2.1 Definitions and basic facts. (Languages and theories of sets)

1. (a) $\mathcal {L}_{\mathrm {set}}=\{=,\in \}$ is the language of $\mathsf {ZF}$ -set theory. We treat $\mathsf {ZF}$ as being axiomatized as usual, except that instead of including the scheme of replacement among the axioms of $\mathsf {ZF}$ , we include the schemes of separation and collection, as in [Reference Chang and Keisler5, Appendix A]. Thus, in our setup the axioms of Zermelo set theory $\mathsf {Z}$ are obtained by removing the scheme of collection from the axioms of $\mathsf {ZF}$ . More generally, for $ \mathcal {L}\supseteq \mathcal {L}_{\mathrm {set}}$ we construe $\mathsf {ZF}( \mathcal {L})$ to be the natural extension of $\mathsf {ZF}$ in which the schemes of separation and collection are extended to $\mathcal {L}$ -formulae; similarly we will use $\mathsf {Z}(\mathcal {L})$ to denote Zermelo set theory over $\mathcal {L}$ , i.e., as the result of extending $\mathsf {Z}$ with the $ \mathcal {L}$ -separation scheme $\mathsf {Sep}(\mathcal {L})$ . Officially speaking, $\mathsf {Sep}(\mathcal {L})$ consists of the universal closures of $ \mathcal {L}$ -formulae of the form

   $$ \begin{align*} \forall v\exists w\forall x(x\in w\longleftrightarrow x\in v\wedge \varphi (x,\vec{y})). \end{align*} $$
   Thus $\mathsf {ZF}(\mathcal {L})$ is the result of augmenting $ \mathsf {Z}(\mathcal {L})$ with the $\mathcal {L}$ -collection scheme $\mathsf { Coll}(\mathcal {L})$ , which consists of the universal closures of $\mathcal {L} $ -formulae of the form:
   $$ \begin{align*} \left( \forall x\in v\text{ }\exists y\text{ }\varphi (x,y,\vec{z})\right) \rightarrow \left( \exists w\text{ }\forall x\in v\text{ }\exists y\in w \text{ }\varphi (x,y,\vec{z})\right). \end{align*} $$
   When X is a new predicate, we write $\mathcal {L}_{\mathrm {set}}( \mathrm {X})$ instead of $\mathcal {L}_{\mathrm {set}}\cup \{\mathrm {X\}}$ , and similarly we write $\mathsf {ZF}(\mathrm {X})$ , $\mathsf {Sep}(\mathrm {X})$ , $ \mathsf {Coll}(\mathrm {X})$ , etc. instead of $\mathsf {ZF}(\mathcal {L})$ , $ \mathsf {Sep}(\mathcal {L})$ , $\mathsf {Coll}(\mathcal {L})$ , etc. (respectively) when $\mathcal {L}=\mathcal {L}_{\mathrm {set}}(\mathrm {X})$ .

2. (b) For $n\in \omega $ , we employ the common notation ( $ \Sigma _{n}, \Pi _{n}, \Delta _{n}$ ) for the Levy hierarchy of $\mathcal { L}_{\mathrm {set}}$ -formulae, as in the standard references in modern set theory by Kunen [Reference Kunen32], Jech [Reference Jech24], and Kanamori [Reference Kanamori25]. In particular, $\Delta _{0}=\Sigma _{0}=\Pi _{0}$ corresponds to the collections of $\mathcal {L}_{\mathrm {set}}$ -formulae all of whose quantifiers are bounded. In addition, we will also consider the Takahashi hierarchy of formulae ( $\Sigma _{n}^{\mathcal {P}}, \Pi _{n}^{ \mathcal {P}}, \Delta _{n}^{\mathcal {P}}$ ) (introduced in [Reference Takahashi46], and further studied by Mathias’ [Reference Mathias35]), where $\Delta _{0}^{ \mathcal {P}}$ is the smallest class of $\mathcal {L}_{\mathrm {set}}$ -formulae that contains all atomic formulae, and is closed both under Boolean connectives and under quantification in the form $Qx\in y$ and $Qx\subseteq y $ where x and y are distinct variables, and Q is $\exists $ or $ \forall $ . The classes $\Sigma _{1}^{\mathcal {P}},\Pi _{1}^{\mathcal {P}}$ , etc. are defined inductively from the class $\Delta _{0}^{\mathcal {P}}$ in the same way that the formula-classes $\Sigma _{1},\Pi _{1}$ , etc. are defined from $\Delta _{0}$ -formulae.

3. (c) For $n\in \omega $ and $\mathcal {L}\supseteq \mathcal {L} _{\mathrm {set}},$ the Levy hierarchy can be naturally extended to $\mathcal {L }$ -formulae ( $\Sigma _{n}(\mathcal {L}), \Pi _{n}(\mathcal {L}), \Delta _{n}(\mathcal {L})$ ), where $\Delta _{0}(\mathcal {L})$ is the smallest family of $\mathcal {L}$ -formulae that contains all atomic $\mathcal {L}$ -formulae and is closed under Boolean operations and bounded quantification.

4. (d) $\mathsf {KPR}$ (Kripke–Platek with ranks) is the subtheory of $\mathsf {ZF}$ whose axioms consist of $\mathsf {KP}$ plus an axiom that asserts that for all ordinals $\alpha , \{x:\mathrm {rank} (x)<\alpha \}$ exists (where $\mathrm {rank}(x)$ is the usual rank function on sets). Following recent practice (initiated by Mathias [Reference Mathias35]), the foundation scheme of $\mathsf {KP}$ is only limited to $\Pi _{1}$ -formulae; thus in this formulation $\mathsf {KP}$ can prove the scheme of $\in $ -induction for $\Sigma _{1}$ -formulae. In contrast to Barwise’s $\mathsf {KP}$ in [Reference Barwise2], which includes the full scheme of foundation, our version of $\mathsf {KP}$ is finitely axiomatizable. Thus $ \mathsf {KPR}$ is finitely axiomatizable.

