论文标题
木质枢机主教的内部模型中的hod
HOD in inner models with Woodin cardinals
论文作者
论文摘要
我们分析了$ m_n（x）[x）[g] $ in $ x $ $ x $ in $ m_n（x）$ in $ m_n（x）$ in $ x $ in $ x $ in $ x $的$ \ operatornage {hod} $，其中$ m_n（x）$是$ n $ woodin cardinals by $ x $和$ g $ collape collape collape collape collape collape collape colling collape collape colluals $ n $ woodin corn（x $ g $ collape）。我们证明，假设$ \boldsymbolπ^1_ {n+2} $ - 确定性，对于真实的图锥$ x $，$ \ operatatorName {hod}^{m_n（x）[g]} [g]} = m_n（\ mathcal {m} _ {m} _ {\ infty} $ \ MATHCAL {M} _ \ infty $是$ m_ {n+1} $的直接限制，$Δ_\ infty $是$ \ Mathcal {m} _ \ infty $ \ n. \ infty $，$κ_\ infty $ inccess -inccess -inccess $ n Mathy $ \ mathy $ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $Δ_\ infty $，$λ$是$ \ Mathcal {M} _ {\ infty} $的部分迭代策略。还将表明，在相同的假设$ \ operatatorName下，{hod}^{m_n（x）[g]} $满足$ \ operatatorName {gch} $。
We analyze the hereditarily ordinal definable sets $\operatorname{HOD}$ in $M_n(x)[g]$ for a Turing cone of reals $x$, where $M_n(x)$ is the canonical inner model with $n$ Woodin cardinals build over $x$ and $g$ is generic over $M_n(x)$ for the Lévy collapse up to its bottom inaccessible cardinal. We prove that assuming $\boldsymbolΠ^1_{n+2}$-determinacy, for a Turing cone of reals $x$, $\operatorname{HOD}^{M_n(x)[g]} = M_n(\mathcal{M}_{\infty} | κ_\infty, Λ),$ where $\mathcal{M}_\infty$ is a direct limit of iterates of $M_{n+1}$, $δ_\infty$ is the least Woodin cardinal in $\mathcal{M}_\infty$, $κ_\infty$ is the least inaccessible cardinal in $\mathcal{M}_\infty$ above $δ_\infty$, and $Λ$ is a partial iteration strategy for $\mathcal{M}_{\infty}$. It will also be shown that under the same hypothesis $\operatorname{HOD}^{M_n(x)[g]}$ satisfies $\operatorname{GCH}$.
