2.1 Generic ultrapowers

Fix a set X. A set $\mathcal {I}\subseteq \mathcal {P}(X)$ is called an ideal if $\emptyset \in \mathcal {I}$ , $X\notin \mathcal {I}$ and $\mathcal {I}$ is closed under subsets and finite unions. Dually, a set $\mathcal {F}\subseteq \mathcal {P}(X)$ is a filter if $\emptyset \notin \mathcal {F}$ , $X\in \mathcal {F}$ and $\mathcal {F}$ is closed under supersets and finite intersections. A filter $\mathcal {U}$ is called an ultrafilter if it satisfies the following additional property: given $A\in \mathcal {P}(X)$ , either $A\in \mathcal {U}$ or $X\setminus A\in \mathcal {U}$ ; equivalently, $\mathcal {U}$ is a $\subseteq $ -maximal filter.

Given an ideal $\mathcal {I}\subseteq \mathcal {P}(X)$ , its dual filter $\mathcal {I}^*$ is defined as $\{X\setminus A\mathrel {|} A\in \mathcal {I}\}$ . A set $A\in \mathcal {P}(X)$ has $\mathcal {I}$ -positive measure if $A\notin I$ , and $\mathcal {I}^+$ denotes the collection of all sets with $\mathcal {I}$ -positive measure. Note that $\mathcal {I}^*$ is a filter, $\mathcal {I}^*\subseteq \mathcal {I}^+$ and $A\in \mathcal {I}^+$ if and only if $A\cap B\neq \emptyset $ for all $B\in \mathcal {I}^*.$ These concepts have natural parallels in the setting of filters $\mathcal {F}\subseteq \mathcal {P}(X)$ as well [Reference JechJec03, §7].

Given an ideal $\mathcal {I}\subseteq \mathcal {P}(X)$ , define an equivalence relation $\sim _{\mathcal {I}}$ on $P(X)$ as follows:

$$ \begin{align*}A \sim_{\mathcal{I}} B\text{ if and only if }A\bigtriangleup B\in\mathcal{I}.\end{align*} $$

This yields a quotient $\mathcal {P}(X)/\mathcal {I}$ , which, endowed with the order

$$ \begin{align*}[X]\leq [Y] \iff X\setminus Y\in\mathcal{I},\end{align*} $$

gives rise to a Boolean algebra. After removing the zero element from $\mathcal {P}(\kappa )/\mathcal {I}$ , the partial ordering $\leq $ becomes a separative order.

There is another presentation of the poset $(\mathcal {P}(X)/\mathcal {I}\setminus \{[\emptyset ]\},\leq )$ as $\mathbb {P}:=(\mathcal {I}^+,\subset )$ . The two posets are forcing equivalent in the sense that they give rise to the same generic extensions. Indeed, the former poset is the separative quotient of the latter. A V-generic filter $G\subseteq \mathbb {P}$ yields a V-ultrafilter on X extending the dual filter $\mathcal {I}^*$ . If $\mathcal {I}$ is $\kappa $ -complete, then G is also V- $\kappa $ -complete.Footnote ² For details concerning these facts, see [Reference JechJec03, §22].

Suppose that $\mathcal {I}\subseteq \mathcal {P}(\kappa )$ is an ideal containing all singletons and that it is $\kappa $ -complete (i.e., $\bigcup _{\alpha <\lambda }A_\alpha \in \mathcal {I}$ provided $\langle A_\alpha \mathrel {|} \alpha <\lambda \rangle \subseteq \mathcal {I}$ with $\lambda <\kappa $ ).

Let $G\subseteq \mathbb {P}$ be a V-generic filter. Working in $V[G]$ , we can define the generic ultrapower of V by G. Namely, in $V[G]$ , one defines the structure

$$ \begin{align*}\operatorname{\mathrm{Ult}}(V,G):=\langle ({}^{X}V)\cap V)/{=}_G, \in_G \rangle,\end{align*} $$

where for each two functions $f, g\colon X\rightarrow V$ (in V)

$$ \begin{align*}f =_G g\text{ if and only if }\{x\in X\mathrel{|} f(x) = g(x)\}\in G\end{align*} $$

and

$$ \begin{align*}[f]_G\in_G [g]_G\text{ if and only if }\{x\in X\mathrel{|} f(x) \in g(x)\}\in G.\end{align*} $$

Here and in the future, we will denote by $[f]_G$ the $=_G$ equivalence class of f, omitting the subscript when there is no chance of confusion.