5. (e) $\mathsf {ZBQC}$ is the subtheory of $\mathsf {Z}$ , obtained by augmenting $\mathsf {M}_{0}$ with $\mathsf {Foundation}$ (as a single axiom), $\mathsf {AC}$ (the axiom of choice), and the axiom of infinity, where $\mathsf {M}_{0}$ is axiomatized by $\mathsf {Extensionality}$ , $\mathsf {Pairing}$ , $\mathsf {Union}$ , $\mathsf {Powerset}$ , and $\Delta _{0} $ - $\mathsf {Separation}$ .Footnote ⁴ Two distinguished strengthenings of $\mathsf {ZBQC}$ are $\mathsf {Mac}:= \mathsf {ZBQC+TC}$ (recall that $\mathsf {TC}$ asserts that every set has a transitive closure) , and $\mathsf {Most}:=\mathsf {ZBQC}+\mathsf {KP}+\Sigma _{1}$ - $ \mathsf {Separation}$ . $\mathsf {ZBQC,} \mathsf {Mac}$ , and $\mathsf {Most}$ are finitely axiomatizable.Footnote ⁵

6. (f) For $\mathcal {L}\supseteq \mathcal {L}_{\mathrm {set}}$ , the dependent choice scheme, denoted $\Pi _{\infty }^{1}$ - $\mathsf {DC}(\mathcal {L})$ , consists of the universal closure of $ \mathcal {L}$ -formulae of the following form:

   $$ \begin{align*} \forall x\exists y\ \varphi (x,y,\vec{z})\longrightarrow \forall x\exists f\ \left[ f:\omega \rightarrow \mathrm{V},\ f(0)=x,\mathrm{and}\ \forall n\in \omega \ \varphi (f(n),f(n+1),\vec{z})\right]. \end{align*} $$
   In the presence of $\mathsf {ZF}(\mathrm {\mathcal {L}})\mathrm {,}$ thanks to the Reflection Theorem (Theorem 2.10), $\Pi _{\infty }^{1}$ - $ \mathsf {DC}\mathrm {(\mathcal {L})}$ is equivalent to the single sentence $ \mathsf {DC}$ below:
   $$ \begin{align*} \forall r\ \forall a\ \left[ \begin{array}{c} \left( \forall x\in a\ \exists y\in a\ \left\langle x,y\right\rangle \in r\right) \longrightarrow \\ \left( \forall x\in a\ \exists f\ \left( f:\omega \rightarrow \mathrm{V},\ f(0)=x,\mathrm{and}\ \forall n\in \omega \ \left\langle f(n),f(n+1)\right\rangle \in r\right) \right) \end{array} \right]. \end{align*} $$
   Note that $\mathsf {DC}$ is provable in $\mathsf {ZFC}$ .

2.2 Definitions and basic facts. (Model theoretic concepts)

In what follows we make the blanket assumption that $ \mathcal {M}$ , ${\mathcal {N}}$ , etc. are $\mathcal {L}$ -structures, where $ \mathcal {L}\supseteq \mathcal {L}_{\mathrm {set}}$ .

1. (a) We follow the convention of using M, $M^{\ast }, M_{0}$ , etc. to denote (respectively) the universes of structures $\mathcal {M }$ , $\mathcal {M}^{\ast }, \mathcal {M}_{0},$ etc. We denote the membership relation of $\mathcal {M}$ by $\in ^{\mathcal {M}}$ ; thus an $\mathcal {L}_{ \mathrm {set}}$ -structure $\mathcal {M}$ is of the form $(M,\in ^{\mathcal {M} }) $ .

2. (b) $\mathrm {Ord}^{\mathcal {M}}$ is the class of ordinals in the sense of $\mathcal {M}$ , i.e.,

   $$ \begin{align*} \mathrm{Ord}^{\mathcal{M}}:=\left\{ m\in M:\mathcal{M}\models \mathrm{Ord} (m)\right\} , \end{align*} $$
   where $\mathrm {Ord}(x)$ expresses “x is transitive and is well-ordered by $\in $ .” More generally, for a formula $ \varphi (\vec {x})$ with parameters from M, where $\vec {m}=\left ( x_{1},\cdot \cdot \cdot ,x_{k}\right ) $ , we write $\varphi ^{\mathcal {M}}$ for
   $$ \begin{align*} \left\{ \vec{m}\in M^{k}:\mathcal{M\models \varphi }\left( m_{1},\cdot \cdot \cdot ,m_{k}\right) \right\}. \end{align*} $$
   A subset D of $M^{k}$ is $\mathcal {M}$ -definable if it is of the form $ \varphi ^{\mathcal {M}}$ for some choice of $\varphi .$ We write $\mathbb { \omega }^{\mathcal {M}}$ for the set of finite ordinals (i.e., natural numbers) of $\mathcal {M}$ , and $\mathbb {\omega }$ for the set of finite ordinals in the real world, whose members we refer to as metatheoretic natural numbers. $\mathcal {M}$ is said to be $ \omega $ -standard if $\left ( \mathbb {\omega },\mathbb {\in }\right ) ^{\mathcal {M}}\cong \left ( \mathbb {\omega },\mathbb {\in }\right ) .$ For $ \alpha \in \mathrm {Ord}^{\mathcal {M}}$ we often use $\mathcal {M}_{\alpha }$ to denote the substructure of $\mathcal {M}$ whose universe is
   $$ \begin{align*} M_{\alpha }:=\left\{ m\in M:\mathcal{M}\models m\in \mathrm{V}_{\alpha }\right\} , \end{align*} $$
   where $\mathrm {V}_{\alpha }$ is defined as usual as $\{x:\mathrm {rank} (x)<\alpha \}$ , where $\mathrm {rank}(x)=\sup \{\mathrm {rank}(y)+1:y\in x\}.$

3. (c) For $c\in M$ , $\mathrm {Ext}_{\mathcal {M}}(c)$ is the $ \mathcal {M}$ -extension of c, i.e., $\mathrm {Ext}_{\mathcal {M}}(c):=\{m\in M:m\in ^{\mathcal {M}}c\}.$

4. (d) We say that $X\subseteq M$ is coded (in $\mathcal {M)}$ if there is some $c\in M$ such that $\mathrm {Ext}_{ \mathcal {M}}(c)=X.$

5. (e) Suppose $\mathcal {M}\models \mathsf {ZF}$ . We say that $X\subseteq M$ is $\mathcal {M}$ -amenable if $(\mathcal {M },X)\models \mathsf {ZF}(\mathrm {X})$ ; here X is the interpretation of $\mathrm {X}$ .

