A note on Woodin's HOD dichotomy

The internal structure of $\mathrm{HOD}^{L[x]}$ up to its Woodin

Assume $\boldsymbolΔ^1_3$-determinacy. It is shown that for any $x \geq_T M_1^{\#}$, $\mathrm{HOD}^{L[x]}$ is a model of GCH, and in fact, it is a Jensen-Steel core model up to $ω_2^{L[x]}$.

HOD in inner models with Woodin cardinals

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 $\mathcal{M}_\infty$…

在所有不可数基数上平方失败比Woodin极限的Woodin基数更弱

The failure of square at all uncountable cardinals is weaker than a Woodin limit of Woodin cardinals

该研究通过强制选择公理，在满足ZF的最小初始段的Nairian模型中展示所有不可数基数上平方失败，并表明正则基数在HOD中为omega-强可测，从而揭示了HOD假设在ZFC中的不可证性。

Strongly compact cardinals and ordinal definability

This paper explores several topics related to Woodin's HOD conjecture. We improve the large cardinal hypothesis of Woodin's HOD dichotomy theorem from an extendible cardinal to a strongly compact cardinal. We show that assuming there is a strongly compact cardinal and the HOD hypothesis holds, there is no elementary embedding from HOD to HOD, settling a question of Woodin. We show that the HOD hypothesis is equivalent to a uniqueness property of elementary embeddings of levels of the cumulative hierarchy. We prove that the HOD hypothesis holds if and only if every regular cardinal above the fi…

超幂的决定论模型作为HOD上的迭代树

Ultrapowers of determinacy models as iteration trees on HOD

本文研究了在大基数假设下，HOD的超幂是否可通过HOD的细结构扩展序列和标准迭代策略表示为HOD上的迭代树，解决了Woodin的问题。

The HOD Dichotomy

This paper provides an accessible introduction to some of the work of Woodin on suitable extender models. We define the HOD conjecture, prove it is equivalent to a formulation in terms of weak extender models for supercompactness, and give some of its consequences.