It turns out that $\operatorname {\mathrm {Ult}}(V,G)$ is a model of ZFC, yet not necessarily well-founded, even if G is V- $\kappa $ -complete for an uncountable cardinal $\kappa $ . As usual, an appropriate version of Łoś’s theorem holds. Namely,

$$ \begin{align*}\operatorname{\mathrm{Ult}}(V,G)\models \varphi([f_1], \dots, [f_{n}])\iff \{x\in X\mathrel{|} V\models \varphi(f_1(x),\dots, f_{n}(x))\}\in G,\end{align*} $$

where $\varphi (v_1,\dots , v_{n})$ is a first-order formula in the language $\{=, \in \}$ .

This ensures that the map $j_G\colon \langle V,\in \rangle \rightarrow \operatorname {\mathrm {Ult}}(V,G)$ given by $a\mapsto [c_a]_G$ is an elementary embedding, where $c_a\colon X\rightarrow V$ is the constant function with value a.

The combinatorial properties of the ideal $\mathcal {I}$ (in V) are related to the properties of the embedding $j_G : V\to \operatorname {\mathrm {Ult}}(V,G)$ in the generic extension $V[G]$ . For example, $\mathcal {I}$ is $\kappa $ -complete if and only if the maximal condition $[X]$ forces that the critical point of $j_G$ is at least $\kappa $ .

The generic ultrapower construction will play a prominent role in the forthcoming §3. We refer the reader to [Reference JechJec03, §22] or Foreman’s excellent handbook chapter [Reference ForemanFor09] for any notion not considered in this account.

Definition 2.2. Given a filter $\mathcal {F}\subseteq \mathcal {P}(X)$ and functions $f,g\colon X\rightarrow V$ , denote

$$ \begin{align*}f<_{\mathcal{F}} g\iff \{x\in X\mathrel{|} f(x)<g(x)\}\in\mathcal{F}.\end{align*} $$

Similarly, if $\mathcal {I}\subseteq \mathcal {P}(X)$ is an ideal,

$$ \begin{align*}f<_{\mathcal{I}}g\iff \{x\in X\mathrel{|} f(x)\geq g(x)\}\in\mathcal{I}.\end{align*} $$

2.2 Ordinal definability and forcing

A set X is called ordinal definable if it is definable by a formula of the language of set theory using ordinals as parameters. More formally, there is $\varphi (x,\vec {y})$ and $\langle \alpha _*,\alpha _0,\dots \alpha _n\rangle \in \operatorname {\mathrm {Ord}}^{<\omega }$ such that

$$ \begin{align*}x\in X\,\Leftrightarrow\, V_{\alpha^*}\models \varphi(x, \alpha_0,\dots, \alpha_n). \end{align*} $$

The class of ordinal definable sets is denoted by $\operatorname {\mathrm {OD}}$ . Since $\operatorname {\mathrm {OD}}$ need not be transitive, one looks at a special subclass of $\operatorname {\mathrm {OD}}$ – the Hereditarily ordinal definable sets, $\operatorname {\mathrm {HOD}}$ . A set X is Hereditarily Ordinal Definable (or, simply, in $\operatorname {\mathrm {HOD}}$ ) if $X\in \operatorname {\mathrm {OD}}$ and the transitive closure of $\{X\}$ is contained in $\operatorname {\mathrm {OD}}$ . It turns that $\operatorname {\mathrm {HOD}}$ is an inner model; namely, it is a transitive class containing the ordinals and satisfying all the ZFC axioms.

At many places in this paper, we shall be preoccupied with the following issue. Suppose that $\mathbb {P}\in \operatorname {\mathrm {OD}}$ is a forcing poset and $G\subseteq \mathbb {P}$ is V-generic – how does $\operatorname {\mathrm {HOD}}^{V[G]}$ compare to $\operatorname {\mathrm {HOD}}^V$ ? Here, $\operatorname {\mathrm {HOD}}^V$ (resp. $\operatorname {\mathrm {HOD}}^{V[G]}$ ) stands for the class $\operatorname {\mathrm {HOD}}$ as computed in V (resp. in $V[G]$ ). In special circumstances, we have $\operatorname {\mathrm {HOD}}^{V[G]}\subseteq \operatorname {\mathrm {HOD}}$ – for example, if $\mathbb {P}$ is cone/weakly homogeneous:

We used $\mathbb {P}/p$ to denote the subposet of $\mathbb {P}$ with universe $\{q\in \mathbb {P}\mathrel {|} q\leq p\}$ . It is clear that every weakly homogeneous forcing is cone-homogeneous.