2.3 Definitions. (‘Geometric shapes’ of extensions)

Suppose $\mathcal {L}\supseteq \mathcal {L}_{\mathrm {set}}$ , $ \mathcal {M}$ and ${\mathcal {N}}$ are $\mathcal {L}$ -structures, and $\mathcal {M} $ a submodel Footnote ⁶ of ${\mathcal {N}}$ (written $ \mathcal {M}\subseteq \mathcal {N)}.$

1. (a) $\mathcal {M}^{\ast }$ is the convex hull of $\mathcal {M}$ in ${\mathcal {N}}$ if $M^{\ast }=\bigcup \limits _{a\in M}\mathrm {Ext}_{{\mathcal {N}}}(a).$

2. (b) Suppose $a\in M. {\mathcal {N}}$ fixes a if $ \mathrm {Ext}_{\mathcal {M}}(a)=\mathrm {Ext}_{{\mathcal {N}}}(a)$ , and $\mathcal {N }$ enlarges a if $\mathrm {Ext}_{\mathcal {M}}(a)\subsetneq \mathrm { Ext}_{{\mathcal {N}}}(a). $

3. (c) ${\mathcal {N}}$ end extends $\mathcal {M}$ (written $\mathcal {M}\subseteq _{\mathrm {end}}{\mathcal {N}}$ ) $,$ if $\mathcal {N }$ fixes every $a\in M$ . End extensions are also referred to in the literature as transitive extensions, and in the old days as outer extensions.

4. (d) ${\mathcal {N}}$ is a powerset-preserving end extension of $\mathcal {M}$ (written $\mathcal {M}\subseteq _{\mathrm {end}}^{ \mathcal {P}}{\mathcal {N}}$ ) if $(i) \mathcal {M}\subseteq _{\mathrm {end}} {\mathcal {N}}$ and $(ii)$ for if $a\in M$ , $b\in N$ , and ${\mathcal {N}}\models (b\subseteq a)$ , then $b\in M$ .

5. (e) ${\mathcal {N}}$ is a rank extension of $\mathcal {N }$ (written $\mathcal {M}\subseteq _{\mathrm {rank}}{\mathcal {N}}$ ) if $\mathcal { N}$ is an end extension of $\mathcal {M}$ , and for all $a\in M$ and all $b\in N\backslash M$ , ${\mathcal {N}}\models \mathrm {rank}(a)\in \mathrm {rank}(b)$ . Here we assume that $\mathcal {M}$ and ${\mathcal {N}}$ are models of a sufficient fragment of $\mathsf {ZF}$ (such as $\mathsf {KP}$ ) in which the rank function is well-defined. ${\mathcal {N}}$ is a topped rank extension of $\mathcal {M}$ if $\mathrm {Ord}^{{\mathcal {N}}}\backslash \mathrm { Ord}^{\mathcal {M}}$ has a least element.Footnote ⁷

6. (f) ${\mathcal {N}}$ is a cofinal extension of $ \mathcal {M}$ (written $\mathcal {M}\subseteq _{\mathrm {cof}}\mathcal {N)}$ if for every $b\in N$ there is some $a\in M$ such that $b\in \mathrm {Ext}_{ {\mathcal {N}}}(a)$ , i.e., ${\mathcal {N}}$ is the convex hull of $\mathcal {M}$ in ${\mathcal {N}}$ .

7. (g) ${\mathcal {N}}$ is taller than $\mathcal {M}$ (written $\mathcal {M}\subseteq _{\mathrm {taller}}\mathcal {N)}$ if there is some $b\in M$ such that $M\subseteq \mathrm {Ext}_{{\mathcal {N}}}(b).$

2.4 Definitions (‘Logical shapes’ of extensions)

Suppose $\mathcal {L}\supseteq \mathcal {L}_{\mathrm {set}}$ , and $ \mathcal {M}$ and ${\mathcal {N}}$ are $\mathcal {L}$ -structures such that $ \mathcal {M}$ is a submodel of ${\mathcal {N}}$ .

1. (a) For a subset $\Gamma $ of $\mathcal {L}$ -formulae, $ \mathcal {M}$ is a $\Gamma $ -elementary submodel of ${\mathcal {N}}$ (written $\mathcal {M}\preceq _{\Gamma }{\mathcal {N}}$ ) if for all n-ary formulae $\gamma \in \Gamma $ and for all $a_{1},\cdot \cdot \cdot ,a_{n}$ in M, $\mathcal {M}\models \gamma (a_{1},\cdot \cdot \cdot ,a_{n})$ iff $ {\mathcal {N}}\models \gamma (a_{1},\cdot \cdot \cdot ,a_{n}). \mathcal {M}$ is an elementary submodel of ${\mathcal {N}}$ (written $\mathcal {M} \preceq {\mathcal {N}}$ ) if $\mathcal {M}\preceq _{\Gamma }{\mathcal {N}}$ for $ \Gamma = \mathcal {L}$ -formulae. We say that $\mathcal {M}$ is an elementary submodel of ${\mathcal {N}}$ if $\mathcal {M}\preceq _{\Gamma } {\mathcal {N}}$ for the set $\Gamma $ of all $\mathcal {L}$ -formulae (written $ \mathcal {M}\preceq {\mathcal {N}}$ ). For a given $\gamma \in \Gamma $ we say that $\gamma $ is absolute between $\mathcal {M}$ and $ {\mathcal {N}}$ if $\mathcal {M}\preceq _{\{\gamma \}}{\mathcal {N}}$ . As usual, we write $\mathcal {M}\prec _{\Gamma }{\mathcal {N}}$ if $\mathcal {M}\preceq _{\Gamma }{\mathcal {N}}$ and $\mathcal {M}\subsetneq {\mathcal {N}}$ (similarly for $\mathcal {M}\prec {\mathcal {N}}$ )

2. (b) ${\mathcal {N}}$ is a self-extension of $\mathcal {M }$ (written $\mathcal {M}\subseteq _{\mathrm {self}}{\mathcal {N}}$ ) if $\mathcal { N}$ is interpretable in $\mathcal {M}$ (i.e., ${\mathcal {N}}$ is isomorphic to a model ${\mathcal {N}}^{\ast }$ whose universe $N^{\ast }$ and the interpretation $R^{{\mathcal {N}}^{\ast }}$ of each $R\in \mathcal {L}$ are all $\mathcal {M}$ -definable), and additionally, there is an $\mathcal {M}$ -definable embedding j of $\mathcal {M}$ into ${\mathcal {N}}^{\ast }$ . In this context, ${\mathcal {N}}$ is an elementary self-extension of $ \mathcal {M}$ if the map j is an elementary embedding.

3. (c) ${\mathcal {N}}$ is a conservative extension of $ \mathcal {M}$ (written $\mathcal {M}\subseteq _{\mathrm {cons}}{\mathcal {N}}$ ) if for every ${\mathcal {N}}$ -definable D, $M\cap D$ is $\mathcal {M}$ -definable.

