* (87) Nir Shlezinger et al.  “UVeQFed: Universal vector quantization for federated learning”  In _IEEE Transactions on Signal Processing_ 69 IEEE, 2020, pp. 500–514
  * (88) Neta Shoham et al.  “Overcoming forgetting in federated learning on non-iid data”  In _arXiv preprint arXiv:1910.07796_ , 2019
  * (89) Saurabh Singh et al.  “A framework for privacy-preservation of IoT healthcare data using Federated Learning and blockchain technology”  In _Future Generation Computer Systems_ 129 Elsevier, 2022, pp. 380–388
  * (90) Kihyuk Sohn, Honglak Lee and Xinchen Yan  “Learning structured output representation using deep conditional generative models”  In _Advances in neural information processing systems_ 28, 2015
  * (91) Nitish Srivastava et al.  “Dropout: a simple way to prevent neural networks from overfitting”  In _The journal of machine learning research_ 15.1 JMLR. org, 2014, pp. 1929–1958
  * (92) Tao Sun, Dongsheng Li and Bao Wang  “Decentralized Federated Averaging”  In _IEEE Trans. Pattern Anal. Mach. Intell._ 45.4 USA: IEEE Computer Society, 2023, pp. 4289–4301  DOI: 10.1109/TPAMI.2022.3196503
  * (93) Ananda Theertha Suresh, X Yu Felix, Sanjiv Kumar and H Brendan McMahan  “Distributed mean estimation with limited communication”  In _International conference on machine learning_ , 2017, pp. 3329–3337  PMLR
  * (94) Mingxing Tan and Quoc Le  “Efficientnet: Rethinking model scaling for convolutional neural networks”  In _International conference on machine learning_ , 2019, pp. 6105–6114  PMLR
  * (95) Yue Tan et al.  “Federated learning from pre-trained models: A contrastive learning approach”  In _Advances in Neural Information Processing Systems_ 35, 2022, pp. 19332–19344
  * (96) Yue Tan et al.  “Fedproto: Federated prototype learning across heterogeneous clients”  In _Proceedings of the AAAI Conference on Artificial Intelligence_ 36.8, 2022, pp. 8432–8440
  * (97) Shay Vargaftik et al.  “Drive: One-bit distributed mean estimation”  In _Advances in Neural Information Processing Systems_ 34, 2021, pp. 362–377
  * (98) Manasi Vartak et al.  “ModelDB: a system for machine learning model management”  In _Proceedings of the Workshop on Human-In-the-Loop Data Analytics_ , 2016, pp. 1–3
  * (99) Hao Wang, Zakhary Kaplan, Di Niu and Baochun Li  “Optimizing Federated Learning on Non-IID Data with Reinforcement Learning”  In _IEEE INFOCOM 2020 - IEEE Conference on Computer Communications_ , 2020, pp. 1698–1707  DOI: 10.1109/INFOCOM41043.2020.9155494
  * (100) Jianyu Wang et al.  “Tackling the objective inconsistency problem in heterogeneous federated optimization”  In _Advances in neural information processing systems_ 33, 2020, pp. 7611–7623
  * (101) Naibo Wang et al.  “Data-Free Diversity-Based Ensemble Selection for One-Shot Federated Learning”  In _Transactions on Machine Learning Research_ , 2023
  * (102) Tongzhou Wang, Jun-Yan Zhu, Antonio Torralba and Alexei A Efros  “Dataset distillation”  In _arXiv preprint arXiv:1811.10959_ , 2018
  * (103) Yujia Wang, Lu Lin and Jinghui Chen  “Communication-efficient adaptive federated learning”  In _International Conference on Machine Learning_ , 2022, pp. 22802–22838  PMLR
  * (104) Zheng Wang et al.  “FedGS: Federated Graph-Based Sampling with Arbitrary Client Availability”  In _Proceedings of the AAAI Conference on Artificial Intelligence_ 37.8, 2023, pp. 10271–10278  DOI: 10.1609/aaai.v37i8.26223
  * (105) Dingzhu Wen, Ki-Jun Jeon and Kaibin Huang  “Federated Dropout–A Simple Approach for Enabling Federated Learning on Resource Constrained Devices”  In _arXiv preprint arXiv:2109.15258_ , 2021
  * (106) Jie Wen et al.  “A survey on federated learning: challenges and applications”  In _International Journal of Machine Learning and Cybernetics_ 14.2 Springer, 2023, pp. 513–535
  * (107) Kok-Seng Wong et al.  “An Empirical Study of Federated Learning on IoT-Edge Devices: Resource Allocation and Heterogeneity”  In _arXiv preprint arXiv:2305.19831_ , 2023
  * (108) Xidong Wu, Feihu Huang, Zhengmian Hu and Heng Huang  “Faster adaptive federated learning”  In _Proceedings of the AAAI Conference on Artificial Intelligence_ 37.9, 2023, pp. 10379–10387
  * (109) Xueyang Wu et al.  “FedNP: Towards Non-IID Federated Learning via Federated Neural Propagation”, 2023
  * (110) Yuexiang Xie et al.  “FederatedScope: A Flexible Federated Learning Platform for Heterogeneity”  In _Proceedings of the VLDB Endowment_ 16.5 VLDB Endowment, 2023, pp. 1059–1072
  * (111) Xiaohang Xu et al.  “Federated Depression Detection from Multi-SourceMobile Health Data”  In _CoRR_ abs/2102.09342, 2021  arXiv: https://arxiv.org/abs/2102.09342
  * (112) He Yang  “H-FL: A hierarchical communication-efficient and privacy-protected architecture for federated learning”  In _arXiv preprint arXiv:2106.00275_ , 2021
  * (113) Mang Ye et al.  “Heterogeneous Federated Learning: State-of-the-art and Research Challenges”  In _ACM Comput. Surv._ 56.3 New York, NY, USA: Association for Computing Machinery, 2023  DOI: 10.1145/3625558
  * (114) Xun Yi et al.  “Homomorphic encryption”  Springer, 2014
  * (115) Yaodong Yu et al.  “TCT: Convexifying federated learning using bootstrapped neural tangent kernels”  In _Advances in Neural Information Processing Systems_ 35, 2022, pp. 30882–30897
  * (116) Binhang Yuan, Song Ge and Wenhui Xing  “A federated learning framework for healthcare iot devices”  In _arXiv preprint arXiv:2005.05083_ , 2020
  * (117) Ram Zamir and Meir Feder  “On universal quantization by randomized uniform/lattice quantizers”  In _IEEE Transactions on Information Theory_ 38.2 IEEE, 1992, pp. 428–436
  * (118) Hangjia Zhang et al.  “Adaptive client selection in resource constrained federated learning systems: A deep reinforcement learning approach”  In _IEEE Access_ 9 IEEE, 2021, pp. 98423–98432
  * (119) Hongyi Zhang, Jan Bosch and Helena Holmström Olsson  “End-to-end federated learning for autonomous driving vehicles”  In _2021 International Joint Conference on Neural Networks (IJCNN)_ , 2021, pp. 1–8  IEEE
  * (120) Jianyi Zhang et al.  “Fed-CBS: A Heterogeneity-Aware Client Sampling Mechanism for Federated Learning via Class-Imbalance Reduction”  In _Proceedings of the 40th International Conference on Machine Learning_ 202, Proceedings of Machine Learning Research  PMLR, 2023, pp. 41354–41381  URL: https://proceedings.mlr.press/v202/zhang23y.html
  * (121) Jie Zhang et al.  “Dense: Data-free one-shot federated learning”  In _Advances in Neural Information Processing Systems_ 35, 2022, pp. 21414–21428
  * (122) Lin Zhang et al.  “Fine-tuning global model via data-free knowledge distillation for non-iid federated learning”  In _Proceedings of the IEEE/CVF conference on computer vision and pattern recognition_ , 2022, pp. 10174–10183
  * (123) Sai Qian Zhang, Jieyu Lin and Qi Zhang  “A Multi-Agent Reinforcement Learning Approach for Efficient Client Selection in Federated Learning”  In _Thirty-Sixth AAAI Conference on Artificial Intelligence, AAAI 2022, Thirty-Fourth Conference on Innovative Applications of Artificial Intelligence, IAAI 2022, The Twelveth Symposium on Educational Advances in Artificial Intelligence, EAAI 2022 Virtual Event, February 22 - March 1, 2022_ AAAI Press, 2022, pp. 9091–9099  DOI: 10.1609/AAAI.V36I8.20894
  * (124) Zihao Zhao et al.  “Towards efficient communications in federated learning: A contemporary survey”  In _Journal of the Franklin Institute_ 360.12, 2023, pp. 8669–8703  DOI: https://doi.org/10.1016/j.jfranklin.2022.12.053
  * (125) Yanlin Zhou et al.  “Distilled one-shot federated learning”  In _arXiv preprint arXiv:2009.07999_ , 2020
  * (126) Zhi-Hua Zhou, Jianxin Wu and Wei Tang  “Ensembling neural networks: many could be better than all”  In _Artificial intelligence_ 137.1-2 Elsevier, 2002, pp. 239–263
  * (127) Zhuangdi Zhu, Junyuan Hong and Jiayu Zhou  “Data-free knowledge distillation for heterogeneous federated learning”  In _International conference on machine learning_ , 2021, pp. 12878–12889  PMLR
  * (128) Alexander Ziller et al.  “Pysyft: A library for easy federated learning”  In _Federated Learning Systems: Towards Next-Generation AI_ Springer, 2021, pp. 111–139


# Strongly compact cardinals and ordinal definability

Gabriel Goldberg

###### Abstract

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 first
strongly compact cardinal carries an ordinal definable $\omega$-Jónsson
algebra. We show that if $\kappa$ is supercompact, the HOD hypothesis holds,
and HOD satisfies the Ultrapower Axiom, then $\kappa$ is supercompact in HOD.

##  1 Introduction

The Jensen covering theorem [2] states that if $0^{\\#}$ does not exist, then
every uncountable set of ordinals is covered by a constructible set of the
same cardinality. This leads to a strong dichotomy for the cardinal
correctness of the constructible universe $L$:

######  Theorem (Jensen).

Exactly one of the following holds:

  1. (1)

For all singular cardinals $\lambda$, $\lambda$ is singular in $L$ and
$(\lambda^{+})^{L}=\lambda^{+}$.

  2. (2)

Every infinite cardinal is strongly inaccessible in $L$.

The proof of this theorem and its generalizations to larger canonical inner
models tend to make heavy use of the special structure of such models. By
completely different techniques, however, Woodin [6] showed that such a
dichotomy holds for the (noncanonical) inner model HOD under large cardinal
hypotheses.

######  Theorem (Woodin’s HOD dichotomy).

If $\kappa$ is extendible, exactly one of the following holds:

  1. (1)

For all singular cardinals $\lambda>\kappa$, $\lambda$ is singular in HOD and
$(\lambda^{+})^{\textnormal{HOD}}=\lambda^{+}$.