Sometimes we will need to assume (see, for example, §3.1) that $\operatorname {\mathrm {HOD}}$ encompasses large cardinals which exist in V. This can be done by forcing $"V=\operatorname {\mathrm {HOD}}$ ” with McAloon iteration coding V into the continuum function. The said iteration preserves large cardinals (see [Reference Fuchs, Hamkins and ReitzFHR15, Reference Bagaria and PovedaBP23]) and produces a model V such that $V\subseteq \operatorname {\mathrm {HOD}}^{V^{\mathbb {Q}}}$ for any set-sized forcing $\mathbb {Q}$ .

3.1 Optimality

In this section, we discuss the optimality of Theorems 3.4 and 3.5. Some of our arguments require rather technical Prikry-type forcings. Instead of elaborating on their precise definitions (which are fairly long), we give appropriate references. Let us begin with Theorem 3.4. The next shows that the cofinality assumption is necessary in Theorem 3.4:

The hypothesis $`\{\delta <\kappa \mathrel {|} \delta ^{+\operatorname {\mathrm {HOD}}}=\delta ^+\}$ is stationary’ is also necessary:

Proof. By preliminarily forcing with McAloon’s iteration, we may assume that $V\subseteq \operatorname {\mathrm {HOD}}^{V^{\mathbb {Q}}}$ for any set-sized forcing $\mathbb {Q}$ . If this iteration is started at a sufficiently large regular cardinal, our hypothesis on $\kappa $ are maintained.

Suppose that $j\colon V\rightarrow M$ is a $\kappa ^{+2}$ -supercompact embedding. Let $\mu <\kappa $ be a regular cardinal and let u be the $(\kappa ,\kappa ^+)$ -measure sequence of length $\mu $ derived from j. Namely, $u=\langle u_\alpha \mathrel {|} \alpha <\mu \rangle $ , where and for $\alpha>0$ . Notice that M contains every $(\kappa ,\kappa ^+)$ -measure sequence of length less than $\mu $ so u above indeed exists. In addition, by the argument in [Reference Cummings, Friedman and GolshaniCFG15, Lemma 3.2], u belongs to $\mathcal {U}^{\sup }_\infty $ .Footnote ⁸

Let $\mathbb {R}^{\sup }_u$ be the supercompact Radin forcing defined from u [Reference Cummings, Friedman and GolshaniCFG15]. Let $G\subseteq \mathbb {R}^{\sup }_u$ a V-generic filter. Combining [Reference Cummings, Friedman and GolshaniCFG15, Corollary 4.2] with our forcing preparation,

$$ \begin{align*}V\subseteq \operatorname{\mathrm{HOD}}^{V[G]}\subseteq V[G^{\phi}],\end{align*} $$

where $G^{\phi }$ is a V-generic for a plain Radin forcing $\mathbb {R}_u$ – hence, for a cardinal-preserving poset. In particular, the following inequalities hold:

$$ \begin{align*}(\kappa^+)^V\leq (\kappa^+)^{\operatorname{\mathrm{HOD}}^{V[G]}}\leq (\kappa^+)^{V[G^\phi]}=(\kappa^+)^V<(\kappa^+)^{V[G]}.\end{align*} $$

The above yields item (2) of the theorem.

Let $\langle w_\alpha \mathrel {|} \alpha <\mu \rangle $ be an injective enumeration of $\{w\mathrel {|} w \text {appears in} p\in G\}$ . Denote and . The increasing enumeration of $\{\kappa _{w_\alpha }\mathrel {|} \alpha <\mu \}$ yields a club $C\subseteq \kappa $ of order-type $\mu $ and by forcing below an appropriate condition $\mu $ remains regular in $V[G]$ . In addition, standard arguments show that $\kappa $ remains a strong limit in $V[G]$ (see [Reference Cummings, Friedman and GolshaniCFG15, Lemma 3.10(6)]). These two observations combined yield item (1) of the theorem. Finally, [Reference Cummings, Friedman and GolshaniCFG15, Lemma 3.10(8)] shows that the only V-cardinals ${\leq }\kappa $ that survive after passing to $V[G]$ are those outside

$$ \begin{align*}\textstyle \bigcup_{\alpha\in \mathrm{Lim}\cap \mu}(\kappa_{w_\alpha},\lambda_{w_\alpha}].\end{align*} $$

Thus, for every $V[G]$ -cardinal $\delta \notin \operatorname {\mathrm {acc}}(C)$ , we have that $(\delta ^+)^V$ does not belong to the above union and thus $(\delta ^+)^V=(\delta ^+)^{V[G]}$ . By our previous observations, this yields $(\delta ^+)^{\operatorname {\mathrm {HOD}}^{V[G]}}=(\delta ^+)^V=(\delta ^+)^{V[G]},$ as claimed.