4. (d) For $\mathcal {M}\models \mathsf {ZF}$ , $ {\mathcal {N}}$ is a faithful extension of $\mathcal {M}$ (written $ \mathcal {M}\subseteq _{\mathrm {faith}}\mathcal {N)}$ , if for every $\mathcal {N }$ -definable D, $M\cap D$ is $\mathcal {M}$ -amenable (as in Definition 2.2(e)).Footnote ⁸

2.5 Remark

The following are readily verifiable:

1. (a) If $\left \langle {\mathcal {N}}_{k}:k\in \omega \right \rangle $ is a chain of models extending $\mathcal {M}$ whose union is $ {\mathcal {N}}$ , and $\mathcal {M}\preceq _{\Pi _{k,\mathrm {cons}}}{\mathcal {N}} _{k}\preceq _{\Pi _{k}}{\mathcal {N}}_{k+1}$ for each $k\in \omega ,$ then $ \mathcal {M}\preceq _{\mathrm {cons}}{\mathcal {N}}$ .Footnote ⁹ Here $\mathcal {M}\preceq _{\Pi _{k,\mathrm {cons}}} {\mathcal {N}}_{k}$ is shorthand for: $\mathcal {M}\preceq _{\Pi _{k}}{\mathcal {N}} _{k}$ and $\mathcal {M}\preceq _{_{\mathrm {cons}}}{\mathcal {N}}_{k}$ .

2. (b) $\left ( \mathcal {M}\subseteq _{\mathrm {self}}{\mathcal {N}} \right ) \Rightarrow \left ( \mathcal {M}\subseteq _{\mathrm {cons}}{\mathcal {N}} \right ). $

3. (c) Assuming $\mathcal {M}\models \mathsf {ZF}$ , $\left ( \mathcal {M}\subseteq _{\mathrm {cons}}{\mathcal {N}}\right ) \Rightarrow \left ( \mathcal {M}\subseteq _{\mathrm {faith}}{\mathcal {N}}\right ) .$

The implications in (b) and (c) are not reversible. More specifically, (b) is not reversible since, e.g., if ${\mathcal {N}}$ is a self-extension of $\mathcal {M},$ then $\left \vert M\right \vert =\left \vert N\right \vert ,$ but it is possible to build a conservative elementary extension ${\mathcal {N}}$ of a model $\mathcal {M}$ with $\left \vert M\right \vert <\left \vert N\right \vert ,$ e.g., if $\mathcal {U}$ is an $ \mathcal {M}$ -ultrafilter, then the ultrapower formation modulo $\mathcal {U}$ can be iterated along any linear order (of arbitrary cardinality) and results in a conservative elementary extension of $\mathcal {M}$ . In particular, it is possible to arrange $\mathcal {M}\subseteq _{\mathrm {cons}} {\mathcal {N}}$ such that M is countable but N is uncountable. Another way to see that $(i)$ is not reversible is to note that by a theorem of Gaifman [Reference Gaifman20] every elementary self-extension of a model $ \mathcal {M}$ of the $\mathsf {ZF}$ is a cofinal extension.Footnote ¹⁰ However, by Theorem 3.1 there are models of $\mathsf {ZF}$ that have taller conservative elementary extensions. To see that (c) is not reversible, note that if $ \kappa $ and $\lambda $ are strongly inaccessible cardinals with $\kappa <\lambda $ , then $\left ( \mathrm {V}_{\kappa },\in \right ) $ is faithfully extended by $\left ( \mathrm {V}_{\lambda },\in \right ) $ , but since the truth predicate for $\left ( \mathrm {V}_{\kappa },\in \right ) $ is parametrically definable in $\left ( \mathrm {V}_{\lambda },\in \right ) $ , by Tarski’s undefinability of truth $\left ( \mathrm {V}_{\lambda },\in \right ) $ is not a conservative extension of $\left ( \mathrm {V}_{\kappa },\in \right ) $ .

It is also noteworthy that the most common examples of definable extensions in the context of models of set theory are elementary extensions. However, the Boolean-valued approach to forcing provides a wealth of definable extensions that are not elementary. More specifically, if $ \mathcal {M}$ is a model of $\mathsf {ZFC}$ , $\mathbb {B}$ is an element of $ \mathcal {M}$ that is a complete Boolean algebra from the point of view of $ \mathcal {M}$ , and $\mathcal {U}$ is an ultrafilter on $\mathbb {B}$ that is in $\mathcal {M}$ , then there is an $\mathcal {M}$ -definable embedding j of $ \mathcal {M}$ into the model $\mathcal {M}^{\mathbb {B}}/\mathcal {U}$ (where $ \mathcal {M}^{\mathbb {B}}$ is the $\mathbb {B}$ -valued forcing extension of the universe, as calculated in $\mathcal {M}$ ). However, in typical cases $ \mathcal {M}$ and $\mathcal {M}^{\mathbb {B}}/\mathcal {U}$ are not elementarily equivalent, e.g., we can always arrange $\mathbb {B}$ so that the truth-value of the continuum hypothesis changes in the transition between the $\mathcal {M }$ and $\mathcal {M}^{\mathbb {B}}/\mathcal {U}$ . Furthermore, even though $ \mathcal {M}$ and $\mathcal {M}^{\mathbb {B}}/\mathcal {U}$ might be elementarily equivalent in some cases, in general the embedding j is not elementary if $\mathbb {B}$ is a nontrivial notion of forcing, since the statement “there is a filter over $\mathbb {B}$ that meets all the dense subsets of $\mathbb {B}$ ” holds in $\mathcal {M }^{\mathbb {B}}/\mathcal {U}$ , but not in $\mathcal {M}$ (assuming $\mathbb {B}$ is a nontrivial notion of forcing).

2.6 Remark

Suppose $\mathcal {M}$ and $\mathcal { N}$ are $\mathcal {L}_{\mathrm {set}}$ -structures with $\mathcal {M}\subseteq {\mathcal {N}}\mathrm {.}$ The following are readily verifiable from the definitions involved:

1. (a) For each $n\in \omega , \mathcal {M}\preceq _{\Sigma _{n}}{\mathcal {N}}$ iff $\mathcal {M}\preceq _{\Pi _{n}}{\mathcal {N}}. $

2. (b) $\mathcal {M}\subseteq _{\mathrm {end}}{\mathcal {N}}$ , implies $\mathcal {M}\preceq _{\Delta _{0}}{\mathcal {N}}$ ; and $\mathcal {M} \preceq _{\Delta _{0}}{\mathcal {N}}$ implies $\mathcal {M}\preceq _{\Delta _{1}} {\mathcal {N}}$ .