  2. (2)

Every regular cardinal greater than or equal to $\kappa$ is measurable in HOD.

Woodin’s HOD conjecture states that conclusion 1, which in this context is
known as the HOD hypothesis,111 In the context of bare ZFC (without large
cardinal axioms), the HOD hypothesis states that there is a proper class of
regular cardinals that are not $\omega$-strongly measurable in HOD.
Conceivably the HOD hypothesis is provable in ZFC.  is provable from large
cardinal axioms.

The first few theorems of this paper (Section 2) show that the HOD dichotomy
in fact takes hold far below the least extendible cardinal.

######  Theorem.

If $\kappa$ is strongly compact, exactly one of the following holds:

  1. (1)

For all singular cardinals $\lambda>\kappa$, $\lambda$ is singular in HOD and
$(\lambda^{+})^{\textnormal{HOD}}=\lambda^{+}$.

  2. (2)

All sufficiently large regular cardinals are measurable in HOD.

We prove this as a corollary of a theorem establishes a similar dichotomy for
a fairly broad class of inner models. An inner model $N$ of ZFC is
$\omega$-club amenable if for all ordinals $\delta$ of uncountable cofinality,
the $\omega$-club filter $F$ on $\delta$ satisfies $F\cap N\in N$.

######  Theorem 2.9.

Assume $N$ is $\omega$-club amenable and $\kappa$ is $\omega_{1}$-strongly
compact. Either all singular cardinals $\lambda>\kappa$ are singular in $N$
and $(\lambda^{+})^{N}=\lambda^{+}$, or all sufficiently large regular
cardinals are measurable in $N$.

Note that we use a somewhat weaker large cardinal hypothesis than strong
compactness: a cardinal $\kappa$ is $\omega_{1}$-strongly compact if every
$\kappa$-complete filter extends to an $\omega_{1}$-complete ultrafilter.

The main theorem of [4] states that if $j_{0},j_{1}:V\to M$ are elementary
embeddings, then
$j_{0}\restriction\textnormal{Ord}=j_{1}\restriction\textnormal{Ord}$. Section
3 turns to the relationship between local forms of this theorem and Woodin’s
HOD conjecture.

######  Theorem 3.5.

Assume $\kappa$ is an extendible cardinal. Then the following are equivalent:

  1. (1)

The HOD hypothesis.

  2. (2)

For all regular $\delta\geq\kappa$ and all sufficiently large $\alpha$, any
elementary embeddings $j_{0},j_{1}:V_{\alpha}\to M$ with
$j_{0}(\delta)=j_{1}(\delta)$ and $\sup j_{0}[\delta]=\sup j_{1}[\delta]$
agree on $\delta$.

We also prove a related theorem that connects the failure of the HOD
hypothesis to definable infinitary partition properties, choiceless large
cardinals, and sharps.

A cardinal $\lambda$ is Jónsson if for all $f:[\lambda]^{<\omega}\to\lambda$,
there is a proper subset $A$ of $\lambda$ of cardinality $\lambda$ such that
$f(s)=s$ for all $s\in[A]^{<\omega}$. A cardinal $\lambda$ is constructibly
Jónsson if this holds for all constructible functions
$f:[\lambda]^{<\omega}\to\lambda$, there is a proper subset $A$ of $\lambda$
of cardinality $\lambda$ such that $f(s)=s$ for all $s\in[A]^{<\omega}$.

######  Proposition.

The following are equivalent:

  * •

$0^{\\#}$ exists.

  * •

Every uncountable cardinal is constructibly Jónsson.

  * •

Some uncountable cardinal is constructibly Jónsson.

A cardinal $\lambda$ is $\omega$-Jónsson if for all
$f:[\lambda]^{\omega}\to\lambda$, there is a proper subset $A$ of $\lambda$ of
cardinality $\lambda$ such that $f(s)=s$ for all $s\in[A]^{\omega}$. Recall
Kunen’s theorem that there is no elementary embedding $j:V\to V$. Kunen proved
his theorem by showing that if $j(\lambda)=\lambda$, then $\lambda$ is
$\omega$-Jónsson. He then cites the following combinatorial theorem:

######  Theorem (Erdös-Hajnal).

There are no $\omega$-Jónsson cardinals.

A cardinal $\lambda$ is definably $\omega$-Jónsson if for all ordinal
definable functions $f:[\lambda]^{\omega}\to\lambda$, there is a proper subset
$A$ of $\lambda$ of cardinality $\lambda$ such that $f(s)=s$ for all
$s\in[A]^{\omega}$.

######  Theorem 3.7.

Assume $\kappa$ is strongly compact. Then the following are equivalent:

  * •

The HOD hypothesis fails.

  * •

All sufficiently large regular cardinals are definably $\omega$-Jónsson.

  * •

Some regular cardinal above $\kappa$ is definably $\omega$-Jónsson.

In Section 3.3, we use techniques drawn from the proof of the meain theorem of
[4] to answer a decade-old question of Woodin (implicit in [6]). Recall
Kunen’s famous theorem relating the rigidity of $L$ to $0^{\\#}$:

######  Theorem (Kunen).

If $0^{\\#}$ does not exist, then there is no nontrivial elementary embedding
from $L$ to $L$.

Here we prove an analog for HOD, replacing the assumption that $0^{\\#}$ does
not exist with the HOD hypothesis.

######  Theorem 3.9.

Assume there is an $\omega_{1}$-strongly compact cardinal. If the HOD
hypothesis holds, then there is no nontrivial elementary embedding from HOD to
HOD.

Section 4.1 turns to the question of the structure of strongly compact
cardinals in the HOD of a model of determinacy. In this context, something
close to the failure of the HOD hypothesis holds, and yet we will show that
HOD still has certain local covering properties in regions of strong
compactness. One consequence of our analysis is the following theorem on the
equivalence of strong compactness and $\omega_{1}$-strong compactness in HOD
assuming Woodin’s HOD ultrafilter conjecture: under AD + $V=L(P(\mathbb{R}))$,
every $\omega_{1}$-complete ultrafilter of HOD generates an
$\omega_{1}$-complete filter in $V$. (The HOD ultrafilter conjecture is true
in $L(\mathbb{R})$ and more generally in all inner models of determinacy
amenable to the techniques of contemporary inner model theory.)

######  Theorem 4.2.

Assume $\textnormal{AD}^{+}$, $V=L(P(\mathbb{R}))$, and the HOD-ultrafilter
conjecture. Suppose $\kappa<\Theta$ is a regular cardinal and
$\delta\geq\kappa$ is a HOD-regular ordinal. Then $\kappa$ is
$\delta$-strongly compact in HOD if and only if $\kappa$ is
$(\omega_{1},\delta)$-strongly compact in HOD.

This should be compared with a theorem of [3] stating that under the
Ultrapower Axiom, for any regular cardinal $\delta$, the least
$(\omega_{1},\delta)$-strongly compact cardinal is $\delta$-strongly compact.
It is natural to conjecture that assuming AD, if $V=L(P(\mathbb{R}))$, then
HOD is a model of the Ultrapower Axiom.

The final section (Section 4.2) explores the absoluteness of large cardinals
to HOD in the context of a strongly compact cardinal. Woodin [6] showed:

######  Theorem (Woodin).

Suppose $\kappa$ is extendible and the HOD hypothesis holds. Then $\kappa$ is
extendible in HOD.

By results of Cheng-Friedman-Hamkins [1], the least supercompact cardinal need
not be weakly compact in HOD. Here we show that nevertheless, HOD is very
close to an inner model with a supercompact cardinal.

######  Theorem 4.12.

Suppose $\kappa$ is supercompact and the HOD hypothesis holds. Then $\kappa$
is supercompact in an inner model $N$ of ZFC such that
$\textnormal{HOD}\subseteq N$ and $\textnormal{HOD}^{<\kappa}\cap
N\subseteq\textnormal{HOD}$.

As a corollary, we show:

######  Theorem 4.13.

Suppose $\kappa$ is supercompact, the HOD hypothesis holds, and HOD satisfies
the Ultrapower Axiom. Then $\kappa$ is supercompact in HOD.

##  2 On the HOD dichotomy

###  2.1 Strongly measurable cardinals

For any regular cardinal $\gamma$ and any ordinal $\delta$ of cofinality
greater than $\gamma$, $\mathcal{C}_{\delta}$ denotes the closed unbounded
filter on $\delta$, $S^{\delta}_{\gamma}$ denotes the set of ordinals less
than $\delta$ of cofinality $\gamma$, and $\mathcal{C}_{\delta,\gamma}$
denotes the filter generated by
$\mathcal{C}_{\delta}\cup\\{S^{\delta}_{\gamma}\\}$, or equivalently the
filter generated by the $\gamma$-closed unbounded subsets of $\delta$.
Similarly, $S^{\delta}_{<\gamma}=\bigcup_{\lambda<\gamma}S^{\delta}_{\lambda}$
and $\mathcal{C}_{\delta,{<}\gamma}$ denotes the filter generated by
$\mathcal{C}_{\delta}\cup\\{S^{\delta}_{<\gamma}\\}$.

A filter $F$ is $\gamma$-saturated if any collection of $F$-almost disjoint
$F$-positive sets has cardinality less than $\gamma$. The saturation of a
filter is a measure of how close the filter is to being an ultrafilter.

######  Lemma 2.1 (Ulam).

Suppose $F$ is a $(2^{\gamma})^{+}$-complete $\gamma$-weakly saturated filter.
Then $F$ is the intersection of fewer than $\gamma$-many ultrafilters.∎

Note that if a $\gamma$-complete filter $F$ is the intersection of fewer than
$\gamma$-many ultrafilters, then the underlying set of $F$ can be partitioned
into atoms of $F$.

If $\nu<\delta$ are regular cardinals, $\delta$ is $\nu$-strongly measurable
in an inner model $N$ if $\mathcal{C}_{\delta,\nu}\cap N$ belongs to $N$ and
is $\gamma$-weakly saturated in $N$ for some $N$-cardinal $\gamma$ such that
$(2^{\gamma})^{N}<\delta$.

######  Lemma 2.2.

Suppose $N$ is an inner model of ZFC, $\nu<\delta$ are regular cardinals, and
$\eta$ is the least $N$-cardinal such that $(2^{\eta})^{N}\geq\delta$. Then
the following are equivalent:

  * •

$\delta$ is $\nu$-strongly measurable in $N$.

  * •

For some $\gamma<\eta$, $\mathcal{C}_{\delta,\nu}\cap N$ can be written as an
intersection of fewer than $\gamma$-many ultrafilters of $N$.

