论文标题
使用荆棘序列强迫更多的DC在Chang模型上
Forcing More DC Over the Chang Model Using the Thorn Sequence
论文作者
论文摘要
在$ \ Mathsf {Zf}+\ Mathsf {Dc} $的上下文中，我们强迫$ \ Mathcal {p}（P}（κ）$ $κ<\ALEPH_Ω$在Chang Model $ \ Mathrm {d.mathrm {dc）上（to）由$ {\ it \ unicode {xfe}} _ 0 =ω$，$ {\ unicode {xfe}} _ {α+1} $作为$ {\ unicode \ unicode {xfe unicode {xfe unicode {xfe iit $ no $ no $ no $ no $ a $ a）定义的刺序列。 \ unicode {xfe}}_α^ω\ rightarrow {\ it \ unicode {xfe}} _ {α+1} $是点的）和$ {\ it \ it \ it \ unicode {xfe}}}}}}}_γ= \ sup_ { \ unicode {xfe}}_α$ for Limit $γ$。这些假设是从确定性背景下约$θ$的结果中激发的，并且可能是考虑张模型的合理方式。明确地，我们假设刺序序列上的红衣主教$λ$是非常规的（含义是规则的，函数$ f：κ^{<κ} \rightArrowλ$每当$κ<λ$在刺上时，都会有限（thorn semence）和合理（含义）（含义是$ \ \ \ \ \ \ \ \ \ \ \ p}（κ^p}） \ Mathrm {l}（\ Mathrm {ord}^ω）\ subseteq \ mathrm {l}_λ（λ^ω，x）$，用于某些$ x \subseteqλ$，用于任何$κ<λ$ the shorn sequence的任何$κ<λ$）。这使我们能够使用Cohen强迫并建立更依赖的选择。
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.
