xFE}}_0=ω$, ${\it \unicode{xFE}}_{α+1}$ as the least ordinal not a surjective image of ${\it \unicode{xFE}}_α^ω$ (i.e. no $f:{\it \unicode{xFE}}_α^ω\rightarrow {\it \unicode{xFE}}_{α+1}$ is surjective) and ${\it \unicode{xFE}}_γ=\sup_{α<γ}{\it \unicode{xFE}}_α$ for limit $γ$. These assumptions are motivated from results about $Θ$ in the context of determinacy, and could be reasonable ways of thinking about the Chang model. Explicitly, we assume cardinals $λ$ on the thorn sequence are strongly regular (meaning regular and functions $f:κ^{<κ}\rightarrow λ$ are bounded whenever $κ<λ$ is on the thorn sequence) and justified (meaning $\mathcal{P}(κ^ω)\cap \mathrm{L}(\mathrm{Ord}^ω)\subseteq \mathrm{L}_λ(λ^ω,X)$ for some $X\subseteq λ$ for any $κ<λ$ on the thorn sequence). This allow us to use Cohen forcing and establish more dependent choice.