  * •

For some $\gamma<\eta$, $\delta$ can be partitioned into $\gamma$-many
$\mathcal{C}_{\delta,\nu}$-positive sets $\langle
S_{\alpha}:\alpha<\nu\rangle\in N$ such that
$(\mathcal{C}_{\delta,\nu}\restriction S_{\alpha})\cap N$ is an ultrafilter in
$N$.∎

###  2.2 The HOD dichotomy

Suppose $M$ is an inner model, $\lambda$ is a cardinal, and $\lambda^{\prime}$
is an $M$-cardinal. An inner model $M$ has the
$(\lambda,\lambda^{\prime})$-cover property if for every set $\sigma\subseteq
M$ of cardinality less than $\lambda$, there is a set $\tau\in M$ of
$M$-cardinality less than $\lambda^{\prime}$ such that $\sigma\subseteq\tau$.
We refer to the $(\lambda,\lambda)$-cover property as the $\lambda$-cover
property.

######  Proposition 2.3.

Suppose $\kappa$ is strongly compact. Then exactly one of the following holds:

  1. (1)

All sufficiently large regular cardinals are measurable in HOD.

  2. (2)

HOD has the $\kappa$-cover property.

###### Proof.

There are two cases.

######  Case 1.

There is a HOD-cardinal $\eta$ such that for all regular cardinals $\delta$,
$\mathcal{C}_{\delta,{<}\kappa}\cap\textnormal{HOD}$ is $\eta$-saturated in
HOD.

In this case, 2.1 implies that all regular cardinals
$\delta>(2^{\eta})^{\textnormal{HOD}}$ are measurable in HOD.

######  Case 2.

For every HOD-cardinal $\eta$, there is a regular cardinal $\delta$ such that
$\mathcal{C}_{\delta,{<}\kappa}\cap\textnormal{HOD}$ is not $\eta$-saturated
in HOD.

We claim that HOD has the $\kappa$-cover property. For this, it suffices to
show that for all $\eta$, there is a $\kappa$-complete fine ultrafilter
$\mathcal{U}$ on $P_{\kappa}(\eta)$ such that $\textnormal{HOD}\cap
P_{\kappa}(\eta)\in\mathcal{U}$. Then for each $\sigma\subseteq\eta$ with
$|\sigma|<\kappa$, the set $\\{\tau\in
P_{\kappa}(\eta):\sigma\subseteq\tau\\}$ belongs to $\mathcal{U}$ by fineness
and $\kappa$-completeness. Thus this set meets $\textnormal{HOD}\cap
P_{\kappa}(\eta)$, which yields a set $\tau\in\textnormal{HOD}$ such that
$|\tau|^{\textnormal{HOD}}<\kappa$ and $\sigma\subseteq\tau$.

Fix an ordinal $\eta$. Let $\delta$ be a regular cardinal such that
$\mathcal{C}_{\delta,{<}\kappa}\cap\textnormal{HOD}$ is not $\eta$-saturated
in HOD. Let $\mathcal{S}=\\{S_{\alpha}:\alpha<\eta\\}$ be an ordinal definable
partition of $S^{\delta}_{<\kappa}$ into stationary sets.

Since $\kappa$ is strongly compact, there is an elementary embedding $j:V\to
M$ with critical point $\kappa$ such that $M$ has the
$(\delta^{+},j(\kappa))$-cover property. Let $\delta_{*}=\sup j[\delta]$. This
implies that $\textnormal{cf}^{M}(\delta_{*})<j(\kappa)$. Fix a closed
unbounded set $C\subseteq\delta_{*}$ such that $C\in M$ and
$|C|^{M}<j(\kappa)$.

Let $\mathcal{T}=\langle T_{\beta}:\beta<j(\eta)\rangle=j(\mathcal{S})$. Let

$\sigma=\\{\beta<j(\eta):T_{\beta}\cap\delta_{*}\text{ is stationary in }M\\}$

Then $\sigma$ is ordinal definable in $M$ since $\mathcal{T}$ is. Notice that
for all $\alpha<\eta$, $j[S_{\alpha}]$ is a stationary subset of $\delta_{*}$:
$j$ is continuous on $S^{\delta}_{<\kappa}$, and so there is a continuous
increasing cofinal function $f:\delta\to\delta_{*}$ such that
$j[S_{\alpha}]=f[S_{\alpha}]$. Since $j[S_{\alpha}]\subseteq T_{j(\alpha)}$,
$j(\alpha)\in\sigma$. On the other hand, for each $\beta\in\sigma$, let
$f(\beta)=\min(T_{\beta}\cap C)$. Then $f:\sigma\to C$ is an injection, so
$|\sigma|^{M}<j(\kappa)$. In other words, $\sigma\in j(P_{\kappa}(\eta))$.

Let $\mathcal{U}$ be the ultrafilter on $P_{\kappa}(\eta)$ derived from $j$
using $\sigma$. Then $\textnormal{HOD}\cap P_{\kappa}(\eta)\in\mathcal{U}$
since $\sigma$ is ordinal definable in $M$. Moreover $\mathcal{U}$ is fine
since $j[\eta]\subseteq\sigma$. Finally $\mathcal{U}$ is $\kappa$-complete
since $\textnormal{crit}(j)=\kappa$.

This finishes the proof that HOD has the $\kappa$-cover property. ∎

The proof of 2.3 shows that if HOD does not cover $V$, then sufficiently large
regular cardinals are measurable in HOD in a strong sense.

######  Proposition 2.4.

Suppose $\kappa$ is strongly compact. Then exactly one of the following holds:

  1. (1)

For some $\gamma$, for all ordinals $\delta$ with
$\textnormal{cf}(\delta)\geq\gamma$,
$\mathcal{C}_{\delta,<\kappa}\cap\textnormal{HOD}$ is the intersection of
fewer than $\gamma$-many ultrafilters in HOD.

  2. (2)

HOD has the $\kappa$-cover property.∎

For any OD set $A\subseteq\textnormal{OD}$, let $|A|^{\textnormal{OD}}$ denote
the minimum ordertype of an ordinal definable wellorder of $A$. For any set
$X$, let $\beta(X)$ denote the set of ultrafilters on $X$.

######  Proposition 2.5.

Suppose $\kappa$ is strongly compact and HOD has the $\kappa$-cover property.
Then for any cardinal $\lambda\geq\kappa$, HOD has the
$({\leq}\lambda,\theta)$-cover property where
$\theta=|P^{\textnormal{OD}}(\beta(\lambda))|^{\textnormal{OD}}$.

###### Proof.

We may assume that $\lambda$ is a regular cardinal. Let $\mathcal{U}$ be a
$\kappa$-complete fine ultrafilter on $P_{\kappa}(\lambda)$. Note that
$\sigma$ is covered by a set of size less than $j_{\mathcal{U}}(\kappa)$ in
$M_{\mathcal{U}}$. Since $\textnormal{HOD}^{M_{\mathcal{U}}}$ has the
$j_{\mathcal{U}}(\kappa)$-cover property in $M_{\mathcal{U}}$, $\sigma$ is in
fact covered by a set of size less than $j_{\mathcal{U}}(\kappa)$ in
$\textnormal{HOD}^{M_{\mathcal{U}}}$. But
$\textnormal{HOD}^{M_{\mathcal{U}}}\subseteq\textnormal{HOD}_{\mathcal{U}}$,
and so $\sigma$ is covered by a set of size less than $(2^{\lambda})^{+}$ in
$\textnormal{HOD}_{\mathcal{U}}$. But $\textnormal{HOD}_{\mathcal{U}}$ is a
$\theta$-cc extension of HOD and $\theta\geq(2^{\lambda})^{+}$, so $\sigma$ is
covered by a set of size less than $\theta$ in HOD. ∎

######  Corollary 2.6.

If $\kappa$ is strongly compact, either all sufficiently large regular
cardinals are measurable in HOD or HOD has the $\lambda$-cover property for
all strong limit cardinals $\lambda\geq\kappa$.∎

In particular, for all strong limit singular cardinals $\lambda\geq\kappa$,
$\lambda$ is singular in HOD and $\lambda^{+\textnormal{HOD}}=\lambda^{+}$.
But in fact one can prove this for arbitrary singular cardinals above
$\kappa$. The proof is more general in two ways. First, it uses a weaker large
cardinal hypothesis: a cardinal $\kappa$ is $\omega_{1}$-strongly compact if
every $\kappa$-complete filter extends to a countably complete ultrafilter.

Second, it applies to a broader class of models than just HOD: an inner model
$N$ of ZFC is $\omega$-club amenable if for every ordinal $\delta$ of
uncountable cofinality, $\mathcal{C}_{\delta,\omega}\cap N\in N$. The proof of
the following lemma is similar to that of 2.3:

######  Lemma 2.7.

Suppose $N$ is $\omega$-club amenable and $\kappa$ is $\omega_{1}$-strongly
compact. Then one of the following holds:

  1. (1)

All sufficiently large regular cardinals are measurable in $N$.

  2. (2)

$N$ has the $(\omega_{1},\kappa)$-cover property.

###### Proof.

The proof splits into cases as in 2.3

######  Case 1.

There is an $N$-cardinal $\eta$ such that for all regular cardinals $\delta$,
$\mathcal{C}_{\delta,\omega}\cap N$ is $\eta$-saturated in $N$.

Then for all sufficiently large regular cardinals $\delta$, $\delta$ is
measurable in $N$ by 2.1, applied in $N$ to $\mathcal{C}_{\delta,\omega}\cap
N$ in $N$.

######  Case 2.

For every $N$-cardinal $\eta$, there is a regular cardinal $\delta$ such that
$\mathcal{C}_{\delta,\omega}\cap N$ is not $\eta$-saturated in $N$.

In this case, one shows that for all $\lambda\geq\kappa$, there is a countably
complete fine ultrafilter $\mathcal{U}$ on $P_{\kappa}(\lambda)$ such that
$N\cap P_{\kappa}(\lambda)\in\mathcal{U}$. It follows that every countable set
$\sigma\subseteq\lambda$ belongs to some $\tau\in N\cap P_{\kappa}(\lambda)$.
∎

######  Theorem 2.8.

Suppose $\nu$ is a regular cardinal and $N$ is an $\omega$-club amenable inner
model with the $(\omega_{1},\nu)$-cover property. Then for all $N$-regular
$\delta\geq\nu$, $\textnormal{cf}(\delta)=|\delta|$.

###### Proof.

Let $A=(S^{\delta}_{<\nu})^{N}$. The $(\omega_{1},\nu)$-cover property of $N$
implies that $S^{\delta}_{\omega}\subseteq A$, which is all we will use. Fix
$\langle c_{\xi}:\xi\in A\rangle\in N$ such that $c_{\xi}$ is a closed cofinal
subset of $\xi$ of ordertype less than $\nu$. For each $\alpha<\delta$, let
$\beta_{\alpha}$ denote the least ordinal such that for a stationary set of
$\xi\in S^{\delta}_{\omega}$,
$c_{\xi}\cap[\alpha,\beta_{\alpha})\neq\emptyset$. One can prove that
$\beta_{\alpha}$ exists for all $\alpha<\delta$ by applying Fodor’s lemma to
the function $f(\xi)=\min(c_{\xi}\setminus\alpha)$. Define a continuous
increasing sequence $\langle\epsilon_{\alpha}:\alpha<\delta\rangle$ by setting
$\epsilon_{\alpha+1}=\beta_{\epsilon_{\alpha}}$, taking suprema at limit
steps; these suprema are always below $\delta$ because the sequence belongs to
$N$ and $\delta$ is regular in $N$.