The following theorem shows that the assumptions of Theorem 3.5 are provably optimal. Starting with appropriate large cardinal assumptions, it is consistent for $\{\delta <\kappa \mathrel {|} \operatorname {\mathrm {cf}}^{\operatorname {\mathrm {HOD}}}(\delta )<\delta \}$ to be non-stationary and $\kappa $ to be inaccessible in $\text {HOD}$ . This is a special case of a recent theorem of the authors [Reference Goldberg, Osinski and PovedaGOP24, Theorem 4.6] which utilizes the Supercompact Radin forcing of Theorem 3.7. We state the theorem without proof:

Theorem 3.9. Suppose that $\delta $ is a supercompact cardinal and the $\mathrm {GCH}$ holds. Then, there is a generic extension where $\delta $ remains supercompact and there is a club $D\subseteq \delta $ consisting of cardinals $\kappa $ for which

$$ \begin{align*}\operatorname{\mathrm{HOD}}\models"\kappa\text{ is regular}".\end{align*} $$

In particular, every $\lambda \in E^\kappa _{\omega _1}\cap \operatorname {\mathrm {acc}}(D)$ is a singular of uncountable cofinality for which $\{\theta <\lambda \mathrel {|} \operatorname {\mathrm {cf}}^{\operatorname {\mathrm {HOD}}}(\theta )=\theta \}$ contains a club and $\operatorname {\mathrm {cf}}^{\operatorname {\mathrm {HOD}}}(\lambda )=\lambda $ .

To obtain the above configuration, it would be natural to utilize Ben-Neria-Unger’s method from [Reference Ben-Neria and UngerBNU17, Theorem 1.3]. However, an apparent drawback of this alternative approach is that it does not seem amenable to preserving large cardinals at the level of strong compactness. This limitation is primarily caused by the club-shooting poset used after the nonstationary support iteration of Prikry-type forcings. It turns out that the preservation of supercompacts becomes interesting in light of the HOD dichotomy theorem proved in [Reference GoldbergGol23].

3.2 On $\omega $ -club amenability

The first author showed that many of the known results on HOD – for example, the $\operatorname {\mathrm {HOD}}$ dichotomy theorem – can actually proved for an arbitrary inner model that is $\omega $ -club amenable [Reference GoldbergGol23].

A set $C\subseteq \delta $ is an $\omega $ -club in $\delta $ (for $\operatorname {\mathrm {cf}}(\delta )\geq \omega _1$ ) if it is unbounded (in $\delta $ ) and whenever S is a countable subset of C, $\sup (S)\in C$ . The $\omega $ -club filter on $\delta $ , denoted by $\mathscr {C}_\delta $ , is the collection of all subsets of $\delta $ that contain an $\omega $ -club.

Not much is known about the size of HOD that does not already hold of any $\omega $ -club amenable model, so it is natural to seek properties that are more specific to $\operatorname {\mathrm {HOD}}$ . In this section, we show that Theorem 3.4 does not generalize to an arbitrary $\omega $ -club amenable model.

Let $\vec {\mathscr C}$ denote the proper class $\{(\delta ,S) \mathrel {|} \operatorname {\mathrm {cf}}(\delta ) \geq \omega _1,S\in \mathscr C_\delta \}$ . If one builds the constructible universe relative to the sequence $\vec {\mathscr C}$ , then one obtains an $\omega $ -club amenable model. More generally,

To construct $\omega $ -club amenable models that do not satisfy the conclusion of Theorem 3.4, we need a mild large cardinal hypothesis. Namely, we will assume that for all sets X, X-sword exists. Let us now define this hypothesis precisely.

If X is a set of rank $\lambda $ , we say that X-sword exists if there is a coarse X-sword mouse, which is an iterable transitive structure $(M, \vec {U},U,W)$ with the following properties:

• • M is an transitive model of $\mathrm {ZFC}^-$ with largest cardinal $\kappa $ .

• • $X\cap M\in M$ and $\vec {U}\in M$ .

• • $M\vDash \vec {U}$ is a coherent sequence of normal ultrafilters of length $\kappa $ .

• • $o^{\vec {U}}(\delta ) = 0$ whenever $\delta \leq \lambda $

• • $o^{\vec {U}}(\delta )\leq 1$ whenever $\lambda < \delta < \kappa $ .

• • W is a weakly amenable M-normal M-ultrafilter on $\kappa $

• • $j_W^M(\vec {U}) \mathbin \upharpoonright \kappa +1 = \vec {U}^{\frown } U$ .