3. (c) Suppose $\mathcal {M}\preceq _{\Delta _{0}}{\mathcal {N}}$ , $ {\mathcal {N}}$ fixes $a\in M$ , and there is bijection f in $\mathcal {M}$ between a and some $b\in M$ . Then ${\mathcal {N}}$ fixes b as well (by $ \Delta _{0}$ -elementarity, f remains a bijection between a and b as viewed from ${\mathcal {N}}$ )

4. (d) Suppose $\mathcal {M}$ and ${\mathcal {N}}$ are models of $ \mathsf {KPR}$ , $\mathcal {M}\preceq _{\Delta _{0}^{\mathcal {P}}}{\mathcal {N}}$ , and $\alpha \in \mathrm {Ord}^{M}$ . Then $\mathrm {V}_{\alpha }^{\mathcal {M}}= \mathrm {V}_{\alpha }^{{\mathcal {N}}}$ ; this follows from the fact that the formula expressing that x is an ordinal and $y=\mathrm {V}_{x}$ as a $ \Sigma _{1}^{\mathcal {P}}$ -formula in $\mathsf {KPR}$ (using the usual recursive definition of the $\mathrm {V}_{\alpha }$ -hierarchy).

5. (e) If $\mathcal {M}\subseteq _{\mathrm {end}}^{\mathcal {P}} {\mathcal {N}}$ , then $\mathcal {M}\preceq _{\Delta _{0}^{\mathcal {P}}}\mathcal {N }$ . On the other hand, since the formula expressing “x is an ordinal and $y=\mathrm {V}_{x}$ ” can be written as a $ \Sigma _{1}^{\mathcal {P}}$ -formula, it is absolute between models $\mathcal {M }$ and ${\mathcal {N}}$ such that $\mathcal {M}\subseteq _{\mathrm {end}}^{ \mathcal {P}}{\mathcal {N}}$ . Thus for models $\mathcal {M}$ and ${\mathcal {N}}$ of $\mathsf {KPR}$ :

   $$ \begin{align*} \mathcal{M}\subseteq _{\mathrm{end}}^{\mathcal{P}}{\mathcal{N}}\Rightarrow \mathcal{M}\subseteq _{\mathrm{rank}}{\mathcal{N}}. \end{align*} $$

   As pointed out by the referee, by [Reference Mathias35, Proposition 6.17], the converse of the above implication also holds provided $\mathcal {M} \models \mathsf {Most}$ and ${\mathcal {N}}\models \mathsf {Mac}$ .

6. (f) If $\mathcal {M}\preceq _{\Delta _{0}}{\mathcal {N}}$ , and $ \mathcal {M}^{\ast }$ is the convex hull of $\mathcal {M}$ in $ {\mathcal {N}}$ , then $\mathcal {M}\subseteq _{\mathrm {cof}}\mathcal {M}^{\ast }\subseteq _{\mathrm {end}}{\mathcal {N}}$ . If $\mathcal {M}$ and ${\mathcal {N}}$ are furthermore assumed to be models of $\mathsf {KPR}$ , and $\mathcal {M} \preceq _{\Delta _{0}^{\mathcal {P}}}{\mathcal {N}}$ , then (e) above implies that $\mathcal {M}\subseteq _{\mathrm {cof}}\mathcal {M}^{\ast }\subseteq _{ \mathrm {rank}}{\mathcal {N}}$ .

7. (g) If both $\mathcal {M}$ and ${\mathcal {N}}$ are models of $ \mathsf {KPR}$ , and $\mathcal {M}\preceq _{\Delta _{0}}{\mathcal {N}}$ , then the statement “ ${\mathcal {N}}$ is taller than $\mathcal {M}$ ” is equivalent to “there is some $\gamma \in \mathrm {Ord}^{{\mathcal {N}}}$ that exceeds each $\alpha \in \mathrm {Ord}^{ \mathcal {M}}$ .”

8. (h) Generic extensions of models of $\mathsf {ZF}$ are end extensions but not rank extensions since they have the same ordinals. However, by (f) above, powerset preserving end extensions of models of $ \mathsf {KPR}$ are rank extensions, thus for models $\mathcal {M}$ and $ {\mathcal {N}}$ of $\mathsf {KPR}$ we have

   $$ \begin{align*} \mathcal{M}\subseteq _{\mathrm{end}}^{\mathcal{P}}\mathcal{N\Longrightarrow M }\subseteq _{\mathrm{rank}}{\mathcal{N}}. \end{align*} $$

   On the other hand, since $\mathcal {P}(x)=y$ is $\Pi _{1}$ -statement, we have

   $$ \begin{align*} \mathcal{M}\preceq _{\Sigma _{1},\mathrm{end}}{\mathcal{N}}\Longrightarrow \mathcal{M}\subseteq _{\mathrm{end}}^{\mathcal{P}}{\mathcal{N}}. \end{align*} $$

   The converse of the above implication need not be true for arbitrary models $ \mathcal {M}$ and ${\mathcal {N}}$ of $\mathsf {KPR}$ . The referee has offered the following counterexample: let ${\mathcal {N}}$ be a model of $\mathsf {ZFC}$ that is $\omega $ -standard but whose $\omega _{1}$ is nonstandard, and let $ \mathcal {M}$ be the well-founded part of ${\mathcal {N}}$ . Then $\mathcal {M} \models \mathsf {KPR}$ , and the property of being equinumerous to a von Neumann ordinal is a $\Sigma _{1}$ -property of $\mathcal {P}^{{\mathcal {N}} }(\omega )$ that holds in ${\mathcal {N}}$ but fails in $\mathcal {M}$ . However, if $\mathcal {M}\models \mathsf {Most}$ and ${\mathcal {N}}\models \mathsf {Mac}$ , then by [Reference Mathias35, Proposition 6.17] the converse of the above implication holds.

9. (i) As we shall see in Lemma 4.3, faithful end extensions of models of $\mathsf {ZF}$ are rank extensions.

10. (j) Thanks to the provability of the induction scheme (over natural numbers) in $\mathsf {ZF}$ , if ${\mathcal {N}}$ is a faithful extension of a model ${\mathcal {N}}$ (where $\mathcal {M}$ is a model of $\mathsf {ZF}$ ), then ${\mathcal {N}}$ fixes each element of $\omega ^{\mathcal {M}}$ (but $ {\mathcal {N}}$ need not fix $\omega ^{\mathcal {M}}$ , e.g., consider the case when ${\mathcal {N}}$ is an internal ultrapower of $\mathcal {M}$ modulo a nonprincipal ultrafilter over $\omega ^{\mathcal {M}}$ ).