For each $\xi\in A$, let

$\sigma_{\xi}=\\{\alpha<\delta:c_{\xi}\cap[\epsilon_{\alpha},\epsilon_{\alpha+1})\neq\emptyset\\}$

Note that $|\sigma_{\xi}|<\nu$ since $|c_{\xi}|<\nu$. Moreover for all
$\alpha$, the set $S_{\alpha}=\\{\xi\in A:\alpha\in\sigma_{\xi}\\}$ is
stationary. Let $C\subseteq\delta$ be a closed cofinal set of ordertype
$\textnormal{cf}(\delta)$. For any $\alpha<\delta$, there is some $\xi\in
S_{\alpha}\cap C$, which means that $\alpha\in\sigma_{\xi}$. This implies that
$\delta=\bigcup_{\xi\in C}\sigma_{\xi}$. Thus $|\delta|=|C|\cdot\sup_{\xi\in
C}|\sigma_{\xi}|$. If $|C|<\nu$, then since $\nu$ is regular, $\sup_{\xi\in
C}|\sigma_{\xi}|<\nu$ and hence $|\delta|<\nu$, contradicting that
$\delta\geq\nu$. Therefore $|C|\geq\nu$. Therefore
$|\delta|=|C|\cdot\sup_{\xi\in
C}|\sigma_{\xi}|=|C|\cdot\nu=|C|=\textnormal{cf}(\delta)$. ∎

######  Corollary 2.9.

Suppose $\kappa$ is $\omega_{1}$-strongly compact and $N$ is an $\omega$-club
amenable inner model. Either all sufficiently large regular cardinals are
measurable in $N$ or every singular cardinal $\lambda$ greater than $\kappa$
is singular in $N$ and $\lambda^{+N}=\lambda^{+}$.∎

We now state some reformulations of the failure of the HOD hypothesis.

######  Theorem 2.10.

If $\kappa$ is strongly compact, the following are equivalent to the failure
of the HOD hypothesis:

  1. (1)

There is a regular cardinal $\delta\geq\kappa$ with a stationary subset
$S\subseteq S^{\delta}_{<\kappa}$ that admits no ordinal definable partition
into $\delta$-many stationary sets.

  2. (2)

There is an $\omega$-strongly measurable cardinal above $\kappa$.

  3. (3)

For some regular $\gamma<\kappa$, there is a $\gamma$-strongly measurable
cardinal above $\kappa$.

  4. (4)

For all regular $\gamma<\kappa$, all sufficiently large regular cardinals are
$\gamma$-strongly measurable.

  5. (5)

For some cardinal $\lambda$, for all regular $\gamma<\kappa$ and all $\nu$
with $\textnormal{cf}(\nu)>\gamma$,
$\mathcal{C}_{\nu,\gamma}\cap\textnormal{HOD}$ is $\lambda$-saturated in HOD.

###### Proof.

We first show that 1 implies that the HOD hypothesis is false. By the proof of
2.3, if the HOD hypothesis fails, then 5 holds. Since each item easily implies
all the previous ones, the theorem easily follows.

It remains to show that 1 implies the failure of the HOD hypothesis, or in
other words, that the HOD hypothesis implies that every stationary subset
$S\subseteq S^{\delta}_{<\kappa}$ splits into $\delta$-many subsets. This
follows from the proof of the Solovay splitting theorem and the fact that
$(S^{\delta}_{<\kappa})^{\textnormal{HOD}}=S^{\delta}_{<\kappa}$. ∎

It is easy to see that if the HOD hypothesis fails, then every regular
cardinal $\delta$ contains an $\omega$-club of ordinals that are strongly
inaccessible in HOD. In fact, this alone implies a stronger conclusion.

######  Theorem 2.11.

Suppose $N$ is an inner model and $\delta$ is a regular cardinal such that
$\textnormal{Reg}^{N}\cap\delta$ contains an $\omega$-club. Then
$\textnormal{Reg}^{N}\cap\delta$ contains a club.

###### Proof.

Assume towards a contradiction that $\textnormal{Sing}^{N}\cap\delta$ is
stationary. Since the cofinality function as computed in $N$ is regressive,
there is an ordinal $\gamma$ that is regular in $N$ such that
$(S^{\delta}_{\gamma})^{N}$ is stationary. Let $\gamma$ be the least such
ordinal and let $S=(S^{\delta}_{\gamma})^{N}$. We claim
$\textnormal{cf}(\gamma)=\omega$.

Choose $\langle c_{\xi}:\xi\in S\rangle\in N$ such that $c_{\xi}$ is a closed
unbounded subset of $\xi$ with ordertype $\gamma$. Consider the set $T$ of
$\nu\in S^{\delta}_{\omega}$ such that there is some $\xi\geq\nu$ in $S$ such
that $c_{\xi}\cap\nu$ is cofinal in $\nu$. Then $T$ is stationary. To see
this, fix a closed unbounded set $C\subseteq\delta$, and we will show that
$C\cap T\neq\emptyset$. Fix $\xi\in S\cap\textnormal{acc}(C)$. Since
$\textnormal{cf}(\xi)>\omega$, $C\cap\textnormal{acc}(c_{\gamma})$ is closed
unbounded in $\gamma$, and so there is some $\nu\in
C\cap\textnormal{acc}(c_{\gamma})\cap S^{\delta}_{\omega}$. By definition,
$\nu\in T$. So $C\cap T\neq\emptyset$, as claimed.

Note that $T\subseteq(S^{\delta}_{<\gamma})^{N}$, so
$(S^{\delta}_{<\gamma})^{N}$ is stationary. Again applying Fodor’s lemma,
there is an $N$-regular ordinal $\gamma^{\prime}<\gamma$ such that
$(S^{\delta}_{\gamma^{\prime}})^{N}$ is stationary, and this contradicts the
minimality of $\gamma$.

Since $(S^{\delta}_{\gamma})^{N}$ is a stationary subset of
$S^{\delta}_{\omega}$, its intersection with any $\omega$-club is stationary.
This implies that $(S^{\delta}_{\gamma})^{N}\cap\textnormal{Reg}^{N}$ is
stationary, which is a contradiction. ∎

######  Corollary 2.12.

Assume there is an $\omega_{1}$-strongly compact cardinal $\kappa$ and $N$ is
an $\omega$-club amenable model such that $\lambda^{+N}<\lambda^{+}$ for some
singular $\lambda>\kappa$. Then all sufficiently large regular cardinals
contain a closed unbounded set of $N$-inaccessible cardinals.

###### Proof.

Suppose $\delta$ is $\omega$-strongly measurable in $N$, and we will show that
there is an $\omega$-club of $N$-regular ordinals below $\delta$. Suppose not.
Then $\textnormal{Sing}^{N}$ is $\mathcal{C}_{\delta,\omega}$-positive. It
follows that there is a set $S\in N$ such that
$S\subseteq\textnormal{Sing}^{N}$ and $\mathcal{C}_{\delta,\omega}\restriction
S$ is a normal ultrafilter in $N$. This is a contradiction since
$\textnormal{Reg}^{N}\cap\delta$ belongs to any normal ultrafilter of $N$ on
$\delta$.

Since all sufficiently large regular cardinals are $\omega$-strongly
measurable in $N$ by 2.9, the desired conclusion follows from 2.11. ∎

######  Corollary 2.13.

Assume there is an $\omega_{1}$-strongly compact cardinal and the HOD
hypothesis fails. Then all sufficiently large regular cardinals contain a
closed unbounded set of HOD-inaccessible cardinals.∎

##  3 Embeddings of HOD

###  3.1 Uniqueness of elementary embeddings

We prove that the HOD Hypothesis is equivalent to a uniqueness property of
elementary embeddings that makes no mention of ordinal definability.

######  Lemma 3.1.

If $\kappa$ is supercompact, then for all regular $\delta\geq\kappa$ and all
$\gamma>\delta$, the set

$\\{\sup j[\bar{\delta}]:\bar{\delta}<\bar{\gamma}<\delta,\
j:V_{\bar{\gamma}}\to V_{\gamma}\text{ is elementary, and
}j(\bar{\delta})=\delta\\}$

is stationary in $\delta$.

######  Lemma 3.2.

Suppose $\kappa$ is strongly compact and for arbitrarily large regular
cardinals $\delta\geq\kappa$, the set
$\\{\alpha<\delta:\textnormal{cf}^{\textnormal{HOD}}(\alpha)<\kappa\\}$ is
stationary. Then the HOD hypothesis holds.

###### Proof.

This follows from 2.4. ∎

######  Definition 3.3.

Suppose $j_{0},j_{1}:M\to N$ are elementary embeddings. For any ordinal
$\delta\in M$, $j_{0}$ and $j_{1}$ are similar at $\delta$ if $\sup
j_{0}[\delta]=\sup j_{1}[\delta]$ and $j_{0}(\delta)=j_{1}(\delta)$. For any
ordinal $\delta^{\prime}\in N$, $j_{0}$ and $j_{1}$ are similar below
$\delta^{\prime}$ if there is some $\delta\in M$ such that
$j_{0}(\delta)=j_{1}(\delta)=\delta^{\prime}$ and $j_{0}$ and $j_{1}$ are
similar at $\delta$.

######  Theorem 3.4.

Suppose $\kappa$ is a supercompact cardinal. Then the following are
equivalent:

  1. (1)

The HOD hypothesis holds.

  2. (2)

For all regular $\delta\geq\kappa$, for all sufficiently large $\gamma$, if
$j_{0},j_{1}:V_{\bar{\gamma}}\to V_{\gamma}$ are similar below $\delta$, then
$j_{0}\restriction\bar{\delta}=j_{1}\restriction\bar{\delta}$.

###### Proof.

Assume 1 holds. Let $\langle S_{\alpha}:\alpha<\delta\rangle$ be the least
partition of $S^{\delta}_{\omega}$ into stationary sets in the canonical
wellorder of HOD. For sufficiently large ordinals $\gamma$, $\langle
S_{\alpha}:\alpha<\delta\rangle$ is definable from $\delta$ in $V_{\gamma}$.
Suppose $j:V_{\bar{\gamma}}\to V_{\gamma}$ and $j(\bar{\delta})=\delta$. Then
$\langle S_{\alpha}:\alpha<\delta\rangle$ is in the range of $j$ since
$\delta$ is. As a consequence, $j[\bar{\delta}]$ is the set of $\alpha<\delta$
such that $S_{\alpha}$ reflects to $\sup j[\bar{\delta}]$. 2 follows
immediately.