The notation for coherent sequences comes from Mitchell’s handbook article [Reference MitchellMit09]. Things are a bit simpler here since all measures on $\vec {U}$ have order 0. One may therefore think of $\vec {U}$ as a partial function with $o^{\vec {U}}(\alpha ) = 0$ if $\vec {U}$ is not defined on $\alpha $ , and $o^{\vec {U}}(\alpha ) = 1$ if it is.

We emphasize that the notion of a coarse X-sword mouse is defined for an arbitrary set X, not necessarily a set of ordinals, and this will be relevant below.

The hypothesis that X-sword exists for every set X follows from the existence of a proper class of measurable cardinals of Mitchell order 2. To see this, assume W is a measure on $\kappa $ of order $1$ and $\vec {U}$ is a sequence of measures of order 0 defined on all measurable cardinals between $\gamma $ and $\kappa $ . Let $U = [\alpha \mapsto U_\alpha ]_W$ . Then $(H_{\kappa ^+},\vec {U}, U,W)$ is a coarse X-sword mouse for every $X\in V_\gamma $ .

In terms of consistency strength, the hypothesis that X-sword exists for every set X is a bit weaker than the existence of a single measurable cardinal of Mitchell order 2: it is not hard to show that it holds in $V_\kappa $ if $\kappa $ is a measurable cardinal of Mitchell order 2.

Suppose $\mathcal M = (M,\vec {U},U,W)$ is a coarse X-sword mouse and $\mathcal H$ is transitive with a $\Sigma _1$ -elementary $\pi :\mathcal H\to \mathcal M$ such that $X\cap M\in \operatorname {\mathrm {ran}}(\pi )$ . Then $\mathcal H$ is a coarse $\pi ^{-1}(X\cap M)$ -sword mouse. Similarly, if $\mathcal M$ is a coarse X-sword mouse and there is a cofinal $\Sigma _0$ -elementary $\pi : \mathcal M\to \mathcal N$ , then $\mathcal N$ is a coarse $\pi (X\cap M)$ -sword mouse if it is iterable. These facts are straightforward, except that the statement that a structure M satisfies $\text {ZFC}^-$ may seem too complicated to be preserved under such weak forms of elementarity. This is not really an issue, however, since the fact that M satisfies $\text {ZFC}^-$ follows from the fact that M satisfies $\Sigma _0$ -Separation and the Well-Ordering Theorem combined with the weak amenability of $W,$ which yields that $M = H(\kappa ^+)^{\text {Ult}_0(M,W)}$ , which is a model of $\text {ZFC}^-$ .

The following lemmas, suggested by one of the anonymous referees, vastly simplify the proof of Lemma 3.14 below.

Lemma 3.13. Suppose X is a set and $\mathcal M = (M,\vec {U},U,W)$ is a coarse X-sword mouse with largest cardinal $\kappa $ . Assume the following hold:

• • $M = \mathrm {Hull}_{\Sigma _1}^{\mathcal M}(\alpha \cup \{p\})$ for some $p\in M$ and $\alpha \geq \operatorname {\mathrm {rank}}(X)$ .

• • $M = \bigcup _{\xi < o(M)} M_\xi $ , where $\langle M_\xi \rangle _{\xi < o(M)}$ is an increasing sequence of transitive sets in M that contain $V_\kappa ^M$ .

Then every ordinal $\delta> \alpha $ that is regular in M satisfies $\operatorname {\mathrm {cf}}(\delta ) = \operatorname {\mathrm {cf}}(o(M))$ .

Proof. For $\xi < o(M)$ , let $\mathcal M_\xi = (M_\xi ,\vec {U},U\cap M_\xi ,W\cap M_\xi )$ and let $H_\xi = \text {Hull}^{\mathcal M_\xi }_{\Sigma _1}(\alpha \cup \{p\})$ . Note that for all $\xi < o(M)$ , $H_\xi \in M$ and $M\vDash |H_\xi | \leq \alpha $ ; also for $\xi \leq \xi '< o(M)$ , $H_\xi \subseteq H_{\xi '}$ . Moreover, for any $\Sigma _1$ -formula $\varphi (x)$ and any $a\in M$ , $\mathcal M\vDash \varphi (a)$ if and only if $\mathcal M_\xi \vDash \varphi (a)$ for all sufficiently large $\xi $ if and only if $\mathcal M_\xi \vDash \varphi (a)$ for some $\xi $ . It follows that

$$\begin{align*}M = \text{Hull}^{\mathcal M}_{\Sigma_1}(\alpha\cup \{p\}) = \bigcup_{\xi < o(M)}H_\xi.\end{align*}$$

Now suppose $\delta> \alpha $ is regular in M. For $\xi < o(M)$ , let $\beta _\xi = \sup (\delta \cap H_\xi )$ . Then $\beta _\xi < \delta $ since $\delta $ is regular in M and $M\vDash |H_\xi | \leq \alpha < \delta $ . Since $\langle \beta _\xi \rangle _{\xi < o(M)}$ is weakly increasing and cofinal in $\delta $ , it follows that $\operatorname {\mathrm {cf}}(\delta ) = \operatorname {\mathrm {cf}}(o(M))$ .