2.7 Definition

Reasoning within $\mathsf {KP}$ , for each object a in the universe of sets, let $\dot {a}$ be a constant symbol denoting a (where the map $a\mapsto \dot {a}$ is $\Delta _{1}$ , e.g., $\dot { a}=\left \langle 3,a\right \rangle $ as in Devlin’s monograph [Reference Devlin7]) $.$ Let $\mathrm {Sent}_{\mathcal {L}_{\mathrm {set}}}(x)$ be the $\mathcal {L}_{\mathrm {set}}$ -formula that defines the proper class $ \mathrm {Sent}_{\mathcal {L}_{\mathrm {set}}}$ of sentences in the language

$$ \begin{align*} \mathcal{L}^{+}= \mathcal{L}\cup \{\dot{a}:a\in \mathrm{V}\}, \end{align*} $$

and let $\mathrm {Sent}_{\mathcal {L}_{\mathrm {set}}}(i,x)$ be the $ \mathcal {L}_{\mathrm {set}}$ -formula that expresses “ $i\in \omega , x\in \mathrm {Sent}_{\mathcal {L}_{\mathrm {set}}},$ and x is a $ \Sigma _{i}$ -sentence.” In (a) and (b) below, $\mathcal {M} \models \mathsf {KP}$ , $S\subseteq M$ , and $k\in \omega ^{\mathcal {M}}$ .Footnote ¹¹

1. (a) Given $k\in \omega $ , S is a $\Sigma _{k}$ -satisfaction class for $\mathcal {M}$ if $\left ( \mathcal {M},S\right ) \models \mathrm {Sat}(k,\mathrm {S})$ , where $\mathrm {Sat}(k,\mathrm {S})$ is the universal generalization of the conjunction of the axioms $(i)$ through $ (iv)$ below. We assume that first order logic is formulated using only the logical constants $\left \{ \lnot ,\vee ,\exists \right \} .$

   1. (i) $\left [ \left ( \mathrm {S}\left ( \dot {x}=\dot {y}\right ) \leftrightarrow x=y\right ) \wedge \left ( \mathrm {S}\left ( \dot {x}\in \dot {y} \right ) \leftrightarrow x\in y\right ) \right ] .$

   2. (ii) $\left [ \mathrm {Sent}_{\mathcal {L}_{\mathrm {set}}}(k,\varphi )\wedge \left ( \varphi =\lnot \psi \right ) \right ] \rightarrow \left [ \mathrm {S}(\varphi )\leftrightarrow \lnot \mathrm {S}\mathsf {(}\psi \mathsf {)} \right ] \mathsf {.}$

   3. (iii) $\left [ \mathrm {Sent}_{\mathcal {L}_{\mathrm {set}}}(k,\varphi )\wedge \left ( \varphi =\psi _{1}\vee \psi _{2}\right ) \right ] \rightarrow \left [ \mathrm {S}(\varphi )\leftrightarrow \left ( \mathrm {S}(\psi _{1})\vee \mathrm {S}(\psi _{2})\right ) \right ] \mathsf {.}$

   4. (iv) $\left [ \mathrm {Sent}_{\mathcal {L}_{\mathrm {set}}}(k,\varphi )\wedge \left ( \varphi =\exists v\ \psi (v)\right ) \right ] \rightarrow \left [ \mathrm {S}(\varphi )\leftrightarrow \exists x\ \mathrm {S}\mathsf {(}\psi \mathsf {(}\dot {x}\mathsf {))}\right ] .$

2. (b) S is a full satisfaction class for $ \mathcal {M}$ if $\left ( \mathcal {M},S\right ) \models \mathrm {Sat}(k,\mathrm {S })$ for every $k\in \omega $ .

2.8 Tarski’s definability and undefinability of truth theorems

Suppose $\mathcal {M}$ is an $\mathcal {L}_{ \mathrm {set}}$ -structure.

1. (a) (Definability/codability of truth). If $\mathcal { M}$ is a structure coded as an element m of a model $ {\mathcal {N}}$ of a sufficiently strong Footnote ¹² fragment of $\mathsf {ZF}$ , then there is $s\in N$ such that ${\mathcal {N}}$ views s as the Tarskian satisfaction relation on m, written $ {\mathcal {N}}\models \mathrm {sat}(s,m).$ Footnote ¹³ Moreover, $\mathrm {sat}(x,y)$ is $ \Delta _{1}$ in ${\mathcal {N}}.$

2. (b) (Undefinability of truth). If S is a full satisfaction class for $\mathcal {M}$ , then S is not $\mathcal {M}$ -definable.

It is a well-known result of Levy that if $\mathcal {M}\models \mathsf {ZF} \mathrm {,}$ then there is a $\Delta _{0}$ -satisfaction class for $ \mathcal {M}$ that is definable in $\mathcal {M}$ both by a $\Sigma _{1}$ -formula and a $\Pi _{1}$ -formula (see ([Reference Jech24], p. 186) for a proof). This makes it clear that for each $n\geq 1,$ there is a $\Sigma _{n}$ -satisfaction class for $\mathcal {M}$ that is definable in $\mathcal {M}$ by a $\Sigma _{n}$ -formula. Levy’s result extends to models of $\mathsf {ZF}\mathrm {(}\mathcal {L }\mathrm {)}$ if $\mathcal {L}$ is finite as follows.

2.9 Levy’s partial definability of truth theorem

Let $\mathcal {L}$ be a finite extension of $\mathcal {L}_{ \mathrm {set}}$ . For each $n\in \omega $ there is an $ \mathcal {L}$ -formula $\mathrm {Sat}_{\Sigma _{n}(\mathcal {L})}$ such that for all models $\mathcal {M}$ of a sufficiently strong Footnote ¹⁴ $\mathsf {ZF}\mathrm {(}\mathcal {L}\mathrm {)}$ , Sat $_{\Sigma _{n}(\mathcal {L})}^{\mathcal {M}}$ is a $\Sigma _{n}( \mathcal {L})$ -satisfaction class for $\mathcal {M}$ . Furthermore, for $n\geq 1$ , $\mathrm {Sat}_{\Sigma _{n}(\mathcal {L})}$ is equivalent to a $\Sigma _{n}(\mathcal {L})$ -formula ( provably in $\mathsf {ZF}(\mathcal {L})$ ) $\mathrm {.}$

The Reflection Theorem is often formulated as a theorem scheme of $\mathsf {ZF }$ (e.g., as in [Reference Jech24]), the proof strategy of the Reflection Theorem applies equally well to $\mathsf {ZF}(\mathcal {L})$ for finite extensions $\mathcal {L}$ of $\mathcal {L}_{\mathrm {set}}$ .