Conversely, assume 2. Fix a regular cardinal $\delta>\kappa$. For each
$\gamma>\delta$, define

$S_{\gamma}=\\{\sup j[\bar{\delta}]:j:V_{\bar{\gamma}}\to V_{\gamma}\text{ is
elementary and }j(\bar{\delta})=\delta\\}$

Fix an ordinal $\gamma$ sufficiently large that if
$j_{0},j_{1}:V_{\bar{\gamma}}\to V_{\gamma}$ are elementary embeddings with
$j_{0}(\bar{\delta})=j_{1}(\bar{\delta})=\delta$ and $\sup
j_{0}[\bar{\delta}]=\sup j_{1}[\bar{\delta}]$, then
$j_{0}\restriction\bar{\delta}=j_{1}\restriction\bar{\delta}$. For $\xi\in
S_{\gamma}$, let $c_{\xi}=j[\bar{\delta}]$ for $j:V_{\bar{\gamma}}\to
V_{\gamma}$ such that $j(\bar{\delta})=\delta$ and $\sup j[\bar{\delta}]=\xi$.
Then $c_{\xi}$ is ordinal definable, so
$\textnormal{cf}^{\textnormal{HOD}}(\xi)\leq\textnormal{ot}(c_{\xi})$. By 3.2,
the HOD hypothesis holds. ∎

As a corollary, one has the following equivalent forms of the local uniqueness
of elementary embeddings at an extendible. We note that the equivalences would
be quite mysterious (and hard to prove) without having the HOD hypothesis to
tie them together.

######  Theorem 3.5.

If $\kappa$ is extendible, then the following are equivalent.

  1. (1)

The HOD hypothesis holds.

  2. (2)

For all regular $\delta\geq\kappa$, for all sufficiently large $\alpha$, if
$j_{0},j_{1}:V_{\alpha}\to M$ are similar at $\delta$, then
$j_{0}\restriction\delta=j_{1}\restriction\delta$.

  3. (3)

For some regular $\delta\geq\kappa$, for all sufficiently large $\alpha$, if
$j_{0},j_{1}:V_{\alpha}\to V_{\alpha^{\prime}}$ are similar at $\delta$, then
$j_{0}\restriction\delta=j_{1}\restriction\delta$.

  4. (4)

Let $\nu$ be the least supercompact cardinal. For all regular $\delta\geq\nu$,
for all sufficiently large $\alpha$, if $j_{0},j_{1}:V_{\bar{\alpha}}\to
V_{\alpha}$ are similar below $\delta$, then
$j_{0}\restriction\delta=j_{1}\restriction\delta$.∎

###  3.2 $\omega$-Jónsson cardinals

Recall that a cardinal $\lambda$ is $\omega$-Jónsson if for all functions
$f:[\lambda]^{\omega}\to\lambda$, there is some $H\subseteq\lambda$ such that
$\textnormal{ot}(H)=\lambda$ and $\textnormal{ran}(f\restriction[H]^{\omega})$
is a proper subset of $\lambda$. The Erdos-Hajnal theorem states that assuming
ZFC there is no such cardinal. One can ask what happens when one places
definability constraints on $f$.

######  Definition 3.6.

A cardinal $\lambda$ is definably $\omega$-Jónsson if for all ordinal
definable functions $f:[\lambda]^{\omega}\to\lambda$, there is some
$H\subseteq\lambda$ such that $\textnormal{ot}(H)=\lambda$ and
$\textnormal{ran}(f\restriction[H]^{\omega})$ is a proper subset of $\lambda$.

######  Theorem 3.7.

Suppose $\kappa$ is strongly compact. Then the following are equivalent:

  1. (1)

The HOD hypothesis fails.

  2. (2)

All sufficiently large regular cardinals are definably $\omega$-Jónsson.

  3. (3)

Some regular cardinal above $\kappa$ is definably $\omega$-Jónsson.

###### Proof.

1 implies 2: The converse rests on Solovay’s published proof of Solovay’s
lemma [5]. Let $R$ be the class of regular cardinals $\delta$ that are not
definably $\omega$-Jónsson. Fix $\delta\in R$, and we will show that HOD has
the $\kappa$-cover property below $\delta$. This completes the proof, because
assuming 2 fails, $R$ is a proper class, and hence HOD has the $\kappa$-cover
property, which, by 2.3, is equivalent to the HOD hypothesis given a strongly
compact cardinal.

Let $f:[\delta]^{\omega}\to\delta$ be an ordinal definable function such that
for all $A\subseteq\delta$ such that $f[[A]^{\omega}]=\delta$. We claim that
any $\omega$-closed unbounded subset $C$ of $\sup j[\delta]$ such that
$j(f)[C]\subseteq C$ contains $j[\delta]$. Let $B=j^{-1}[C]$. Then $B$ is
unbounded below $\delta$ and $B$ is closed under $f$, and therefore
$B\supseteq f[B]=\delta$. In other words, $j[\delta]\subseteq C$.

Working in $M$, let $A$ be the intersection of all $\omega$-closed unbounded
subsets of $\sup j[\delta]$ that are closed under $j(f)$. Then $A\in M$ and
$j[\delta]\subseteq A$ and $A$ is ordinal definable from $j(f)$ in $M$.
Therefore $A\in\textnormal{HOD}^{M}$. It follows from the proof of 2.3 that
HOD has the $\kappa$-cover property below $\delta$.

2 implies 3: Trivial.

3 implies 1: Suppose $\delta\geq\kappa$ is regular and definably
$\omega$-Jónsson, and assume towards a contradiction that the HOD hypothesis
holds. Applying 2.10, let $\langle
S_{\alpha}:\alpha<\delta\rangle\in\textnormal{HOD}$ partition
$S^{\delta}_{\omega}$ into stationary sets. Let $f:[\delta]^{\omega}\to\delta$
be defined by $f(s)=\alpha$ where $\alpha<\delta$ is the unique ordinal such
that $\sup(s)\in S_{\alpha}$. Then for any unbounded set $T\subseteq\delta$,
for each $\alpha<\delta$, there is some $s\in[T]^{\omega}$ such that $\sup
s\in S_{\alpha}$, and therefore $f(s)=\alpha$. It follows that
$f[[T]^{\omega}]]=\delta$. ∎

The usual characterizations of Jónsson cardinals in terms of elementary
embeddings yields the following equivalence.

######  Corollary 3.8.

Assume there is a strongly compact cardinal. Then exactly one of the following
holds.

  1. (1)

The HOD hypothesis.

  2. (2)

For all sufficiently large regular cardinals $\delta$, for all ordinals
$\alpha>\delta$, there is an elementary embedding $j:M\to\textnormal{HOD}\cap
V_{\alpha}$ such that $\textnormal{crit}(j)<\delta$, $j(\delta)=\delta$, and
$j$ is continuous at ordinals of cofinality $\omega$.∎

###  3.3 The rigidity of HOD

This section is devoted to a proof of the following theorem:

######  Theorem 3.9.

Assume there is an $\omega_{1}$-strongly compact cardinal. If the HOD
hypothesis holds, then there is no nontrivial elementary embedding from HOD to
HOD.

We need to prove some preliminary lemmas.

######  Lemma 3.10.

If $j:\textnormal{HOD}\to\textnormal{HOD}$ is a nontrivial elementary
embedding, then $j$ has a proper class of generators.∎

######  Lemma 3.11.

Suppose $\kappa$ is $\omega$-strongly compact and $H$ is an $\omega$-club
amenable model such that arbitrarily large regular cardinals are not
measurable in $H$. Then the Singular Cardinals Hypothesis holds in $H$ above
$\kappa^{+}\cdot(\kappa^{\omega})^{H}$.

###### Proof.

By Silver’s theorem, it suffices to show that in $H$, for all singular
cardinals $\lambda>2^{<\kappa}$ with $\textnormal{cf}(\lambda)=\omega$,
$\lambda^{\omega}=\lambda^{+}$. In fact, we will show that for all $H$-regular
$\delta\geq\kappa^{+}$, there is a set $C\subseteq P_{\kappa}(\delta)\cap H$
in $H$ such that $|C|^{H}=\delta$ and every countable $\sigma\subseteq\delta$
is contained in some $\tau\in H$. Then in particular,
$([\delta]^{\omega})^{H}\subseteq\bigcup_{\sigma\in C}([\sigma]^{\omega})^{H}$
and hence $(\delta^{\omega})^{H}=(\kappa^{\omega})^{H}\cdot\delta$.

To define $C$, we first split $S=(S^{\delta}_{<\kappa})^{H}$ into
$\delta$-many $\mathcal{C}_{\delta,\omega}$-positive sets in $H$. Note that
$S$ contains $S^{\delta}_{\omega}$ by 2.7, and hence $S$ has full measure with
respect to $\mathcal{C}_{\delta,\omega}$. Now one can use a standard argument
to partition $S$ into $\delta$-many $\mathcal{C}_{\delta,\omega}$-positive
sets in $H$. Working in $H$, for each $\alpha\in S$, fix a club
$c_{\alpha}\subseteq\alpha$ of ordertype less than $\kappa$. For $\xi<\kappa$,
let $f_{\xi}(\alpha)=c_{\alpha}(\xi)$ for any $\alpha$ such that
$\xi<\textnormal{ot}(c_{\alpha})$.

We claim that there is some $\xi<\kappa$ such that for unboundedly many
$\beta<\delta$, $f_{\xi}^{-1}\\{\beta\\}$ is
$\mathcal{C}_{\delta,\omega}$-positive. Otherwise, using that
$C_{\delta,\omega}$ is weakly normal and $f_{\xi}$ is regressive, for each
$\xi<\kappa$, there is an ordinal $\gamma_{\xi}$ such that
$f_{\xi}(\alpha)<\gamma_{\xi}$ for an $\omega$-club of $\alpha\in S$. Since
$\textnormal{cf}(\delta)>\kappa$, there is then a single $\omega$-club of
$\alpha$ such that for all $\xi<\kappa$, $f_{\xi}(\alpha)<\gamma_{\xi}$ for
all appropriate $\alpha$. Letting
$\gamma=\sup_{\xi<\kappa}\gamma_{\xi}<\delta$, we see that for an
$\omega$-club of $\alpha$, for all appropriate $\xi<\kappa$,
$f_{\xi}(\alpha)<\gamma$. But if $\alpha>\gamma$ belongs to this club, then
the fact that $c_{\alpha}(\xi)=f_{\xi}(\alpha)<\gamma$ for all
$\xi<\textnormal{ot}(c_{\alpha})$ contradicts that $c_{\alpha}$ was chosen to
be unbounded in $\alpha$.

Now fix $\xi<\kappa$ such that such that for unboundedly many $\beta<\delta$,
$f_{\xi}^{-1}\\{\beta\\}$ is $\mathcal{C}_{\delta,\omega}$-positive. Since $H$
is $\omega$-club amenable and $f_{\xi}\in N$, the set $B$ of such $\beta$
belongs to $H$, and so since $\delta$ is regular in $H$, the ordertype of $B$
is $\delta$. Finally, let $S_{\beta}=f_{\xi}^{-1}\\{\beta\\}$. Then $\langle
S_{\beta}:\beta\in B\rangle$ is the desired stationary partition.