Lemma 3.14. Assume that for all sets X, X-sword exists. Then for any cardinal $\lambda $ and any set $A\subseteq \lambda $ , $L[A,\vec {\mathscr {C}}]$ does not correctly compute $\lambda ^+$ .

Proof. Fix a set $A\subseteq \lambda $ . Using a pairing function on $\lambda $ , it is not hard to construct a family X of subsets of $\lambda $ such that for any class E, $L[X, E] = L[A,\mathscr {C}\mathbin \upharpoonright (\lambda +1),E]$ . In particular,

$$\begin{align*}L[X,\vec{\mathscr{C}}\mathbin\upharpoonright (\lambda,\infty)] = L[A,\vec{\mathscr{C}}],\end{align*}$$

where $(\lambda ,\infty ) = \{\xi \in \operatorname {\mathrm {Ord}} : \xi> \lambda \}$ .

Let $\mathcal M = (M,\vec {U},U,W)$ be a coarse X-sword mouse such that $o(M)$ is as small as possible. Note that letting $\bar M = L_{o(M)}[X,\vec {U},U, W]$ , the structure $\bar {\mathcal M} = (\bar M, \vec {U}\cap \bar M, U\cap \bar M, W\cap \bar M)$ is also a coarse X-sword mouse, and so we may assume $\mathcal M = \bar {\mathcal M}$ . This guarantees that $M = \bigcup _{\xi < o(M)} M_\xi $ , where $\langle M_\xi \rangle _{\xi < o(M)}$ is an increasing sequence of transitive sets in M that contain $V_\kappa ^M$ : take $M_\xi = L_{\gamma +\xi }[X,\vec {U},U,W]$ where $\gamma < o(M)$ is least such that $V_\kappa ^M \subseteq L_\gamma [X,\vec {U},U,W]$ . Similarly, we may assume that

$$\begin{align*}\mathcal M = \mathrm{Hull}_{\Sigma_1}^{\mathcal M}(\lambda\cup \{X\cap M\})\end{align*}$$

since the transitive collapse of $\mathrm {Hull}_{\Sigma _1}^{\mathcal M}(\lambda \cup \{X\cap M\})$ is again a coarse X-sword mouse. Thus, we may ensure that $\mathcal M$ satisfies the hypotheses of Lemmas 3.12 and 3.13. In particular, every M-regular cardinal $\delta> \lambda $ has countable cofinality in $V.$

We claim that $L[A,\vec {\mathscr {C}}]$ is contained in a proper initial segment N of a proper class iterate of $\mathcal M$ . Granting this, we have $\lambda ^{+L[A,\vec {\mathscr {C}}]} \leq \lambda ^{+N} = \lambda ^{+M} < \lambda ^+$ . (The final inequality comes from the fact that $\lambda ^{+M}$ has countable cofinality in V.) Thus, our claim suffices to finish the proof.

The idea is to iterate $\mathcal M$ to a model $\mathcal N$ with the following property. For each ordinal $\delta> \lambda $ , $o^{\mathcal N}(\delta )> 0$ if and only if $\delta $ is regular in $\mathcal N$ and has uncountable cofinality in V; moreover, in this case, the unique measure on $\delta $ on the sequence of $\mathcal N$ is equal to $\mathscr C_\delta \cap \mathcal N$ .

The iteration is defined by selecting at each stage the first total measure on the sequence of the current iterate that lies on an ordinal of countable cofinality. More formally, we define an iterated ultrapower

$$\begin{align*}\langle (\mathcal M_\alpha, U_\alpha) \mathrel{|} \alpha\in \operatorname{\mathrm{Ord}}\rangle\end{align*}$$

of $\mathcal M$ by setting $U_\alpha $ equal to the first measure on the sequence $\vec {U}^{\mathcal M_\alpha \frown } U^{\mathcal M_\alpha }$ that lies on an ordinal $\kappa _\alpha $ of countable cofinality in V; if there is no such measure, set $U_\alpha $ equal to the top measure $W^{\mathcal M_\alpha }$ . For $\alpha \leq \beta \in \text {Ord}$ , let

$$\begin{align*}j_{\alpha\beta} : \mathcal M_\alpha\to \mathcal M_\beta\end{align*}$$

denote the iterated ultrapower embedding.

