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Unlike L, it is consistent with
$ZF$
that
$HOD$
contains large cardinals. Unfortunately, the theory of
$HOD$
is neither absolute nor so easy to ascertain as that of L. But in many models of
$ZF + AD$
, it is possible to construct
$HOD$
as a fine structural model, proving
$HOD \models GCH$
and enabling a detailed study of
$HOD$
. HOD as a Core Model provides such a construction in a few important special cases by extending the technique of
$HOD^{L(\mathbb {R})}$
is a Core Model below
$\Theta $
,
The Bulletin of Symbolic Logic
, vol. 1 (1995), pp. 75–84. This earlier paper of Steel’s characterizes 
as a direct limit of all countable iterates of
$M_\omega $
cut to its least cardinal which is strong up to a Woodin cardinal. The direct limit is ordinal definable with height 
. And conversely, Steel shows 
is contained in the direct limit.
Another precursor to HOD as a Core Model is Woodin’s proof that
$Con(ZF + AD) \iff Con(ZFC + \text {"exist } \omega \text { Woodin cardinals"})$
. The forward direction can be done using a Prikry forcing with the Martin measure. A proof of this theorem, and related results, appear in Large Cardinals from Determinacy,
The Handbook of Set Theory
(Foreman, Kanamori, eds.), pp. 1951–2120.
The main result of HOD as a Core Model is an extension of the above techniques to characterize the full
$HOD^{L(\mathbb {R})}$
as
$L[M_{\infty }, \Lambda ]$
, where
$M_{\infty }$
is the direct limit of all countable iterates of
$M_\omega $
by iteration trees below its least Woodin cardinal and
$\Lambda $
is a portion of the iteration strategy for
$M_{\infty }$
. An ordinal definable definition of
$M_{\infty }$
below its least Woodin cardinal is derived using that the iteration strategy for
$M_\omega $
is definable from indiscernibles. The full direct limit is isolated in
$L(\mathbb {R})$
as (essentially) a Prikry generic premouse over
$M_{\infty }$
cut to its least Woodin cardinal.
Implicit in the argument outlined above is that
$M_{\infty }$
is an iterate of
$M_\omega $
by a stack of normal trees, each of which can be approximated in
$L(\mathbb {R})$
. Recent work of Steel strengthened this to the following:
Theorem (Steel).
There is a normal tree
$\mathcal {T}$
of length
$\Theta +1$
on
$M_\omega $
below its least Woodin cardinal such that the last model of
$\mathcal {T}$
is
$M_{\infty }$
and
$\mathcal {T}\upharpoonright \Theta \in L(\mathbb {R})$
.
Normalizing iteration trees also plays a role in the author’s efforts to analyze
$HOD$
in larger models of determinacy.
Another major contribution of HOD as a Core Model is an adaptation of this direct limit argument to identify
$HOD^{L[x,G]}$
for a real x such that
$L[x] \vDash $
“OD-determinacy” and G which is
$L[x]$
-generic for
$Col(\omega ,<\kappa )$
, where
$\kappa $
is the least inaccessible cardinal of
$L[x]$
. The authors show
$HOD^{L[x,G]} = L[M_{\infty },\Lambda ]$
, where
$M_{\infty }$
is the direct limit of all iterates of
$M_1$
which are countable in
$L[x,G]$
and
$\Lambda $
is a portion of the iteration strategy for
$M_{\infty }$
. The generic G ensures that the last model of a coiteration of a pair of countable,
$M_1$
-like mice in
$L[x]$
is countable in
$L[x,G]$
. The inability to prove the last model of this coiteration is countable in
$L[x]$
inhibits the authors from resolving the more natural goal of characterizing
$HOD^{L[x]}$
.
The analysis of
$HOD^{L(\mathbb {R})}$
required the property of mouse capturing (MC): If x and y are countable, transitive sets,
$x\subseteq y$
, and x is ordinal definable from parameters in
$y \cup \{y\}$
, then there is a y-premouse M with an
$\omega _1$
-iteration strategy such that
$x\in M$
. The last chapter of the paper considers the problem of analyzing
$HOD$
from the axiom
$AD^+$
. It is open whether
$AD^+$
alone implies
$MC$
, but it does under certain smallness assumptions. For example, Sargsyan proves
$MC$
in the minimal model of
$AD_{\mathbb {R}} + “\Theta \text { is regular"}$
in Hod Mice and the Mouse Set Conjecture, Memoirs of the American Mathematical Society, vol. 236 (2015). In this context,
$HOD$
can be constructed as a fine structural model in which extenders are added while simultaneously coding iteration strategies for initial segments of the model built so far.
Models of this form are now referred to as hod mice. The study of
$HOD$
below the minimal model of
$AD_{\mathbb {R}} + “\Theta \text { is regular"}$
proceeds inductively by associating each pointclass
$\Gamma $
of Wadge rank equal to some level
$\Theta _\alpha $
in the Solovay sequence to a hod mouse with an iteration strategy in this pointclass, and identifying
$HOD|\Theta _{\alpha }$
as a direct limit of this hod mouse. A different technique is needed if
$\Theta _\alpha $
is the largest Suslin cardinal below
$\Theta _{\alpha +1}$
. In this case, a set of Wadge rank
$\Theta _\alpha $
cannot code a new iteration strategy for a hod mouse, but Sargsyan and Trang showed a strategy restricted to short trees can have Wadge rank
$\Theta _\alpha $
and characterized
$HOD$
in the minimal model of
$AD^+ + $
“The largest Suslin cardinal exists and is in the Solovay sequence.”
Performing an analysis of
$HOD$
from
$AD^+$
without any smallness assumption is a central question of descriptive inner model theory. Towards this goal, Steel proves a comparison lemma for hod mice in
A Comparison Process for Mouse Pairs
, Lecture Notes in Logic, vol. 51 (2022). But for now, proving the existence of hod mice large enough to analyze
$HOD$
in a more general setting is unfinished.