Finally, for each $\xi<\delta$ of uncountable cofinality, let $\sigma_{\xi}$
be the set of $\beta\in B$ such that $S_{\beta}$ is
$\mathcal{C}_{\delta,\xi}$-positive. Then
$\langle\sigma_{\xi}:\xi<\delta\rangle$ is in $H$ by $\omega$-club
amenability, and the argument of 2.7 shows that every countable subset of
$\delta$ is contained in $\sigma_{\xi}$ for some $\xi<\delta$. Therefore
$C=\\{\sigma_{\xi}:\xi<\delta\\}$ is as desired. ∎

######  Lemma 3.12.

Suppose $\delta$ is a cardinal, $N$ and $H$ are models with the
$(\omega_{1},\delta)$-cover property, and $(2^{\lambda})^{H}=\lambda^{+H}$ for
all sufficiently large strong limit cardinals of countable cofinality. Then
any elementary embedding $k:N\to H$ with $\textnormal{crit}(k)\geq\delta$ is
an extender embedding.

###### Proof.

Note that $k$ is continuous at ordinals of cofinality $\omega$, since these
have $N$-cofinality less than $\delta$. We will show that there cannot exist a
singular strong cardinal $\lambda$ with the following properties:

  * •

$\textnormal{cf}(\lambda)=\omega$.

  * •

$\lambda$ is a limit of generators of $k$.

  * •

$k(\lambda)=\lambda$.

  * •

$(2^{\lambda})^{H}=\lambda^{+H}$.

We first point out that $\lambda^{+H}=\lambda^{+N}=\lambda^{+}$. Using the
$(\omega_{1},\delta)$-cover property, $\lambda^{+}\leq|P_{\delta}(\lambda)\cap
H|\leq(2^{\lambda})^{H}=\lambda^{+H}$. The argument for $N$ is similar, using
that $(2^{\lambda})^{N}=\lambda^{+N}$ by elementarity.

Let $\sigma\subseteq\lambda$ be a countable cofinal set, and let $\tau\in H$
be a cover of $\sigma$ of size less than $\delta$. Let $U$ be the
$N$-ultrafilter on $P_{\delta}(\lambda)\cap N$ derived from $k$ using $\tau$.
Let $j_{U}:N\to P$ be the ultrapower embedding and $i:P\to H$ be the factor
embedding. Then $\tau=i([\textnormal{id}]_{U})\in\textnormal{ran}(i)$, and so
since $|\tau|<\delta$ and $\textnormal{crit}(i)<\delta$,
$\tau\subseteq\textnormal{ran}(i)$. As a consequence, $\lambda$ is a limit of
generators of $j_{U}$, which implies $\lambda_{U}\geq\lambda$. But
$(2^{\lambda})^{H}=\lambda^{+H}$, and so by elementarity
$(2^{\lambda})^{N}=\lambda^{+N}$. Thus
$\lambda_{U}\leq|P_{\delta}(\lambda)\cap
N|\leq(2^{\lambda})^{N}\leq\lambda^{+N}$. Since
$\lambda\leq\lambda_{U}\leq\lambda^{+N}$, we must have
$j_{U}(\lambda)>\lambda$ or $j_{U}(\lambda^{+N})>\lambda^{+N}$. This implies
$k(\lambda)>\lambda$ or $k(\lambda^{+N})>\lambda^{+N}$, but contradicting
either that $k(\lambda)=\lambda$ (by assumption) or that
$k(\lambda^{+N})=\lambda^{+H}=\lambda^{+N}$ (by the previous paragraph).

Since there is no such $\lambda$, the class of limits of generators of $k$
does not intersect the $\omega$-closed unbounded class of singular strong
limit fixed points of $k$ of countable cofinality satisfying
$(2^{\lambda})^{H}=\lambda^{+H}$. It follows that the class of limits of
generators of $k$ not closed unbounded, which means that the class of
generators of $k$ is bounded, or in other words, $k$ is an extender embedding.
∎

######  Lemma 3.13.

Suppose $i:H\to N$ is an extender embedding of length $\lambda$ where
$\lambda$ is an $H$-cardinal of uncountable cofinality. If $H$ has the
$(\omega_{1},\lambda)$-cover property, then so does $N$.

###### Proof of 3.9.

Suppose $j:\textnormal{HOD}\to\textnormal{HOD}$ is an elementary embedding and
assume towards a contradiction that it is nontrivial. Let $\kappa$ be the
least $\omega_{1}$-strongly compact cardinal. By 2.7, HOD has the
$(\omega_{1},\kappa)$-cover property. Let $i:\textnormal{HOD}\to N$ be given
by the extender of length $\kappa$ derived from $j$, and let
$k:N\to\textnormal{HOD}$ be the factor embedding, so
$\textnormal{crit}(k)\geq\kappa$.

By 3.13, $N$ has the $(\omega_{1},\kappa)$-cover property. 3.11 implies that
HOD satisfies the cardinal arithmetic condition of 3.12, and so $k$ is an
extender embedding. Thus $j=k\circ i$ is an extender embedding, contradicting
3.10. ∎

##  4 Large cardinals in HOD

###  4.1 HOD under AD

Recall that a cardinal $\kappa$ is $(\nu,\delta)$-strongly compact if there is
an elementary embedding $j:V\to M$ such that $\textnormal{crit}(j)\geq\nu$ and
$M$ has the $(\delta^{+},j(\kappa))$-cover property. An ultrafilter $U$
witnesses that $\kappa$ is $(\nu,\delta)$-strongly compact if
$\textnormal{crit}(j_{U})\geq\nu$ and $M_{U}$ has the
$(\delta^{+},j(\kappa))$-cover property. If $N$ is an inner model and $U$ is
an ultrafilter, then $U$ witnesses that $\kappa$ is $\delta$-strongly compact
in $N$ if $U\cap N\in N$ and $U\cap N$ witnesses that $\kappa$ is
$\delta$-strongly compact in $N$.

We say an inner model $N$ has the $\kappa$-cover property below $\delta$ if
$P_{\kappa}(\delta)\cap N$ is cofinal in $P_{\kappa}(\delta)$; $N$ has the
$\kappa$-approximation property below $\delta$ if $N$ contains every
$A\subseteq\delta$ such that $A\cap\sigma\in N$ for all $\sigma\in
P_{\kappa}(\delta)$.

######  Theorem 4.1.

Assume $\textnormal{AD}^{+}$ and $V=L(P(\mathbb{R}))$. Suppose $\kappa<\Theta$
is a regular cardinal and $\delta\geq\kappa$ is a HOD-regular ordinal. Then
the following are equivalent:

  1. (1)

$(S^{\delta}_{<\kappa})^{\textnormal{HOD}}$ is stationary.

  2. (2)

HOD has the $\kappa$-cover property below $\delta$.

  3. (3)

HOD has the $\kappa$-cover and $\kappa$-approximation properties below
$\delta$.

  4. (4)

$\mathcal{C}_{\delta,\omega}$ witnesses that $\kappa$ is $\delta$-strongly
compact in HOD.

  5. (5)

Some countably complete ultrafilter witnesses $\kappa$ is
$(\omega_{1},\delta)$-strongly compact in HOD.

  6. (6)

HOD has the $(\omega_{1},\kappa)$-cover property below $\delta$.

  7. (7)

$S^{\delta}_{\omega}\subseteq(S^{\delta}_{<\kappa})^{\textnormal{HOD}}$.

###### Proof.

1 implies 2 Let $T=(S^{\delta}_{<\kappa})^{\textnormal{HOD}}$. We claim that
$S^{\delta}_{\omega}\cap T$ is stationary. Fix a sequence $\langle
c_{\alpha}:\alpha\in T\rangle\in\textnormal{HOD}$ such that $c_{\alpha}$ is a
closed cofinal subset of $\alpha$ of ordertype
$\textnormal{cf}^{\textnormal{HOD}}(\alpha)$. Let

$S=\\{\beta\in S^{\delta}_{\omega}:\exists\alpha\in
T\,\sup(c_{\alpha}\cap\beta)=\beta\\}$

Suppose $C$ is closed unbounded, and we will show $C\cap S\neq\emptyset$.
Since $T$ is stationary, there is some $\beta$ in $\textnormal{acc}(C)$ such
that $\sup(c_{\alpha}\cap\beta)=\beta$. We claim that the least such $\beta$
has countable cofinality. If not, then $\beta$ has uncountable cofinality, so
$C\cap c_{\alpha}\cap\beta$ is closed unbounded. But fixing any
$\beta^{\prime}\in\textnormal{acc}(C\cap c_{\alpha}\cap\beta)$, we have
$\beta^{\prime}\in\textnormal{acc}(C)$ and
$\sup(c_{\alpha}\cap\beta^{\prime})=\beta^{\prime}$, contrary to the
minimality of $\beta$.

Next, we construct a sequence $\langle\sigma_{\xi}:\xi\in T\rangle$ such that
$\sigma_{\xi}\in P_{\kappa}(\xi)$ and for all $\alpha$, for
$\mathcal{C}_{\delta,\omega}$-almost all $\xi$, $\alpha\in\sigma_{\xi}$. This
proceeds as in 2.8, using however that $\mathcal{C}_{\delta,\omega}$ is an
ultrafilter. As in 2.8, one can use this sequence to show
$\textnormal{cf}(\delta)=|\delta|$. Let $C\subseteq\delta$ be a closed cofinal
set of ordertype $\textnormal{cf}(\delta)$. Then $\delta=\bigcup_{\xi\in
C}\sigma_{\xi}$. Therefore $|\delta|=|C|\cdot\sup_{\xi\in C}\sigma_{\xi}$, so
as in 2.8, $|\delta|=|C|=\textnormal{cf}(\delta)$.

Finally, suppose $\sigma\in P_{\kappa}(\delta)$. The set $\\{\xi\in
T:\sigma\subseteq\sigma_{\xi}\\}$ is the intersection of fewer than
$\kappa$-many sets in $\mathcal{C}_{\delta,\omega}$, and so it belongs to
$\mathcal{C}_{\delta,\omega}$. It follows that there is some $\xi\in T$ such
that $\sigma\subseteq\sigma_{\xi}$. This shows that HOD has the $\kappa$-cover
property below $\delta$.