For any ordinal $\xi $ , the structure $M_\alpha \cap V_\xi $ is eventually constant, and therefore, we can define an inner model N of ZFC such that for all ordinals $\xi $ , $N\cap V_\xi $ is equal to the eventual value of $M_\alpha \cap V_\xi $ . Similarly, we can define a sequence of N-ultrafilters $\vec {U}^{\mathcal N} : \operatorname {\mathrm {Ord}} \to N$ by setting $\vec {U}^{\mathcal N}(\delta )$ equal to the eventual value of $U^{\mathcal M_\alpha }(\delta )$ . We let $\mathcal N = (N,\vec {U}^{\mathcal N})$ .

By the definition of the iteration, it is clear that if for some ordinal $\delta $ , $o^{\mathcal N}(\delta )> 0$ , then $\delta $ has uncountable cofinality.

We claim that conversely, if $\delta> \lambda $ is a regular cardinal of N that has uncountable cofinality in V, then $o^{\mathcal N}(\delta ) = 1$ and $\vec {U}^{\mathcal N}(\delta ) = \mathscr {C}_\delta \cap N$ . Since the models $\mathcal M_\alpha $ converge to $\mathcal N$ , to prove the claim, it suffices to show, by induction on $\alpha \in \operatorname {\mathrm {Ord}}$ , that if $\delta> \lambda $ is a regular cardinal of $M_\alpha $ that has uncountable cofinality in V, then either $o^{\vec {U}^{\mathcal M_\alpha }}(\delta ) = 1$ and $\vec {U}^{\mathcal M_\alpha }(\delta ) = \mathscr {C}_\delta \cap M_\alpha $ or $\delta $ is the largest cardinal of $\mathcal M_\alpha $ and $U^{\mathcal M_\alpha } = \mathscr {C}_\delta \cap M_\alpha $ .

For the case that $\alpha = 0$ , note that our choice of $\mathcal M_0 = \mathcal M$ , we have $\mathcal M = \text {Hull}^{\mathcal M}_{\Sigma _1}(\lambda \cup \{X\cap M\})$ , and so by the referee’s Lemmas 3.12 and 3.13, if $\delta> \lambda $ is a regular cardinal of M, then $\delta $ has countable cofinality in V. Thus, the base case holds vacuously.

Now assume the induction hypothesis holds for $\mathcal M_\alpha $ , and we claim it is true for $\mathcal M_{\alpha +1}$ . Suppose therefore that $\delta> \lambda $ is a regular cardinal of $M_{\alpha +1}$ . Note that $\mathcal M_\alpha | \kappa _\alpha = \mathcal M_{\alpha +1} |\kappa _\alpha $ , so for ordinals $\delta $ in the open interval $(\lambda ,\kappa _\alpha )$ , the induction hypothesis for $\mathcal M_\alpha $ easily implies the induction hypothesis for $\mathcal M_{\alpha +1}$ . A slightly more complicated variation of this argument establishes the induction hypothesis for $\mathcal M_{\alpha +1}$ in the case that $\delta = \kappa _\alpha $ : by the definition of the iteration, either $\kappa _\alpha $ has countable cofinality, in which case we have nothing to show, or else $U_\alpha = W^{\mathcal {M}_\alpha }$ , in which case the fact that $U^{\mathcal M_\alpha } = \vec {U}^{\mathcal M_{\alpha +1}}(\kappa _\alpha )$ implies the induction hypothesis holds for $\mathcal M_{\alpha +1}$ with respect to $\delta = \kappa _\alpha $ .

To finish the successor case, we show that if $\delta> \kappa _\alpha $ is regular in $M_{\alpha +1}$ , then $\operatorname {\mathrm {cf}}(\delta ) = \omega $ , so the induction hypothesis holds vacuously in the interval $(\kappa _\alpha ,o(M_{\alpha +1}))$ . Since $M_{\alpha +1}$ is generated by the critical points of the iteration along with the range of $j_{0\alpha +1}$ , $\mathcal M_{\alpha +1} = \text {Hull} ^{\mathcal {M}_{\alpha +1}}_{\Sigma _1}((\kappa _{\alpha }+1)\cup \{X\cap M\})$ . Since there is a cofinal embedding from M to $M_{\alpha +1}$ , Lemma 3.12 implies that $\operatorname {\mathrm {cf}}(o(M_{\alpha +1})) = \omega $ . Therefore, we can apply Lemma 3.13 to obtain that $\operatorname {\mathrm {cf}}(\delta ) = \omega $ , as desired.