2.10 Montague–Vaught–Levy reflection theorem

Let $\mathcal {L}$ be a finite extension of $\mathcal {L}_{\mathrm {set} }.$ For each $n\in \omega $ , $\mathsf {ZF}(\mathcal {L})$ proves that there are arbitrarily large ordinals $\gamma $ such that the submodel of the universe determined byV $_{\gamma } $ is a $\Sigma _{n}(\mathcal {L})$ -elementary submodel of the universe.Footnote ¹⁵

2.11 $\boldsymbol {\Sigma }_{\mathbf {n}}$ -elementary chains theorem

Suppose $n\in \omega , \mathcal {L}\supseteq \mathcal {L}_{\mathrm {set}}$ , $(I,<_{I})$ is a linear order, and $ \left \{ \mathcal {M}_{i}:i\in I\right \} $ is a collection of $ \mathcal {L}$ -structures such that $\mathcal {M}_{i}\ \mathcal { \preceq }_{\Sigma _{n}(\mathcal {L})}\ \mathcal {M}_{i^{\prime }}$ whenever $i<_{I}i^{\prime }.$ Then we have

$$ \begin{align*} \mathcal{M}_{i}\ \mathcal{\preceq }_{\Sigma _{n}(\mathcal{L})}\ \bigcup\limits_{i\in I}\mathcal{M}_{i}. \end{align*} $$

The following is a special case of a result of Gaifman ([Reference Gaifman20], Theorem 1, p.54). In the statement and the proof, $\mathcal {M}^{\ast }\prec _{_{\Delta _{0}(\mathcal {L}),\mathrm {end}}}{\mathcal {N}}$ is shorthand for the conjunction of $\mathcal {M}^{\ast }\prec _{\Delta _{0}(\mathcal {L)}} {\mathcal {N}}$ and $\mathcal {M}^{\ast }\prec _{\mathrm {end}}{\mathcal {N}}$ .

2.12 Gaifman splitting theorem

Let $ \mathcal {L}\supseteq \mathcal {L}_{\mathrm {set}}.$ Suppose $\mathcal { M}\models \mathsf {ZF}(\mathcal {L})$ , ${\mathcal {N}}$ is an $\mathcal {L }$ -structure with $\mathcal {M}\prec _{\Delta _{0}(\mathcal {L})} {\mathcal {N}}$ , and $\mathcal {M}^{\ast }$ is the convex hull of $\mathcal {M}$ in ${\mathcal {N}}$ . Then the following hold:

1. (a) $\mathcal {M}\preceq _{\mathrm {cof}}\mathcal {M}^{\ast }\preceq _{\Delta _{0}(\mathcal {L}),\mathrm {end}}{\mathcal {N}}$ .

2. (b) $\mathcal {M\prec N}\Rightarrow \mathcal {M}\preceq _{ \mathrm {cof}}\mathcal {M}^{\ast }\preceq _{\mathrm {end}}{\mathcal {N}}$ .

Proof. To simplify the notation, we present the proof for $\mathcal {L}=\mathcal {L}_{\mathrm {set}}$ since the same reasoning handles the general case. In our proof we use the notation $\varphi ^{v}$ , where $ \varphi $ is an $\mathcal {L}_{\mathrm {set}}$ -formula and v is a parameter, to refer to the $\Delta _{0}$ -formula obtained by restricting all the quantifiers of $\varphi $ to the elements of v. A straightforward induction on the complexity of formulae shows that:

(1) If $\mathcal {K}$ is an $\mathcal {L}_{\mathrm {set}}$ -structure, and $m\in K$ , then $\mathcal {K}\models \varphi ^{m}$ iff $(m,\in )^{\mathcal { K}}\models \varphi $ .

Note that in the above, $\varphi $ is allowed to contain parameters in $\mathrm {Ext}_{\mathcal {K}}(m).$ Suppose $\mathcal {M}\prec _{\Delta _{0}}{\mathcal {N}},$ where $\mathcal {M}\models \mathsf {ZF.}$ Note that we are not assuming that ${\mathcal {N}}\models \mathsf {ZF}$ . Putting (1) together with $\mathcal {M}\prec _{\Delta _{0}}{\mathcal {N}}$ implies that $(m,\in )^{\mathcal {M}}\preceq (m,\in )^{\mathcal {M}^{\ast }}$ for all $m\in M.$ On the other hand, the definition of $\mathcal {M}^{\ast }$ makes it clear that $(m,\in )^{{\mathcal {N}}}=(m,\in )^{\mathcal {M}^{\ast }}$ if $m\in M.$ Putting all this together, we have:

(2) For all $m\in M$ , $(m,\in )^{\mathcal {M}}\preceq (m,\in )^{ \mathcal {M}^{\ast }}=(m,\in )^{{\mathcal {N}}}$ .

It is easy to see, using the definition of $\mathcal {M}^{\ast }$ , that $\mathcal {M}\subseteq _{\mathrm {cof}}\mathcal {M}^{\ast }\subseteq _{ \mathrm {end}}{\mathcal {N}}$ , and in particular $\mathcal {M}^{\ast }\preceq _{\Delta _{0}}{\mathcal {N}}$ by Remark 2.6(b). Thus the proof of (a) is complete once we show that $\mathcal {M}\preceq _{\Sigma _{n}}\mathcal {M} ^{\ast }$ for each $n\in \omega .$ Towards this aim, given $n\in \omega $ , we can invoke the Reflection Theorem in $\mathcal {M}$ to get hold of some $ U\subseteq \mathrm {Ord}^{\mathcal {M}}$ such that:

(3) $(\mathcal {M},U)$ satisfies “U is unbounded in $\mathrm {Ord},$ and $\left [ \left ( \mathrm {V}_{\alpha },\in \right ) \prec _{\Sigma _{n}}\left ( \mathrm {V},\in \right ) \right ] $ for every $\alpha \in U$ .”

For each $\alpha \in U$ let $m_{\alpha }\in M$ such that $ m_{\alpha }=\mathrm {V}_{\alpha }^{\mathcal {M}}.$ By (3), we have:

(4) If $\alpha ,\beta \in U$ with $\alpha <\beta $ , then $ (m_{\alpha },\in )^{\mathcal {M}}\prec _{\Sigma _{n}}(m_{\beta },\in )^{ \mathcal {M}}.$

Next, we observe that if $a,b$ are in M with $(a,\in )^{\mathcal { M}}\prec _{\Sigma _{n}}(b,\in )^{\mathcal {M}}$ , then for each k-ary $ \Sigma _{n}$ -formula $\sigma (\vec {x})$ , $\mathcal {M}$ satisfies the $\Delta _{0}$ -statement $\delta _{\sigma }(a,b),$ where:

$$ \begin{align*} \delta _{\sigma }(a,b):=\forall \vec{x}\in a\ \left[ \sigma ^{a}(\vec{x} )\leftrightarrow \sigma ^{b}(\vec{x})\right], \end{align*} $$

where the prefix $\forall \vec {x}\in a$ is shorthand for $\forall x_{0}\in a\cdot \cdot \cdot \forall x_{k-1}\in a.$ Therefore, by putting (4) and (2) together with the assumption $\mathcal {M}\prec _{\Delta _{0}} {\mathcal {N}}$ we can conclude:

(5) If $\alpha ,\beta \in U$ with $\alpha <\beta $ , then $ (m_{\alpha },\in )^{\mathcal {M}^{\ast }}\prec _{\Sigma _{n}}(m_{\beta },\in )^{\mathcal {M}^{\ast }}.$

On the other hand, since U is unbounded in $\mathrm {Ord}^{ \mathcal {M}}$ , the definition of $m_{\alpha }$ , together with the fact that $ \mathcal {M}\subseteq _{\mathrm {cof}}\mathcal {M}^{\ast }$ , imply:

(6) $\ \ \mathcal {M}=\bigcup \limits _{\alpha \in U}(m_{\alpha },\in )^{\mathcal {M}}$ and $\mathcal {M}^{\ast }=\bigcup \limits _{\alpha \in U}(m_{\alpha },\in )^{\mathcal {M}^{\ast }}.$

Thanks to (2), (5), (6), and Theorem 2.11 ( $\Sigma _{n}$ -Elementary Chains Theorem) we can conclude:

(7) If $\alpha \in U$ , then $(m_{\alpha },\in )^{\mathcal {M} }\preceq (m_{\alpha },\in )^{\mathcal {M}^{\ast }}\preceq _{\Sigma _{n}} \mathcal {M}^{\ast }\mathrm {.}$

We can conclude that $\mathcal {M}\preceq _{\Sigma _{n}}\mathcal {M} ^{\ast }$ by (4), (7), and Theorem 2.11. This concludes the proof of (a).

To prove (b), suppose $\mathcal {M}\prec {\mathcal {N}}.$ With (a) at hand, we only need to show that $\mathcal {M}^{\ast }\preceq _{\Sigma _{n}}{\mathcal {N}}$ for each $n\in \omega $ . We observe that if $\alpha \in U,$ then by (5) $ (m_{\alpha },\in )^{\mathcal {M}^{\ast }}\prec _{\Sigma _{n}}\mathcal {M} ^{\ast },$ and by (2), $(m_{\alpha },\in )^{\mathcal {M}^{\ast }}=(m_{\alpha },\in )^{{\mathcal {N}}}.$ On the other hand, the assumption $\mathcal {M}\prec {\mathcal {N}}$ together with (3) makes it clear that if $\alpha \in U, (m_{\alpha },\in )^{{\mathcal {N}}}\prec _{\Sigma _{n}}{\mathcal {N}}$ . Thus $ \mathcal {M}^{\ast }\preceq _{\Sigma _{n}}{\mathcal {N}}$ by Theorem 2.11.

2.13 Remark

In general, if $\mathcal {M}\subseteq {\mathcal {N}}$ , where $\mathcal {M}$ and ${\mathcal {N}}$ are both models of $ \mathsf {ZF}$ , the condition “ $\mathrm {Ord}^{\mathcal {M}}$ is cofinal in $\mathrm {Ord}^{{\mathcal {N}}}$ ” does not imply that $\mathcal {M}$ is cofinal in ${\mathcal {N}}$ (e.g., consider $ \mathcal {M}$ and ${\mathcal {N}}$ where ${\mathcal {N}}$ is a set-generic extension of $\mathcal {M}$ ). However, the conjunction of $\Sigma _{1}$ -elementarity and “ $\mathrm {Ord}^{\mathcal {M}}$ is cofinal in $\mathrm {Ord}^{{\mathcal {N}}}$ ,” where $\mathcal {M} \models \mathsf {ZF}$ (or even $\mathsf {KPR}$ ) and ${\mathcal {N}}$ is an $ \mathcal {L}_{\mathrm {set}}$ -structure implies that $\mathcal {M}$ is cofinal in ${\mathcal {N}}$ , since the statement $(y\in \mathrm {Ord}\wedge x=\mathrm {V} _{y})$ can be written as a $\Pi _{1}$ -statement within $\mathsf {KPR}$ , and therefore it is absolute between $\mathcal {M}$ and ${\mathcal {N}}$ if $ \mathcal {M}\prec _{\Sigma _{1}}{\mathcal {N}}.$ Footnote ¹⁶ This is how elementary embeddings between inner models are usually formalized within $\mathsf {ZF}$ , as in Kanamori’s monograph [Reference Kanamori25, Proposition 5.1].

Recall that for models of $\mathsf {ZF}$ , the $\mathcal {L}_{\mathrm { set}}$ -sentence $\exists p\left ( \mathrm {V}=\mathrm {HOD}(p)\right ) $ expresses: “there is some p such that every set is first order definable in some structure of the form $\left ( \mathrm {V}_{\alpha },\in ,p,\beta \right ) _{\beta <\alpha }$ with $p\in \mathrm {V}_{\alpha }$ .” The following result is classical.

2.14 Myhill–Scott theorem [Reference Myhill, Scott and Scott38]

The following statements are equivalent for $\mathcal {M}\models \mathsf {ZF}$ :

1. (a) $\mathcal {M}\models \exists p\left ( \mathrm {V}=\mathrm { HOD}(p)\right ).$

2. (b) $\mathcal {M}$ carries a definable global well-ordering, i.e., for some $p\in M$ and some set-theoretic formula $\varphi (x,y,z)$ , $\mathcal {M}$ satisfies “ $\varphi (x,y,p)$ well-orders the universe.”

3. (c) $\mathcal {M}$ carries a definable global choice function, i.e., for some $p\in M$ and some set-theoretic formula $\psi (x,y,z)$ , $\mathcal {M}$ satisfies “ $\psi (x,y,p)$ is the graph of a global choice function.”