2 implies 3. In HOD, let $\langle X_{\alpha}:\alpha<\delta\rangle$ be a
$\kappa$-independent family of subsets of some set $S$. Suppose
$A\subseteq\delta$ and $A\cap\sigma\in\textnormal{HOD}$ for all $\sigma\in
P_{\kappa}(\delta)\cap\textnormal{HOD}$. For $\alpha<\delta$, let

$Y_{\alpha}=\begin{cases}X_{\alpha}&\text{if }\alpha\in A\\\ S\setminus
X_{\alpha}&\text{otherwise}\end{cases}$

Suppose $\sigma\subseteq\delta$ are disjoint. We will show that
$\bigcap_{\alpha\in\sigma}Y_{\alpha}\neq\emptyset$. First, let
$\tau\in\textnormal{HOD}$ cover $\sigma$. Then by our assumption on $A$,
$\tau\cap A\in\textnormal{HOD}$. It follows that $\langle
Y_{\alpha}:\alpha\in\tau\rangle\in\textnormal{HOD}$, and therefore since
$\langle X_{\alpha}:\alpha<\delta\rangle$ is $\kappa$-independent in HOD,
$\bigcap_{\alpha\in\tau}Y_{\alpha}\neq\emptyset$. Since $\sigma\subseteq\tau$,
$\bigcap_{\alpha\in\sigma}Y_{\alpha}\neq\emptyset$. Let $F$ be the filter
generated by $\\{Y_{\alpha}:\alpha<\delta\\}$. Then $F$ is a $\kappa$-complete
filter on a wellorderable set, and so $F$ extends to a countably complete
ultrafilter $U$. Applying AD, $U\cap\textnormal{HOD}\in\textnormal{HOD}$, and
therefore $A=\\{\alpha<\delta:X_{\alpha}\in U\\}$ belongs to HOD as well.

3 implies 4.
$(S^{\delta}_{<\kappa})^{\textnormal{HOD}}\in\mathcal{C}_{\delta,\omega}$ and
$C_{\delta,\omega}$ is $\kappa$-complete since
$\textnormal{cf}(\delta)\geq\kappa$. Therefore $\mathcal{C}_{\delta,\omega}$
is a $\kappa$-complete, $(\kappa,\delta)$-regular ultrafilter on $\delta$, and
so it witnesses that $\kappa$ is $\delta$-strongly compact.

4 implies 5. Trivial.

5 implies 6. Let $U$ be a countably complete ultrafilter on $\delta$ such that
$U\cap\textnormal{HOD}$ witnesses that $\kappa$ is
$(\omega_{1},\delta)$-strongly compact in HOD. Let $f:\delta\to
P_{\kappa}(\delta)\cap\textnormal{HOD}$ push $U\cap\textnormal{HOD}$ forward
to a fine ultrafilter in HOD. Let $\mathcal{U}=f_{*}(U)$. Then $\mathcal{U}$
is a fine countably complete ultrafilter, and therefore for all $\sigma\in
P_{\omega_{1}}(\delta)$, the set $\\{\tau\in
P_{\kappa}(\delta):\sigma\subseteq\tau\\}$ belongs to $\mathcal{U}$. Since
$P_{\kappa}(\delta)\cap\textnormal{HOD}\in\mathcal{U}$, it follows that there
is some $\tau\in P_{\kappa}(\delta)\cap\textnormal{HOD}$ such that
$\sigma\subseteq\tau$. This shows that HOD has the $(\omega_{1},\delta)$-cover
property.

6 implies 7. Trivial.

7 implies 1. Trivial. ∎

In the context of $\textnormal{AD}^{+}+V=L(P(\mathbb{R}))$, Woodin’s HOD-
ultrafilter conjecture asserts that every countably complete ultrafilter of
$\textnormal{HOD}\cap V_{\Theta}$ extends to a countably complete ultrafilter.

######  Theorem 4.2.

Assume $\textnormal{AD}^{+}$, $V=L(P(\mathbb{R}))$, and the HOD-ultrafilter
conjecture. Suppose $\kappa<\Theta$ is a regular cardinal and
$\delta\geq\kappa$ is a HOD-regular ordinal. Then $\kappa$ is
$\delta$-strongly compact in HOD if and only if $\kappa$ is
$(\omega_{1},\delta)$-strongly compact in HOD.∎

###  4.2 Weak extender models

######  Definition 4.3.

If $U$ is a $\kappa$-complete ultrafilter on $\lambda$ and $\sigma\in
P_{\kappa}(P(\lambda))$, then $A_{U}(\sigma)=\bigcap_{A\in U\cap\sigma}A$ and
$\chi_{U}(\sigma)=\min A_{U}$. Suppose $\mathcal{F}$ is a filter on
$P_{\kappa}(P(\lambda))$ and $U,W$ are $\kappa$-complete ultrafilters on
$\lambda$. Then $U<_{\mathcal{F}}W$ if $\chi_{U}(\sigma)<\chi_{W}(\sigma)$ for
$\mathcal{F}$-almost all $\sigma\in P_{\kappa}(P(\lambda))$.

######  Lemma 4.4.

Suppose $\mathcal{F}$ is a filter on $P_{\kappa}(P(\lambda))$. Then
$<_{\mathcal{F}}$ is a strict partial order. If $\mathcal{F}$ is countably
complete, then $<_{\mathcal{F}}$ is wellfounded. If $\mathcal{F}$ is an
ultrafilter, then $<_{\mathcal{F}}$ is linear.

Thus $<_{\mathcal{F}}$ is a wellorder if $\mathcal{F}$ is a countably complete
ultrafilter on $P_{\kappa}(P(\lambda))$. The Ultrapower Axiom, which we will
not discuss here, implies that if $\mathcal{F}$ is the closed unbounded filter
on $P_{\omega_{1}}(P(\lambda))$, then $<_{\mathcal{F}}$ is a wellorder. This
is also a consequence of $\textnormal{AD}_{\mathbb{R}}$ if $\lambda<\Theta$.

We will use the order $<_{\mathcal{F}}$ in the proof of the following theorem.

######  Theorem 4.5.

Suppose $\kappa$ is strongly compact and $N$ is an inner model of ZFC with the
$\kappa$-cover property. Then there is a minimum extension of $M$ to a model
of ZFC with the $\kappa$-approximation property.

The proof uses the following lemmas.

######  Lemma 4.6.

Suppose $\delta$ is a regular cardinal, $\kappa$ is $\delta$-strongly compact,
$N$ is an inner model of ZFC with the $\kappa$-cover property, and $M$ is an
inner model containing $N$ such that every $\kappa$-complete $N$-ultrafilter
on $\delta$ belongs to $M$. Then every subset of $\delta$ that is
$\kappa$-approximated by $N$ belongs to $M$.∎

######  Theorem 4.7 (Hamkins).

Suppose $\kappa<\lambda$ are cardinals, $\textnormal{cf}(\lambda)\geq\kappa$
and $M$ and $M^{\prime}$ are inner models with the $\kappa$-approximation and
cover properties below $\lambda$ such that $P_{\kappa}(\kappa^{+})\cap
M=P_{\kappa}(\kappa^{+})\cap M^{\prime}$. Then
$P_{\textnormal{bd}}(\lambda)\cap M=P_{\textnormal{bd}}(\lambda)\cap
M^{\prime}$.∎

###### Proof of 4.5.

Suppose $\mathcal{U}$ is a $\kappa$-complete fine ultrafilter on
$P_{\kappa}(\lambda)$ where $\lambda$ is a strong limit cardinal $\lambda$ of
cofinality at least $\kappa$ and $P_{\kappa}(\lambda)\cap N\in\mathcal{U}$.
Suppose $f:\lambda\to P_{\textnormal{bd}}(\lambda)\cap N$ is a surjection in
$N$. Let $\tilde{\mathcal{U}}$ denote the pushforward of $\mathcal{U}$ by the
function $\tilde{f}:P_{\kappa}(\lambda)\to
P_{\kappa}(P_{\textnormal{bd}}(\lambda))$ given by
$\tilde{f}(\sigma)=f[\sigma]$. Let $\vec{U}=\vec{U}_{\mathcal{U},f}=\langle
U_{\alpha}:\alpha<\lambda\rangle$ enumerate the $\kappa$-complete
$N$-ultrafilters on ordinals less than $\lambda$ in the wellorder
$<_{\tilde{\mathcal{U}}}$. Let $N^{\prime}$ be the inner model $L[\vec{U},f]$
and let $f^{\prime}$ be the increasing enumeration of
$P_{\textnormal{bd}}(\lambda)\cap N^{\prime}$ in the following order: for
$a,b\in P_{\textnormal{bd}}(\lambda)\cap N^{\prime}$, set $a<b$ if either
$\sup(a)<\sup(b)$ or $a$ precedes $b$ in the canonical wellorder of
$L[\vec{U},f]$ and $\sup(a)=\sup(b)$.

Note that any model $M$ with the $\kappa$-approximation property that contains
$N$ must contain the set of $\kappa$-complete $N$-ultrafilters on ordinals
less than $\lambda$ and the order on it induced by $\tilde{\mathcal{U}}$.
Hence $N^{\prime}\subseteq M$.

Iterating this procedure yields, for each ordinal $\gamma$, an inner model
$N_{\gamma}=N_{\mathcal{U},f,\gamma}$ of ZFC, a function $f_{\gamma}$, and a
$\lambda$-sequence $\vec{U}_{\gamma}$ enumerating the $\kappa$-complete
$N_{\gamma}$-ultrafilters on ordinals less than $\lambda$. Specifically
(although still somewhat informally), let
$N_{\gamma}=L[\langle\vec{U}_{\xi},f_{\xi}:\xi<\gamma\rangle]$, let
$f_{\gamma}:\lambda\to P_{\textnormal{bd}}(\lambda)\cap N_{\gamma}$ be the
increasing enumeration of $P_{\textnormal{bd}}(\lambda)\cap N_{\gamma}$ in a
wellorder similar to the one described in the first paragraph, and finally
define $\vec{U}_{\gamma}=\vec{U}_{\mathcal{U},f_{\gamma}}$ as in the first
paragraph.

The uniformity of this procedure guarantees that any model $M$ such that
$N\subseteq M$ and $\mathcal{U}\cap M\in M$ contains $N_{\gamma}$ for all
ordinals $\gamma$.

The sequence $\langle N_{\gamma}:\gamma<\kappa^{+}\rangle$ is increasing, so
let $\gamma=\gamma_{\mathcal{U},f}$ be the least ordinal $\alpha<\kappa^{+}$
such that $N_{\alpha}\cap P_{\textnormal{bd}}(\kappa)=N_{\alpha+1}\cap
P_{\textnormal{bd}}(\kappa)$. We claim that $N_{\gamma+1}$ has the
$\kappa$-approximation property below $\lambda$.

Notice that $P_{\kappa}(\lambda)\cap N_{\gamma}=P_{\kappa}(\lambda)\cap
N_{\gamma+1}$: if $\sigma\in P_{\kappa}(\lambda)\cap N_{\gamma+1}$, then there
is some $\tau\in P_{\kappa}(\lambda)\cap N$ such that $\sigma\subseteq\tau$;
then $\sigma\in N_{\gamma}$ and since $P_{\textnormal{bd}}(\kappa)\cap
N_{\gamma}=P_{\textnormal{bd}}(\kappa)\cap N_{\gamma+1}$, $P(\tau)\cap
N_{\gamma}=P(\tau)\cap N_{\gamma+1}$; the latter set contains $\sigma$, and
hence $\sigma\in N_{\gamma}$ as desired. 4.6 and the definition of
$N_{\gamma+1}$, every bounded subset of $\lambda$ that is
$\kappa$-approximated by $N_{\gamma}$ belongs to $N_{\gamma+1}$. Since
$P_{\kappa}(\lambda)\cap N_{\gamma}=P_{\kappa}(\lambda)\cap N_{\gamma+1}$,
every subset of $\lambda$ that is $\kappa$-approximated by $N_{\gamma+1}$ is
$\kappa$-approximated by $N_{\gamma}$. Thus $N_{\gamma+1}$ has the
$\kappa$-approximation property below $\lambda$.

Now for each pair $(\mathcal{U},f)$ such that there is a strong limit cardinal
$\lambda$ of cofinality at least $\kappa$ such that $\mathcal{U}$ is a
$\kappa$-complete fine ultrafilters on $P_{\kappa}(\lambda)$ and $f:\lambda\to
P_{\textnormal{bd}}(\lambda)$ is a surjection in $N$, let
$M_{\mathcal{U},f}=N_{\mathcal{U},f,\gamma}\cap H(\lambda)$, where
$\gamma=\gamma_{\mathcal{U},f}$. Reiterating what we have proved above, any
inner model $M$ of ZFC such that $N\subseteq M$ and $\mathcal{U}\cap M\in M$
must contain $M_{\mathcal{U},f}$, and therefore any inner model $M$ with the
$\kappa$-approximation property contains
$\bigcup_{\mathcal{U},f}M_{\mathcal{U},f}$.

Fix $S\subseteq P_{\kappa}(\kappa^{+})$ such that for a proper class of
appropriate $\mathcal{U}$ and $f$, $M_{\mathcal{U},f}\cap
P_{\kappa}(\kappa^{+})=S$. Let $C$ be the class of pairs $(\mathcal{U},f)$
such that $M_{\mathcal{U},f}\cap P_{\kappa}(\kappa^{+})=S$. Then by the
Hamkins uniqueness theorem (4.7), for all $u,v\in C$, either $M_{u}\subseteq
M_{v}$ or $M_{v}\subseteq M_{u}$. Thus $M=\bigcup_{u\in C}M_{u}$ is an inner
model of ZFC, and since each $M_{u}$ has the $\kappa$-approximation property
below $\lambda$, $M$ has the $\kappa$-approximation property. Finally,
$N\subseteq M$ since for all but a set of $u\in C$, $N\cap H(\lambda)\subseteq
M_{u}$. ∎

The construction of the previous theorem is much simpler when $N$ is an inner
model that not only has the $\kappa$-cover property (i.e., is positive for the
fine filter) but also is positive for the supercompactness filters
$\mathcal{N}_{\kappa,\lambda}$, defined for all $\kappa\leq\lambda$ as the
intersection of the $\kappa$-complete normal fine ultrafilters on
$P_{\kappa}(\lambda)$.

######  Definition 4.8.

An inner model $N$ is $(\kappa,\lambda)$-supercompact if $N\cap
P_{\kappa}(\lambda)$ is $\mathcal{N}_{\kappa,\lambda}$-positive; $N$ is
$(\kappa,\infty)$-supercompact if it is $(\kappa,\lambda)$-supercompact for
all cardinals $\lambda$.

Note that $N$ is $(\kappa,\lambda)$-supercompact if and only if $N\cap
P_{\kappa}(\lambda)$ belongs to some $\kappa$-complete normal fine ultrafilter
on $P_{\kappa}(\lambda)$, and if $N$ is $(\kappa,\lambda)$-supercompact for
some ordinal $\lambda$, then $N$ is $(\kappa,\alpha)$-supercompact for all
$\alpha<\lambda$. The notion bears an obvious resemblance to Woodin’s weak
extender models.

######  Definition 4.9.

An inner model $N$ of ZFC is a weak extender model of $\kappa$ is
$\lambda$-supercompact if there is a $\kappa$-complete normal fine ultrafilter
$\mathcal{U}$ on $P_{\kappa}(\lambda)$ such that $\mathcal{U}\cap N\in N$ and
$P_{\kappa}(\lambda)\cap N\in\mathcal{U}$; $N$ is a weak extender model of
$\kappa$ is supercompact if it is a weak extender model of $\kappa$ is
$\lambda$-supercompact for all cardinals $\lambda$.

The substantive part of the following characterization of weak extender models
is due to Woodin and Usuba independently.

######  Lemma 4.10.

An inner model of ZFC is a weak extender model of $\kappa$ is supercompact if
and only if it is $(\kappa,\infty)$-supercompact and has the
$\kappa$-approximation property.∎

The following theorem shows that the $(\kappa,\infty)$-supercompact inner
models are precisely the ${<}\kappa$-closed inner models of weak extender
models.

######  Theorem 4.11.

If $N$ is a $(\kappa,\infty)$-supercompact inner model of ZFC, then there is a
weak extender model of $\kappa$ is supercompact that contains $N$ as a
${<}\kappa$-closed inner model.

###### Proof.

The reverse direction is obvious, so we focus on the forwards direction. Let
$\lambda$ be a cardinal and let $\mathcal{U}$ be a $\kappa$-complete normal
fine ultrafilter on $P_{\kappa}(\lambda)$ such that $P_{\kappa}(\lambda)\cap
N\in N$, and let $j:V\to M$ be the associated ultrapower embedding. Let
$W=j(N)$. We claim $W$ contains every subset $A$ of $\lambda$ that is
$\kappa$-approximated by $N$.

To see this, note that since $W$ has the $j(\kappa)$-cover property in $M$, we
can find $\sigma\in P_{j(\kappa)}(j(\lambda))$ such that
$j[A]\subseteq\sigma$. But $j(A)$ is $j(\kappa)$-approximated by $W$, so
$j(A)\cap\sigma\in W$. Since $P_{\kappa}(\lambda)\cap N\in\mathcal{U}$,
$j[\lambda]\in W$, and this implies $A=j^{-1}[j(A)\cap\sigma]\in W$, as
desired.

Next, a familiar argument shows that since $N\cap P(\lambda)\subseteq W$,
$N\cap V_{\kappa}=W\cap V_{\kappa}$, and $N$ has the $\kappa$-cover property,
in fact, $P_{\kappa}(\lambda)\cap W\subseteq N$. This implies that $W$ has the
$\kappa$-approximation property below $\lambda$.

It follows that for any $\kappa$-complete normal fine ultrafilters
$\mathcal{U}$ and $\mathcal{U}^{\prime}$ with $P_{\kappa}(\lambda)\cap
N\in\mathcal{U}\cap\mathcal{U}^{\prime}$, $j_{\mathcal{U}}(N)\cap
P(\lambda)=j_{\mathcal{U}^{\prime}}(N)\cap P(\lambda)$ by the preceding
remarks and the Hamkins uniqueness theorem. Letting $X$ be the union of
$j_{\mathcal{U}}(N)\cap P(\lambda)$ for all $\mathcal{U}$ such that
$P_{\kappa}(\lambda)\cap N\in\mathcal{U}$, it follows that $M=L(X)$ is a model
of ZFC with the $\kappa$-approximation and cover properties and $N$ is a
${<}\kappa$-closed innder model of $M$. Since $N$ is
$(\kappa,\infty)$-supercompact and $M$ has the $\kappa$-approximation
property, 4.10 implies that $N$ is a weak extender model of $\kappa$ is
supercompact. ∎

A cardinal $\kappa$ is distributively supercompact if for all cardinals
$\lambda$, there is a $\kappa$-distributive partial order $\mathbb{P}$ such
that $\kappa$ is $\lambda$-supercompact in $V^{\mathbb{P}}$.

######  Corollary 4.12 (HOD Hypothesis).

Suppose $\kappa$ is supercompact. Then there is a unique weak extender model
of $\kappa$ is supercompact that contains HOD as a ${<}\kappa$-closed inner
model. In particular, HOD satisfies that $\kappa$ is distributively
supercompact.∎

The following corollary shows that the first-order theory of HOD has an
influence on the question of whether the least supercompact cardinal is
supercompact in HOD. One could actually replace UA in the argument with any
$\Pi_{2}$ sentence that implies that every countably complete ultrafilter on
an ordinal is ordinal definable.

######  Theorem 4.13.

Suppose $\kappa$ is a supercompact cardinal and the HOD hypothesis holds. If
$V_{\kappa}\cap\textnormal{HOD}$ satisfies the Ultrapower Axiom, then $\kappa$
is supercompact in HOD.

###### Proof.

Let $N$ be the inner model of 4.12. Then $N$ is definable without parameters
and $V_{\kappa}\cap\textnormal{HOD}=V_{\kappa}\cap N\preceq_{\Sigma_{2}}N$
since $\kappa$ is supercompact in $N$. Since UA is $\Pi_{2}$, $N$ satisfies
UA. Now working in $N$, we apply the following consequence of UA: if $\kappa$
is supercompact and $A$ is a set such that
$V_{\kappa}\subseteq\textnormal{HOD}_{A}$, then $V=\textnormal{HOD}_{A}$.
Therefore $N=(\textnormal{HOD}_{A})^{N}$ for any set of ordinals $A\in N$ such
that $V_{\kappa}\cap N\subseteq(\textnormal{HOD}_{A})^{N}$. Let
$A\in\textnormal{HOD}$ be a set of ordinals such that
$V_{\kappa}\cap\textnormal{HOD}\subseteq L[A]$. Then $A\in N$ and
$N=(\textnormal{HOD}_{A})^{N}$, so there is a wellorder of $N$ definable over
$N$ from $A$. Since $N$ is definable without parameters and $A$ is ordinal
definable, this wellorder of $N$ is definable from an ordinal parameter. Any
ordinal definably wellordered transitive class is contained in HOD, so
$N\subseteq\textnormal{HOD}$. Therefore $\textnormal{HOD}=N$, and so $\kappa$
is supercompact in HOD. ∎

## References

  * [1] Yong Cheng, Sy-David Friedman, and Joel David Hamkins.  Large cardinals need not be large in HOD.  Annals of Pure and Applied Logic, 166(11):1186 – 1198, 2015.
  * [2] Keith Devlin and Ronald Jensen.  Marginalia to a theorem of silver.  In ISILC Logic Conference, pages 115–142. Springer, 1975.
  * [3] Gabriel Goldberg.  The Ultrapower Axiom.  PhD thesis, Harvard University, 2019.
  * [4] Gabriel Goldberg.  The uniqueness of elementary embeddings.  To appear.
  * [5] Robert M. Solovay.  Strongly compact cardinals and the GCH.  In Proceedings of the Tarski Symposium (Proc. Sympos. Pure Math., Vol. XXV, Univ. California, Berkeley, Calif., 1971), pages 365–372. Amer. Math. Soc., Providence, R.I., 1974.
  * [6] W. Hugh Woodin.  In search of Ultimate-$L$: the 19th Midrasha Mathematicae Lectures.  Bull. Symb. Log., 23(1):1–109, 2017.