Finally, we consider the limit case. Suppose $\alpha $ is a limit ordinal and $\delta> \lambda $ is a regular cardinal of $M_\alpha $ . Let $\gamma = \sup _{\beta < \alpha }\kappa _\beta $ . As in the previous case, if $\delta < \gamma $ , the desired conclusion is immediate from the induction hypothesis. Moreover, since $\mathcal M_\alpha = \text {Hull}^{\mathcal M_\alpha }_{\Sigma _1}(\gamma \cup \{X\cap M\})$ , if $\delta> \gamma $ , then by Lemma 3.13, $\operatorname {\mathrm {cf}}(\delta ) = \omega $ , and there is nothing to prove.

It therefore suffices to consider the case that $\delta = \gamma $ .

For each $\beta < \alpha $ , let $\delta _\beta = j_{\beta \alpha }^{-1}[\delta ]$ . Note that $\delta _\beta \geq \kappa _\beta $ for all $\beta < \alpha $ . Moreover, for sufficiently large $\beta < \alpha $ , $j_{\beta \alpha }(\delta _\beta ) = \delta $ , and therefore for any $\beta ' \geq \beta $ with $\beta ' < \alpha $ , $j_{\beta \beta '}(\delta _\beta ) = \delta _{\beta '}$ .

Assume first that for sufficiently large $\beta < \alpha $ , $\delta _\beta \neq \kappa _\beta $ . Then in fact $\delta _\beta> \kappa _\beta $ , and so by Lemma 3.13, $\operatorname {\mathrm {cf}}(\delta _\beta ) = \omega $ . Moreover, it follows that for sufficiently large $\beta < \alpha $ , $j_{\beta \alpha }$ is continuous at $\delta _\beta $ and $j_{\beta \alpha }(\delta _\beta ) = \delta $ , so $\delta $ has countable cofinality as well, and we are done.

To finish the induction, assume instead that $\kappa _\beta = \delta _\beta $ for cofinally many $\beta < \alpha $ . In this case, the set $C = \{\kappa _\beta : \beta < \alpha \text { and }j_{\beta \alpha }(\kappa _\beta ) = \delta \}$ is unbounded in $\delta $ . Moreover, it is $\omega $ -closed in $\delta $ , since if $\langle \beta _n\rangle _{n < \omega }$ is an increasing sequence of ordinals with $\kappa _{\beta _n} \in C$ and $\beta $ is their supremum, then we claim $\kappa _\beta = \sup _{n < \omega } \kappa _{\beta _n}$ . Clearly, $\kappa _\beta \geq \sup _{n < \omega } \kappa _{\beta _n}$ , and the reverse inequality follows from the choice of $\kappa _\beta $ in our construction of the iterated ultrapower, noting that $\sup _{n < \omega }\kappa _{\beta _n}$ has countable cofinality and carries a measure on the $\mathcal M_\beta $ -sequence. The latter fact uses that $\sup _{n < \omega } \kappa _{\beta _n} = j_{\beta _0\beta }(\kappa _{\beta _0})$ .

Similarly, $o^{\mathcal M_\alpha }(\delta ) \geq 1$ . Let $Z = \vec {U}^{\mathcal M_\alpha }(\delta )$ if $\delta $ is less than the largest cardinal of $\mathcal M_\alpha $ , and let $Z = U^{\mathcal M_\alpha }$ otherwise. The standard argument (going back to Kunen) shows that for any $A\in P(\delta )\cap M_\alpha $ , $A\in Z$ if and only if $C\setminus \eta \subseteq A$ for some $\eta < \delta $ . Therefore, if $\delta $ has uncountable cofinality, then $Z = \mathscr {C}_\delta \cap \mathcal M_\alpha $ .

This completes our transfinite induction and establishes that if $\delta> \lambda $ is a regular cardinal of N that has uncountable cofinality in V, then $o^{\mathcal N}(\delta ) = 1$ and $\vec {U}^{\mathcal N}(\delta ) = \mathscr {C}_\delta \cap N$ .

From this, it follows that $L[A,\vec {\mathscr {C}}]$ is a definable inner model of $\mathcal N$ , since in fact $L[A,\vec {\mathscr {C}}] = L[X,\vec {\mathscr {C}}\mathbin \upharpoonright (\lambda ,\infty )]$ is definable over $\mathcal N$ using the parameter $X\cap N$ and the sequence $\vec {U}^{\mathcal N}$ . As explained above, it follows that $\lambda ^{+L[A,\vec {\mathscr C}]} < \lambda ^+,$ which completes the proof.

Putting everything together, we arrive at the following corollary: