2 Background

Let $\mathcal {L}$ be the language of set theory—the language whose only non-logical symbol is the binary relation $\in $ . Let $\mathcal {L}^\prime $ be a language that contains $\mathcal {L}$ and let $\Gamma $ be a collection of $\mathcal {L}^\prime $ -formulae.

• • $\Gamma \textsf {-Separation}$ is the scheme that consists of the sentences

  $$ \begin{align*}\forall \vec{z} \forall w \exists y \forall x (x \in y \iff (x \in w \land \phi(x, \vec{z})),\end{align*} $$
  for all formulae $\phi (x, \vec {z})$ in $\Gamma $ . $\textsf {Separation}$ is the scheme that consists of these sentences for every formula $\phi (x, \vec {z})$ in $\mathcal {L}$ .

• • $\Gamma \textsf {-Collection}$ is the scheme that consists of the sentences

  $$\begin{align*}\forall \vec{z} \forall w ((\forall x \in w) \exists y \phi(x, y, \vec{z}) \Rightarrow \exists c (\forall x \in w) (\exists y \in c)\phi(x, y, \vec{z})), \end{align*}$$
  for all formulae $\phi (x, y, \vec {z})$ in $\Gamma $ . $\textsf {Collection}$ is the scheme that consists of these sentences for every formula $\phi (x, y, \vec {z})$ in $\mathcal {L}$ .

• • $\textsf {Strong }\Gamma \textsf {-Collection}$ is the scheme that consists of the sentences

  $$\begin{align*}\forall \vec{z} \forall w \exists c (\forall x \in w)( \exists y \phi(x, y, \vec{z}) \Rightarrow (\exists y \in c) \phi(x, y, \vec{z})), \end{align*}$$
  for all formulae $\phi (x, y, \vec {z})$ in $\Gamma $ . $\textsf {Strong Collection}$ is the scheme that consists of these sentences for every formula $\phi (x, y, \vec {z})$ in $\mathcal {L}$ .

• • $\Gamma \textsf {-Foundation}$ is the scheme that consists of the sentences

  $$\begin{align*}\forall \vec{z} (\exists x \phi(x, \vec{z}) \Rightarrow \exists y (\phi(y, \vec{z}) \land (\forall w \in y) \neg \phi(w, \vec{z}))), \end{align*}$$
  for all formulae $\phi (x, \vec {z})$ in $\Gamma $ . If $\Gamma = \{x \in z\}$ , then the resulting axiom is referred to as $\textsf {Set-Foundation}$ . $\textsf {Foundation}$ is the scheme that consists of these sentences for every formula $\phi (x, \vec {z})$ in $\mathcal {L}$ .

In addition to the Lévy classes of $\mathcal {L}$ -formulae, $\Delta _0$ , $\Sigma _1$ , $\Pi _1$ , …, we will also make reference to the class $\Delta _0^{\mathcal {P}}$ , introduced by Takahashi [Reference Takahashi and Hodges17], that consists of $\mathcal {L}$ -formulae whose quantifiers are bounded either by the membership relation ( $\in $ ) or the subset relation ( $\subseteq $ ), and the classes $\Sigma _1^{\mathcal {P}}$ , $\Pi _1^{\mathcal {P}}$ , $\Sigma _2^{\mathcal {P}}$ , …that are defined from $\Delta _0^{\mathcal {P}}$ in the same way that the classes $\Sigma _1$ , $\Pi _1$ , $\Sigma _2$ , …are defined from $\Delta _0$ . Let T be a theory in a language, $\mathcal {L}^\prime $ , that includes $\mathcal {L}$ . Let $\Gamma $ be a class of $\mathcal {L}^\prime $ -formulae. A formula is $\Gamma $ in T or $\Gamma ^T$ if it is provably equivalent in T to a formula in $\Gamma $ . A formula is $\Delta _n$ in T or $\Delta _n^T$ if it is both $\Sigma _n^T$ and $\Pi _n^T$ .

• • $\Delta _n\textsf {-Separation}$ is the scheme that consists of the sentences

  $$ \begin{align*}\forall \vec{z}(\forall v (\phi(v, \vec{z}) \iff \psi(v, \vec{z})) \Rightarrow \forall w \exists y \forall x(x \in y \iff (x \in w \land \phi(x, \vec{z}))))\end{align*} $$
  for all $\Sigma _n$ -formulae $\phi (x, \vec {z})$ and $\Pi _n$ -formulae $\psi (x, \vec {z})$ .

• • $\Delta _n\textsf {-Foundation}$ is the scheme that consists of the sentences

  $$ \begin{align*}\forall \vec{z}(\forall v(\phi(x, \vec{z}) \iff \psi(x, \vec{z})) \Rightarrow (\exists x \phi(x, \vec{z}) \Rightarrow \exists y (\phi(y, \vec{z}) \land (\forall w \in y) \neg \phi(w, \vec{z}))))\end{align*} $$
  for all $\Sigma _n$ -formulae $\phi (x, \vec {z})$ and $\Pi _n$ -formulae $\psi (x, \vec {z})$ .

We use $\mathsf {S}_1$ to denote the $\mathcal {L}$ -theory with axioms: Extensionality, Emptyset, Pair, Union, Set Difference, and Powerset. Following [Reference Mathias13], we take Kripke–Platek Set Theory ( $\mathsf {KP}$ ) to be the theory obtained from $\mathsf {S}_1$ by removing Powerset and adding $\Delta _0\textsf {-Separation}$ , $\Delta _0\textsf {-Collection}$ and $\Pi _1\textsf {-Foundation}$ . Note that this differs from [Reference Barwise2, Reference Friedman, Mathias and Rogers6], which defines Kripke–Platek Set Theory to include Foundation. The theory $\mathsf {KPI}$ is obtained from $\mathsf {KP}$ by adding the axiom Infinity, which states that a superset of the von Neumann ordinal $\omega $ exists. We use $\mathsf {M}^-$ to denote the theory that is obtained from $\mathsf {KPI}$ by replacing $\Pi _1\textsf {-Foundation}$ with $\textsf {Set-Foundation}$ and removing $\Delta _0\textsf {-Collection}$ , and adding an axiom $\mathsf {TCo}$ asserting that every set is contained in a transitive set. The theory $\mathsf {M}$ is obtained from $\mathsf {M}^-$ by adding $\textsf {Powerset}$ . The theory $\mathsf {MOST}$ is obtained from $\mathsf {M}$ by adding $\textsf {Strong } \Delta _0\textsf {-Collection}$ and the Axiom of Choice ( $\mathsf {AC}$ ). Zermelo Set Theory ( $\mathsf {Z}$ ) is obtained from $\mathsf {M}$ by removing $\mathsf {TCo}$ and adding $\textsf {Separation}$ . The theory $\mathsf {KP}^{\mathcal {P}}$ is obtained from $\mathsf {M}$ by adding $\Delta _0^{\mathcal {P}}\textsf {-Collection}$ and $\Pi _1^{\mathcal {P}}\textsf {-Foundation}$ .

The theory $\mathsf {KP}$ proves $\mathsf {TCo}$ (see, for example, [Reference Barwise2, Theorem I.6.1]). Both $\mathsf {KP}$ and $\mathsf {M}$ prove that every set x is contained in a least transitive set that is called the transitive closure of x, and denoted $\mathsf {TC}(x)$ . The following are some important relationships between axiom schemes over the theory $\mathsf {M}^-$ :

• • In the theory $\mathsf {M}^-$ , $\Gamma \textsf {-Separation}$ implies $\Gamma \textsf {-Foundation}$ .

• • The proof of [Reference Barwise2, Theorem I.4.4] generalises to show that, in the theory $\mathsf {M}^-$ , $\Pi _n\textsf {-Collection}$ implies $\Sigma _{n+1}\textsf {-Collection}$ .

• • [Reference Friedman, Li and Wong7, Lemma 4.13] shows that, over $\mathsf {M}^-$ , $\Pi _n\textsf {-Collection}$ implies $\Delta _{n+1}\textsf {-Separation}$ .

• • It is noted in [Reference Friedman, Li and Wong7, Proposition 2.4] that if T is $\mathsf {M}^-+\Pi _n\textsf {-Collection}$ , then the classes $\Sigma _{n+1}^T$ and $\Pi _{n+1}^T$ are closed under bounded quantification.

• • [Reference McKenzie12, Lemma 2.4], for example, shows that, over $\mathsf {M}^-$ , $\textsf {Strong }\Pi _n\textsf {-Collection}$ is equivalent to $\Pi _n\textsf {-Collection}+\Sigma _{n+1}\textsf {-Separation}$ .

Let $\mathcal {L}^\prime $ be a language that contains $\mathcal {L}$ . Let $\mathcal {M}= \langle M, \in ^{\mathcal {M}}, \ldots \rangle $ be an $\mathcal {L}^\prime $ -structure. If $a \in M$ , then we will use $a^*$ to denote the set $\{x \in M \mid \mathcal {M} \models (x \in a)\}$ , as long as $\mathcal {M}$ is clear from the context. Let $\Gamma $ be a collection of $\mathcal {L}^\prime $ -formulae. We say $X \subseteq M$ is $\Gamma $ over $\mathcal {M}$ if there is a formula $\phi (x, \vec {z})$ in $\Gamma $ and $\vec {a} \in M$ such that $X= \{x \in M \mid \mathcal {M} \models \phi (x, \vec {a})\}$ . In the special case that $\Gamma $ is all $\mathcal {L}^\prime $ -formulae, we say that X is a definable subclass of $\mathcal {M}$ . A set $X \subseteq M$ is $\Delta _n$ over $\mathcal {M}$ if it is both $\Sigma _n$ over $\mathcal {M}$ and $\Pi _n$ over $\mathcal {M}$ .

A structure $\mathcal {N}= \langle N, \in ^{\mathcal {N}} \rangle $ is an end extension of $\mathcal {M}= \langle M, \in ^{\mathcal {M}} \rangle $ , written $\mathcal {M} \subseteq _e \mathcal {N}$ , if $\mathcal {M}$ is a substructure of $\mathcal {N}$ and for all $x \in M$ and for all $y \in N$ , if $\mathcal {N} \models (y \in x)$ , then $y \in M$ . An end extension $\mathcal {N}$ of $\mathcal {M}$ is proper if $M \neq N$ . If $\mathcal {N}= \langle N, \in ^{\mathcal {N}} \rangle $ is an end extension of $\mathcal {M}= \langle M, \in ^{\mathcal {M}} \rangle $ and for all $x \in M$ and for all $y \in N$ , if $\mathcal {N} \models (y \subseteq x)$ , then $y \in M$ , then we say that $\mathcal {N}$ is a powerset-preserving end extension of $\mathcal {M}$ and write $\mathcal {M} \subseteq _e^{\mathcal {P}} \mathcal {N}$ . We say that $\mathcal {N}$ is a $\Sigma _n$ -elementary end extension of $\mathcal {M}$ , and write $\mathcal {M} \prec _{e, n} \mathcal {N}$ , if $\mathcal {M} \subseteq _e \mathcal {N}$ and $\Sigma _n$ properties are preserved between $\mathcal {M}$ and $\mathcal {N}$ .

We use $\mathsf {Ord}$ to denote the class of ordinals. The construction of Gödel’s constructible universe (L) presented in [Reference Barwise2, Chapter II] invokes no more than $\Pi _1\textsf {-Foundation}$ and can therefore be carried out in the theory $\mathsf {KP}$ . For all sets X,

$$\begin{align*}\mathsf{Def}(X)= \{Y \subseteq X \mid Y \textrm{ is a definable subclass of } \langle X, \in \rangle \}, \end{align*}$$

which can be seen to be a set in the theory $\mathsf {KP}$ using a formula for satisfaction in set structures such as the one described in [Reference Barwise2, Section III.1]. The levels of L are then defined by the recursion:

$$ \begin{align*}L_0= \emptyset \textrm{ and } L_\alpha= \bigcup_{\beta < \alpha} L_\beta \textrm{ if } \alpha \textrm{ is a limit ordinal,}\end{align*} $$

$$ \begin{align*}L_{\alpha+1}= L_\alpha \cup \mathsf{Def}(L_\alpha), \textrm{ and}\end{align*} $$

$$ \begin{align*}L= \bigcup_{\alpha \in \mathsf{Ord}} L_\alpha.\end{align*} $$

The function $\alpha \mapsto L_\alpha $ is total in $\mathsf {KP}$ and $\Delta _1^{\mathsf {KP}}$ . The axiom $\mathsf {V=L}$ asserts that every set is the member of some $L_\alpha $ . A transitive set M such that $\langle M, \in \rangle $ satisfies $\mathsf {KP}$ is said to be an admissible set. An ordinal $\alpha $ is said to be an admissible ordinal if $L_\alpha $ is an admissible set.

The theory $\mathsf {KP}^{\mathcal {P}}$ proves that the function $\alpha \mapsto V_\alpha $ is total and $\Delta _1^{\mathcal {P}}$ . Mathias [Reference Mathias13, Proposition Scheme 6.12] refines the relationships between the classes $\Delta _0^{\mathcal {P}}$ , $\Sigma _1^{\mathcal {P}}$ , $\Pi _1^{\mathcal {P}}$ , …, and the Lévy classes by showing that $\Sigma _1 \subseteq (\Delta _1^{\mathcal {P}})^{\textsf {MOST}}$ and $\Delta _0^{\mathcal {P}} \subseteq \Delta _2^{\mathsf {S}_1}$ . Therefore, the function $\alpha \mapsto V_\alpha $ is $\Delta _2^{\mathsf {KP}^{\mathcal {P}}}$ . It also follows from this analysis that $\mathsf {KP}^{\mathcal {P}}$ is a subtheory of $\mathsf {M}+\Pi _1\textsf {-Collection}+\Pi _2\textsf {-Foundation}$ .

Let T be an $\mathcal {L}$ -theory. A transitive set M is said to be the minimum model of T if $\langle M, \in \rangle \models T$ and for all transitive sets N with $\langle N, \in \rangle \models T$ , $M \subseteq N$ . For example, $L_{\omega _1^{\mathrm {CK}}}$ is the minimum model of $\mathsf {KPI}$ . For an $\mathcal {L}$ -theory T to have a minimum model it is sufficient that the conjunction of the following conditions hold:

1. (I) There exists a transitive set M such that $\langle M, \in \rangle \models T$ ;

2. (II) for all transitive M with $\langle M, \in \rangle \models T$ , $\langle L^M, \in \rangle \models T$ .

Gostanian [Reference Gostanian8, Section 1] shows that all sufficiently strong subsystems of $\mathsf {ZF}$ and $\mathsf {ZF}^-$ obtained by restricting the separation and collection schemes to formulae in the Lévy classes have minimum models. In particular:

Gostanian’s analysis also yields:

The fact that $\mathsf {KP}$ is able to define satisfaction in set structures also facilitates the definition of formulae expressing satisfaction, in the universe, for formulae in any given level of the Lévy hierarchy.

Definition 2.1. The formula $\mathsf {Sat}_{\Delta _0}(q, x)$ is defined as

$$ \begin{align*}\begin{array}{c} (q \in \omega) \land (q= \ulcorner \phi(v_1, \ldots, v_m) \urcorner \textrm{ where } \phi \textrm{ is } \Delta_0) \land (x= \langle x_1, \ldots, x_m \rangle) \land\\ \exists N \left( \bigcup N \subseteq N \land (x_1, \ldots, x_m \in N) \land (\langle N, \in \rangle \models \phi[x_1, \ldots, x_m]) \right) \end{array}\!\!.\end{align*} $$

We can now inductively define formulae $\mathsf {Sat}_{\Sigma _n}(q, x)$ and $\mathsf {Sat}_{\Pi _n}(q, x)$ that express satisfaction for formulae in the classes $\Sigma _n$ and $\Pi _n$ .

Definition 2.2. The formulae $\mathsf {Sat}_{\Sigma _n}(q, x)$ and $\mathsf {Sat}_{\Pi _n}(q, x)$ are defined recursively for $n>0$ . $\mathsf {Sat}_{\Sigma _{n+1}}(q, x)$ is defined as the formula

$$ \begin{align*}\exists \vec{y} \exists k \exists b \left( \begin{array}{c} (q= \ulcorner\exists \vec{u} \phi(\vec{u}, v_1, \ldots, v_l)\urcorner \textrm{ where } \phi \textrm{ is } \Pi_n)\land (x= \langle x_1, \ldots, x_l \rangle)\\ \land (b= \langle \vec{y}, x_1, \ldots, x_l \rangle) \land (k= \ulcorner \phi(\vec{u}, v_1, \ldots, v_l) \urcorner) \land \mathsf{Sat}_{\Pi_n}(k, b) \end{array}\right);\end{align*} $$

and $\mathsf {Sat}_{\Pi _{n+1}}(q, x)$ is defined as the formula

$$ \begin{align*}\forall \vec{y} \forall k \forall b \left( \begin{array}{c} (q= \ulcorner\forall \vec{u} \phi(\vec{u}, v_1, \ldots, v_l) \urcorner \textrm{ where } \phi \textrm{ is } \Sigma_n)\land (x= \langle x_1, \ldots, x_l \rangle)\\ \land ((b= \langle \vec{y}, x_1, \ldots, x_l \rangle) \land (k= \ulcorner\phi(\vec{u}, v_1, \ldots, v_l)\urcorner) \Rightarrow \mathsf{Sat}_{\Sigma_n}(k, b)) \end{array}\right).\end{align*} $$

Theorem 2.3. Suppose $n \in \omega $ and $m=\max \{ 1, n \}$ . The formula $\mathsf {Sat}_{\Sigma _n}(q, x)$ (respectively $\mathsf {Sat}_{\Pi _n}(q, x)$ ) is $\Sigma _m^{\mathsf {KP}}$ ( $\Pi _m^{\mathsf {KP}}$ , respectively). Moreover, $\mathsf {Sat}_{\Sigma _n}(q, x)$ (respectively $\mathsf {Sat}_{\Pi _n}(q, x)$ ) expresses satisfaction for $\Sigma _n$ -formulae ( $\Pi _n$ -formulae, respectively) in the theory $\mathsf {KP}$ , i.e., if $\mathcal {M} \models \mathsf {KP}$ , $\phi (v_1,\ldots ,v_k)$ is a $\Sigma _n$ -formula, and $x_1,\ldots ,x_k$ are in M, then for $q = \ulcorner \phi ( v_1, \ldots , v_k) \urcorner $ , $\mathcal {M}$ satisfies the universal generalisation of the following formula:

$$ \begin{align*}x= \langle x_1, \ldots,x_k \rangle \Rightarrow \left( \phi(x_1,\ldots,x_k) \iff \mathrm{Sat}_{\Sigma_{n}}(q, x) \right).\end{align*} $$

Kaufmann [Reference Kaufmann9] identifies necessary and sufficient conditions for models of $\mathsf {KP}$ to have proper $\Sigma _n$ -elementary end extensions.

It should be noted that Kaufmann proves that (I) implies (II) in the above under the weaker assumption that $\mathcal {M}$ is a resolvable model of $\mathsf {M}^-$ . A model $\mathcal {M}=\langle M, \in ^{\mathcal {M}} \rangle $ of $\mathsf {M}^-$ is resolvable if there is a function F that is $\Delta _1$ over $\mathcal {M}$ such that for all $x \in M$ , there exists $\alpha \in \mathsf {Ord}^{\mathcal {M}}$ such that $x \in F(\alpha )$ . The function $\alpha \mapsto L_\alpha $ witnesses the fact that every model of $\mathsf {KP}+\mathsf {V=L}$ is resolvable.

3 Limitations of Kaufmann’s theorem

In this section we show that there are limitations on the amount of the theory of the base model that can be transferred to the partially-elementary end extension guaranteed by Theorem 2.4. We utilise a generalisation of a result, due to Simpson and that is mentioned in [Reference Kaufmann9, Remark 2], showing that if a $\mathcal {M}$ satisfies $\mathsf {KP}+\mathsf {V=L}$ and has a $\Sigma _n$ -elementary end extension that satisfies enough set theory and contains no least new ordinal, then $\mathcal {M}$ must satisfy $\Pi _{n}\textsf {-Collection}$ . The proof of this generalisation, Theorem 3.1, is based on Enayat’s proof of a refinement of Simpson’s result (personal communication) that corresponds to the specific case where $n=1$ and $\mathcal {M}$ is transitive.

Proof. Assume that $\mathcal {N}= \langle N, \in ^{\mathcal {N}} \rangle $ is such that

1. (I) $\mathcal {M} \prec _{e, n} \mathcal {N}$ ;

2. (II) $\mathcal {N} \models \mathsf {KP}$ ;

3. (III) $\mathsf {Ord}^{\mathcal {N}} \backslash \mathsf {Ord}^{\mathcal {M}}$ is nonempty and has no least element.

Note that, since $\mathcal {M} \prec _{e, 1} \mathcal {N}$ and $\mathcal {M} \models \mathsf {V=L}$ , for all $\beta \in \mathsf {Ord}^{\mathcal {N}} \backslash \mathsf {Ord}^{\mathcal {M}}$ , $M \subseteq (L_\beta ^{\mathcal {N}})^*$ . We need to show that if either $\Pi _{n-1}\textsf {-Collection}$ or $\Pi _{n+2}\textsf {-Foundation}$ hold in $\mathcal {N}$ , then $\mathcal {M} \models \Pi _n\textsf {-Collection}$ . Let $\phi (x, y, \vec {z})$ be a $\Pi _n$ -formula. Let $\vec {a}, b \in M$ be such that

$$\begin{align*}\mathcal{M} \models (\forall x \in b) \exists y \phi(x, y, \vec{a}). \end{align*}$$

So, for all $x \in b^*$ , there exists $y \in M$ such that

$$ \begin{align*}\mathcal{M} \models \phi(x, y, \vec{a}).\end{align*} $$

Therefore, since $\mathcal {M} \prec _{e, n} \mathcal {N}$ , for all $x \in b^*$ , there exists $y \in M$ such that

$$ \begin{align*}\mathcal{N} \models \phi(x, y, \vec{a}).\end{align*} $$

Now, $\phi (x, y, \vec {z})$ can be written as $\forall w \psi (w, x, y, \vec {z})$ where $\psi (w, x, y, \vec {z})$ is $\Sigma _{n-1}$ . Let $\xi \in \mathsf {Ord}^{\mathcal {N}} \backslash \mathsf {Ord}^{\mathcal {M}}$ . So, for all $\beta \in \mathsf {Ord}^{\mathcal {N}} \backslash \mathsf {Ord}^{\mathcal {M}}$ and for all $x \in b^*$ , there exists $y \in (L_\beta ^{\mathcal {N}})^*$ such that

$$ \begin{align*}\mathcal{N} \models (\forall w \in L_\xi)\psi(w, x, y, \vec{a}).\end{align*} $$

Therefore, for all $\beta \in \mathsf {Ord}^{\mathcal {N}} \backslash \mathsf {Ord}^{\mathcal {M}}$ ,

(1) $$ \begin{align} \mathcal{N} \models (\forall x \in b)(\exists y \in L_\beta)(\forall w \in L_\xi) \psi(w, x, y, \vec{a}). \end{align} $$

Now, define $\theta (\beta , \xi , b, \vec {a})$ to be the formula

$$ \begin{align*}(\forall x \in b)(\exists y \in L_\beta)(\forall w \in L_\xi) \psi(w, x, y, \vec{a}).\end{align*} $$

If $\Pi _{n-1}\textsf {-Collection}$ holds in $\mathcal {N}$ , then $\theta (\beta , \xi , b, \vec {a})$ is equivalent to a $\Sigma _{n-1}$ -formula. Without $\Pi _{n-1}\textsf {-Collection}$ , $\theta (\beta , \xi , b, \vec {a})$ can be written as a $\Pi _{n+2}$ -formula. Therefore, $\Pi _{n-1}\textsf {-Collection}$ or $\Pi _{n+2}\textsf {-Foundation}$ in $\mathcal {N}$ will ensure that there is a least $\beta _0 \in \mathrm {Ord}^{\mathcal {N}}$ such that $\mathcal {N} \models \theta (\beta _0, \xi , b, \vec {a})$ . Moreover, by (1), $\beta _0 \in M$ . Therefore,

$$ \begin{align*}\mathcal{N} \models (\forall x \in b) (\exists y \in L_{\beta_0})(\forall w \in L_\xi) \psi(w, x, y, \vec{a}).\end{align*} $$

So, for all $x \in b^*$ , there exists $y \in (L_{\beta _0}^{\mathcal {M}})^*$ , for all $w \in (L_\xi ^{\mathcal {N}})^*$ ,

$$ \begin{align*}\mathcal{N} \models \psi(w, x, y, \vec{a}).\end{align*} $$

Which, since $\mathcal {M} \prec _{e, n} \mathcal {N}$ , implies that for all $x \in b^*$ , there exists $y \in (L_{\beta _0}^{\mathcal {M}})^*$ , for all $w \in M$ ,

$$ \begin{align*}\mathcal{M} \models \psi(w, x, y, \vec{a}).\end{align*} $$

Therefore, $\mathcal {M} \models (\forall x \in b)(\exists y \in L_{\beta _0}) \phi (x, y, \vec {a})$ . This shows that $\Pi _n\textsf {-Collection}$ holds in $\mathcal {M}$ .

Enayat (personal communication) uses a specific case of Theorem 3.1 to show that the $\langle L_{\omega _1^{\mathrm {CK}}}, \in \rangle $ has no proper $\Sigma _1$ -elementary end extension that satisfies $\mathsf {KP}$ . We now turn to generalising this result to show that for all $n \geq 1$ , the minimum model of $\mathsf {Z}+\Pi _n\textsf {-Collection}$ has no proper $\Sigma _{n+1}$ -elementary end extension that satisfies either $\mathsf {KP}+\Pi _{n+3}\textsf {-Foundation}$ or $\mathsf {KP}+\Pi _n\textsf {-Collection}$ . However, by Theorem 2.4, for all $n \geq 1$ , the minimum model of $\mathsf {Z}+\Pi _n\textsf {-Collection}$ does have a proper $\Sigma _{n+1}$ -elementary end extension.

The following result follows from [Reference McKenzie12, Theorem 4.4].

Proof. The fact that $\langle M, \in \rangle $ has a proper $\Sigma _{n+1}$ -elementary end extension follows from Theorem 2.4. Let $\mathcal {N}= \langle N, \in ^{\mathcal {N}} \rangle $ be such that $\mathcal {N} \models \mathsf {KP}$ , $N \neq M$ and $\langle M, \in \rangle \prec _{e, n+1} \mathcal {N}$ . Since M is the minimal model of $\mathsf {Z}+\Pi _{n}\textsf {-Collection}$ , $\langle M, \in \rangle \models \neg \sigma $ where $\sigma $ is the sentence

$$ \begin{align*}\exists x (x \textrm{ is transitive} \land \langle x, \in \rangle \models \mathsf{Z}+\Pi_{n}\textsf{-Collection}).\end{align*} $$

Since $\sigma $ is $\Sigma _1^{\mathsf {KP}}$ and $\langle M, \in \rangle \prec _{e, 1} \mathcal {N}$ , $\mathcal {N} \models \neg \sigma $ . Since $\mathcal {N} \models \mathsf {KP}$ and $M \neq N$ , $\mathsf {Ord}^{\mathcal {N}} \backslash \mathsf {Ord}^{\langle M, \in \rangle }$ is nonempty. If $\gamma $ is the least element of $\mathsf {Ord}^{\mathcal {N}} \backslash \mathsf {Ord}^{\langle M, \in \rangle }$ , then

$$ \begin{align*}\mathcal{N} \models (\langle L_{\gamma}, \in \rangle \models \mathsf{Z}+\Pi_{n}\textsf{-Collection}),\end{align*} $$

which contradicts the fact that $\mathcal {N} \models \neg \sigma $ . Therefore, $\mathsf {Ord}^{\mathcal {N}} \backslash \mathsf {Ord}^{\langle M, \in \rangle }$ is nonempty and contains no least element. Therefore, by Theorem 3.1 and Corollary 3.3, there must be both an instance of $\Pi _n\textsf {-Collection}$ and an instance of $\Pi _{n+3}\textsf {-Foundation}$ that fails in $\mathcal {N}$ .

We can also obtain an analog of Theorem 3.4 for the minimum models of $\mathsf {KPI}+\Pi _n\textsf {-Collection}$ that allow us to recover Enayat’s result. [Reference Gostanian8, Theorem 2.3] yields the following analog of Corollary 3.3.

Using Theorems 3.1 and 3.5, and the same argument used in the proof of Theorem 3.4 now yields:

4 Building partially-elementary end extensions

In this section we will show that if $\mathcal {M}$ is a countable model of $\mathsf {KP}+\Pi _n\textsf {-Collection}+\Sigma _{n+1}\textsf {-Foundation}$ and T is a recursively enumerable theory that holds in $\mathcal {M}$ , then there exists a proper $\Sigma _n$ -elementary end extension $\mathcal {N}$ of $\mathcal {M}$ such that $\mathcal {N}$ satisfies T (Theorem 4.15). The special case of this result for $\mathcal {M}$ transitive can be proved using the Barwise Compactness theorem. The more general result is obtained using Barwise’s machinery of admissible covers that facilitate the application of Barwise compactness arguments to nonstandard models. In order to motivate the proof of Theorem 4.15, we begin by sketching the proof of the special case that applies only to countable transitive models.

Theorem 4.1. Let T be a recursively enumerable $\mathcal {L}$ -theory such that

$$\begin{align*}T \vdash \mathsf{KP}+\Pi_n\textsf{-Collection}, \end{align*}$$

and let M be countable and transitive with $\langle M, \in \rangle \models T$ . Then there exists $\mathcal {N}= \langle N, \in ^{\mathcal {N}} \rangle $ such that $\langle M, \in \rangle \prec _{e, n} \mathcal {N} \models T$ and there exists $d \in N$ such that for all $x \in M$ , $\mathcal {N} \models (x \in d)$ .

Barwise [Reference Barwise, Fenstad and Hinman1] and [Reference Barwise2, Appendix] introduces the machinery of admissible covers to apply infinitary compactness arguments, such as the one used in the proof sketch of Theorem 4.1, to nonstandard countable models. The proof of [Reference Barwise2, Theorem A.4.1] shows that for any countable model $\mathcal {M}$ of $\mathsf {KP}+\textsf {Foundation}$ and for any recursively enumerable $\mathcal {L}$ -theory T that holds in $\mathcal {M}$ , $\mathcal {M}$ has proper end extension that satisfies T. By calibrating [Reference Barwise2, Appendix], Ressayre [Reference Ressayre15, Theorem 2.15] shows that this result also holds for countable models of $\mathsf {KP}+\Sigma _1\textsf {-Foundation}$ .

In [Reference Ressayre15, 2.17 Remarks], Ressayre notes, without providing the details, that if $\mathcal {M}$ satisfies $\mathsf {KP}+\Pi _n\textsf {-Collection}+\Pi _{n+1}\cup \Sigma _{n+1}\textsf {-Foundation}$ , then the end extension obtained in Theorem 4.2 can be guaranteed to be $\Sigma _n$ -elementary. In this section, we work through the details of this result showing that the assumption that the model $\mathcal {M}$ being extended satisfies $\Pi _{n+1}\textsf {-Foundation}$ is not necessary. Our main result (Theorem 4.15) can be viewed as a generalisation of [Reference Enayat and McKenzie5, Theorem 5.3], where admissible covers are used to build powerset-preserving end extension of countable models of set theory. Here we follow the presentation of admissible covers presented in [Reference Enayat and McKenzie5].

In order to present admissible covers of (not necessarily well-founded) models of extensions of $\mathsf {KP}$ we need to describe extensions of Kripke–Platek Set Theory that allow structures to appear as urelements in the domain of discourse. Let $\mathcal {L}^*$ be obtained from $\mathcal {L}$ by adding a new unary predicate $\mathsf {U}$ , binary relation $\mathsf {E}$ and unary function symbol $\mathsf {F}$ . Let $\mathcal {L}^*_{\mathsf {S}}$ be obtained from $\mathcal {L}^*$ by adding a new binary predicate $\mathsf {S}$ . The intention is that $\mathsf {U}$ distinguishes objects that are urelements from objects that are sets, the urelements together with $\mathsf {E}$ form an $\mathcal {L}$ -structure, and $\in $ is a membership relation between sets or urelments and sets. That is, the $\mathcal {L}^*$ - and $\mathcal {L}^*_{\mathsf {S}}$ -structures we will consider will be structures in the form $\mathfrak {A}_{\mathcal {M}}= \langle \mathcal {M}; A, \in ^{\mathfrak {A}}, \mathsf {F}^{\mathfrak {A}}\rangle $ or $\mathfrak {A}_{\mathcal {M}}= \langle \mathcal {M}; A, \in ^{\mathfrak {A}}, \mathsf {F}^{\mathfrak {A}}, \mathsf {S}^{\mathfrak {A}}\rangle $ , where $\mathcal {M}= \langle M, \mathsf {E}^{\mathfrak {A}} \rangle $ , M is the extension of $\mathsf {U}$ , $\mathsf {E}^{\mathfrak {A}} \subseteq M \times M$ , A is the extension of $\neg \mathsf {U}$ and $\in ^{\mathfrak {A}} \subseteq (M \cup A) \times A$ .

Following [Reference Barwise2] we simplify the presentation of $\mathcal {L}^*$ - and $\mathcal {L}^*_{\mathsf {S}}$ -formulae by treating these languages as two-sorted instead of one-sorted and using the following conventions:

• • The variables p, q, r, $p_1$ , …range over elements of the domain that satisfy $\mathsf {U}$ ;

• • the variables a, b, c, $a_1$ , …range over elements of the domain that satisfy $\neg \mathsf {U}$ ;

• • the variables x, y, z, w, $x_1$ , …range over all elements of the domain.

So, $\forall a (\cdots )$ is an abbreviation of $\forall x(\neg \mathsf {U}(x) \Rightarrow \cdots )$ , $\exists p(\cdots )$ is an abbreviation of $\exists x( \mathsf {U}(x) \land \cdots )$ , etc. These conventions are used in the following $\mathcal {L}^*_{\mathsf {S}}$ -axioms and -axiom schemes:

• (Extensionality for sets) $\forall a \forall b(a=b \iff \forall x(x \in a \iff x \in b))$ .

• (Pair) $\forall x \forall y \exists a \forall z(z \in a \iff z= x \lor z=y)$ .

• (Union) $\forall a \exists b (\forall y \in b)(\forall x \in y)(x \in b)$ .

Let $\Gamma $ be a class of $\mathcal {L}^*_{\mathsf {S}}$ -formulae.

• ( $\Gamma $ -Separation) For all $\phi (x, \vec {z})$ in $\Gamma $ ,

  $$ \begin{align*}\forall \vec{z} \forall a \exists b \forall x(x \in b \iff (x \in a) \land \phi(x, \vec{z})).\end{align*} $$

• ( $\Gamma $ -Collection) For all $\phi (x, y, \vec {z})$ in $\Gamma $ ,

  $$ \begin{align*}\forall \vec{z} \forall a ((\forall x \in a) \exists y \phi(x, y, \vec{z}) \Rightarrow \exists b (\forall x \in a)(\exists y \in b) \phi(x, y, \vec{z})).\end{align*} $$

• ( $\Gamma $ -Foundation) For all $\phi (x, \vec {z})$ in $\Gamma $ ,

  $$ \begin{align*}\forall \vec{z}(\exists x \phi(x, \vec{z}) \Rightarrow \exists y(\phi(y, \vec{z}) \land (\forall w \in y) \neg \phi(w, \vec{z}))).\end{align*} $$

The interpretation of the function symbol $\mathsf {F}$ will map urelements, p, to sets, a, such that the $\mathsf {E}$ -extension of p is equal to the $\in $ -extension of a. This is captured by the following axiom:

• ( $\dagger $ ) $\forall p \exists a(a= \mathsf {F}(p) \land \forall x( x \mathsf {E} p \iff x \in a)) \land \forall b(\mathsf {F}(b)= \emptyset )$ .

The following theory is the analog of $\mathsf {KP}$ in the language $\mathcal {L}^*$ :

• • $\mathsf {KPU}_{\mathbb {C}\mathsf {ov}}$ is the $\mathcal {L}^*$ -theory with axioms: $\exists a (a=a)$ , $\forall p \forall x(x \notin p)$ , Extensionality for sets, Pair, Union, $\Delta _0(\mathcal {L}^*)\textsf {-Separation}$ , $\Delta _0(\mathcal {L}^*)\textsf {-Collection}$ , $\Pi _1(\mathcal {L}^*)\textsf {-Foundation}$ and ( $\dagger $ ).

An order pair $\langle x, y \rangle $ is coded in $\mathsf {KPU}_{\mathbb {C}\mathsf {ov}}$ by the set $\{\{x\}, \{x, y\}\}$ , and we write $\mathsf {OP}(x)$ for the usual $\Delta _0$ -formula that says that z is an order pair and that also works in this theory. We write $\mathsf {fst}$ for the function $\langle x, y \rangle \mapsto x$ and $\mathsf {snd}$ for the function $\langle x, y \rangle \mapsto y$ . The usual $\Delta _0$ definitions of the graphs of these functions also work in $\mathsf {KPU}_{\mathbb {C}\mathsf {ov}}$ . The rank function, $\rho $ , and support function, $\mathsf {sp}$ , are defined in $\mathsf {KPU}_{\mathbb {C}\mathsf {ov}}$ by recursion:

$$ \begin{align*}\rho(p)= 0 \textrm{ for all urelements } p, \textrm{ and } \rho(a)= \sup \{\rho(x)+1 \mid x \in a\} \textrm{ for all sets } a;\end{align*} $$

$$ \begin{align*}\mathsf{sp}(p)=\{p\} \textrm{ for all urelements } p, \textrm{ and } \mathsf{sp}(a)= \bigcup_{x \in a} \mathsf{sp}(x) \textrm{ for all sets } a.\end{align*} $$

The theory $\mathsf {KPU}_{\mathbb {C}\mathsf {ov}}$ proves that both $\mathsf {sp}$ and $\rho $ are total functions and their graphs are $\Delta _1(\mathcal {L}^*)$ . We say that x is a pure set if $\mathsf {sp}(x)=\emptyset $ . The following $\Delta _0(\mathcal {L}^*)$ -formulae assert that ‘x is transitive’ and ‘x is an ordinal (a hereditarily transitive pure set)’:

$$ \begin{align*}\mathsf{Transitive}(x) \iff \neg\mathsf{U}(x) \land (\forall y \in x)(\forall z \in y) (z \in x);\end{align*} $$

$$ \begin{align*}\mathsf{Ord}(x) \iff (\mathsf{Transitive}(x) \land (\forall y \in x) \mathsf{Transitive}(y)).\end{align*} $$

We will consider $\mathcal {L}^*_{\mathsf {S}}$ -structures in which the predicate $\mathsf {S}$ is a satisfaction class for the $\Sigma _n$ -formulae of the $\mathcal {L}$ -structure $\mathcal {M}$ . Let $\mathsf {KPU}^\prime _{\mathbb {C}\mathsf {ov}}$ be obtained from $\mathsf {KPU}_{\mathbb {C}\mathsf {ov}}$ by adding axioms asserting that the $\mathcal {L}$ -structure formed by the urelements and the binary relation $\mathsf {E}$ satisfies $\mathsf {KP}$ . For $n \in \omega $ , define

1. ( $n\textsf {-Sat}$ ) $\mathsf {S}(m, x)$ if and only if $\mathsf {U}(m)$ and $\mathsf {U}(x)$ and $\mathsf {Sat}_{\Sigma _n}(m, x)$ holds in the $\mathcal {L}$ -structure defined by $\mathsf {U}$ and $\mathsf {E}$ .

We can now define a family of $\mathcal {L}^*_{\mathsf {S}}$ -theories extending $\mathsf {KPU}_{\mathbb {C}\mathsf {ov}}$ that assert that the $\mathcal {L}$ -structure defined by $\mathsf {U}$ and $\mathsf {E}$ satisfies $\mathsf {KP}$ and $\mathsf {S}$ is a satisfaction class on this structure for $\Sigma _n$ -formulae, and $\mathsf {S}$ can be used in the separation, collection and foundation schemes.

• • For all $n \in \omega $ , define $\mathsf {KPU}^n_{\mathbb {C}\mathsf {ov}}$ to be the $\mathcal {L}^*_{\mathsf {S}}$ -theory extending $\mathsf {KPU}^\prime _{\mathbb {C}\mathsf {ov}}$ with the axiom $n\textsf {-Sat}$ and the schemes $\Delta _0(\mathcal {L}^*_{\mathsf {S}})\textsf {-Separation}$ , $\Delta _0(\mathcal {L}^*_{\mathsf {S}})\textsf {-Collection}$ and $\Pi _1(\mathcal {L}^*_{\mathsf {S}})\textsf {-Foundation}$ .

The arguments used in [Reference Barwise2, Theorems I.4.4 and I.4.5] show that $\mathsf {KPU}_{\mathbb {C}\mathsf {ov}}$ proves the schemes of $\Sigma _1(\mathcal {L}^*)\textsf {-Collection}$ and $\Delta _1(\mathcal {L}^*)\textsf {-Separation}$ , and for all $n \in \omega $ , $\mathsf {KPU}^n_{\mathbb {C}\mathsf {ov}}$ proves the schemes of $\Sigma _1(\mathcal {L}^*_{\mathsf {S}})\textsf {-Collection}$ and $\Delta _1(\mathcal {L}^*_{\mathsf {S}})\textsf {-Separation}$ .

Definition 4.1. Let $\mathcal {M}= \langle M, \mathsf {E}^{\mathcal {M}} \rangle $ be an $\mathcal {L}$ -structure. An admissible set covering $\mathcal {M}$ is an $\mathcal {L}^*$ -structure

$$ \begin{align*}\mathfrak{A}_{\mathcal{M}} = \langle \mathcal{M}; A, \in^{\mathfrak{A}}, \mathsf{F}^{\mathfrak{A}} \rangle \models \mathsf{KPU}_{\mathbb{C}\mathsf{ov}}\end{align*} $$

such that $\in ^{\mathfrak {A}}$ is well-founded. If $\mathcal {M} \models \mathsf {KP}$ and $n \in \omega $ , then an n-admissible set covering $\mathcal {M}$ is an $\mathcal {L}^*_{\mathsf {S}}$ -structure

$$ \begin{align*}\mathfrak{A}_{\mathcal{M}} = \langle \mathcal{M}; A, \in^{\mathfrak{A}}, \mathsf{F}^{\mathfrak{A}}, \mathsf{S}^{\mathfrak{A}} \rangle \models \mathsf{KPU}^n_{\mathbb{C}\mathsf{ov}}\end{align*} $$

such that $\in ^{\mathfrak {A}}$ is well-founded. Note that if $\mathfrak {A}_{\mathcal {M}} = \langle \mathcal {M}; A, \in ^{\mathfrak {A}}, \mathsf {F}^{\mathfrak {A}}, \ldots \rangle $ is an (n-)admissible set covering $\mathcal {M}$ , then $\mathfrak {A}_{\mathcal {M}}$ is isomorphic to a structure whose membership relation ( $\in $ ) is the membership relation of the metatheory. The admissible cover of $\mathcal {M}$ , denoted $\mathbb {C}\mathrm {ov}_{\mathcal {M}}= \langle \mathcal {M}; A_{\mathcal {M}}, \in , \mathsf {F}_{\mathcal {M}} \rangle $ , is the smallest admissible set covering $\mathcal {M}$ whose membership relation ( $\in $ ) coincides with the membership relation of the metatheory. If $\mathcal {M} \models \mathsf {KP}$ and $n \in \omega $ , the n-admissible cover of $\mathcal {M}$ , denoted $\mathbb {C}\mathrm {ov}^n_{\mathcal {M}}= \langle \mathcal {M}; A_{\mathcal {M}}, \in , \mathsf {F}_{\mathcal {M}}, \mathsf {S}_{\mathcal {M}} \rangle $ , is the smallest n-admissible set covering $\mathcal {M}$ whose membership relation ( $\in $ ) coincides with the membership relation of the metatheory.

Definition 4.2. Let $\mathcal {M}= \langle M, \mathsf {E}^{\mathcal {M}} \rangle $ be an $\mathcal {L}$ -structure, and let $\mathfrak {A}_{\mathcal {M}} = \langle \mathcal {M}; A, \in , \mathsf {F}^{\mathfrak {A}}, \ldots \rangle $ be an $\mathcal {L}^*$ - or $\mathcal {L}^*_{\mathsf {S}}$ -structure. We use $\mathrm {WF}(A)$ to denote the largest $B \subseteq A$ such that $\langle B, \in ^{\mathfrak {A}} \rangle \subseteq _e \langle A, \in ^{\mathfrak {A}} \rangle $ and $\langle B, \in ^{\mathfrak {A}} \rangle $ is well-founded. The well-founded part of $\mathfrak {A}_{\mathcal {M}}$ is the $\mathcal {L}^*$ - or $\mathcal {L}^*_{\mathsf {S}}$ -structure

$$ \begin{align*}\mathrm{WF}(\mathfrak{A}_{\mathcal{M}})= \langle \mathcal{M}; \mathrm{WF}(A), \in^{\mathfrak{A}}, \mathsf{F}^{\mathfrak{A}}, \ldots \rangle.\end{align*} $$

Note that $\mathrm {WF}(\mathfrak {A}_{\mathcal {M}})$ is always isomorphic to a structure whose membership relation $\in $ coincides with the membership relation of the metatheory.

Let $\mathcal {M}= \langle M, \mathsf {E}^{\mathcal {M}} \rangle $ be such that $\mathcal {M} \models \mathsf {KP}$ . Let $\mathcal {L}^{\mathtt {ee}}$ be the language obtained from $\mathcal {L}$ by adding new constant symbols $\bar {a}$ for each $a \in M$ and a new constant symbol $\mathbf {c}$ . Let $\mathfrak {A}_{\mathcal {M}}= \langle \mathcal {M}; A, \in , \mathsf {F}^{\mathfrak {A}}, \mathsf {S}^{\mathfrak {A}} \rangle $ be an n-admissible set covering $\mathcal {M}$ . There is a coding $\ulcorner \cdot \urcorner $ of a fragment of the infinitary language $\mathcal {L}_{\infty \omega }^{\mathtt {ee}}$ in $\mathfrak {A}_{\mathcal {M}}$ with the property that the classes of codes of atomic formulae, variables, constants, well-formed formulae, sentences, etc. are all $\Delta _1(\mathcal {L}^*)$ -definable over $\mathfrak {A}_{\mathcal {M}}$ (see [Reference Enayat and McKenzie5, p. 9] for an explicit definition of such a coding). We write $\mathcal {L}_{\mathfrak {A}_{\mathcal {M}}}^{\mathtt {ee}}$ for the fragment of $\mathcal {L}_{\infty \omega }^{\mathtt {ee}}$ whose codes appear in $\mathfrak {A}_{\mathcal {M}}$ . In order to apply compactness arguments to $\mathcal {L}_{\mathfrak {A}_{\mathcal {M}}}^{\mathtt {ee}}$ -theories where $\mathfrak {A}_{\mathcal {M}}$ is an n-admissible set, we will use the following specific version of the Barwise Compactness theorem ([Reference Barwise2, Theorem III.5.6]):

The work in [Reference Barwise2, Appendix] and [Reference Ressayre15, Chapter 2] shows that if $\mathcal {M}$ satisfies $\mathsf {KP}+\Sigma _1\textsf {-Foundation}$ , then $\mathbb {C}\mathrm {ov}_{\mathcal {M}}$ exists. In particular, $\mathbb {C}\mathrm {ov}_{\mathcal {M}}$ can be obtained from $\mathcal {M}$ by first defining a model of $\mathsf {KPU}_{\mathbb {C}\mathsf {ov}}$ inside $\mathcal {M}$ and then considering the well-founded part of this model. We now turn to reviewing the construction of $\mathbb {C}\mathrm {ov}_{\mathcal {M}}$ from $\mathcal {M}$ and showing that if $\mathcal {M}$ satisfies $\mathsf {KP}+\Pi _n\textsf {-Collection}+\Sigma _{n+1}\textsf {-Foundation}$ , then $\mathbb {C}\mathrm {ov}_{\mathcal {M}}$ can be expanded to an $\mathcal {L}^*_{\mathsf {S}}$ -structure corresponding to $\mathbb {C}\mathrm {ov}^n_{\mathcal {M}}$ .

Let $n \geq 1$ . Fix a model $\mathcal {M}= \langle M, \mathsf {E}^{\mathcal {M}} \rangle $ that satisfies $\mathsf {KP}+\Pi _n\textsf {-Collection}+\Sigma _{n+1}\textsf {-Foundation}$ . Working inside $\mathcal {M}$ , define unary relations $\mathsf {N}$ and $\mathsf {Set}$ , binary relations $\mathsf {E}^\prime $ , $\mathcal {E}$ and $\bar {\mathsf {S}}$ , and unary function $\bar {\mathsf {F}}$ by:

$$ \begin{align*}\mathsf{N}(x) \textrm{ iff } \exists y(x = \langle 0, y \rangle);\end{align*} $$

$$ \begin{align*}x \mathsf{E}^\prime y \textrm{ iff } \exists w \exists z(x= \langle 0, w \rangle \land y= \langle 0, z \rangle \land w \in z);\end{align*} $$

$$ \begin{align*}\mathsf{Set}(x)= \exists y (x = \langle 1, y \rangle \land (\forall z \in y) (\mathsf{N}(z) \lor \mathsf{Set}(z)));\end{align*} $$

$$ \begin{align*}x \mathcal{E} y \textrm{ iff } \exists z (y= \langle 1, z \rangle \land x \in z);\end{align*} $$

$$ \begin{align*}\bar{\mathsf{F}}(x)= \langle 1, X \rangle \textrm{ where } X= \{ \langle 0, y \rangle \mid \exists w (x= \langle 0, w \rangle \land y \in w)\};\end{align*} $$

$$ \begin{align*}\bar{\mathsf{S}}(x, y) \textrm{ iff } \exists z \exists w(x= \langle 0, w \rangle \land y= \langle 0, z \rangle \land \mathsf{Sat}_{\Sigma_n}(w, z)).\end{align*} $$

It is noted in [Reference Barwise2, Appendix Section 3] that $\mathsf {N}$ , $\mathsf {E}^\prime $ , $\mathcal {E}$ and $\bar {\mathsf {F}}$ are defined by $\Delta _0$ -formulae in $\mathcal {M}$ . The Second Recursion theorem ([Reference Barwise2, Theorem V.2.3]), provable in $\mathsf {KP}+\Sigma _1\textsf {-Foundation}$ as note in [Reference Ressayre15], ensures that $\mathsf {Set}$ can be expressed as a $\Sigma _1$ -formula in $\mathcal {M}$ . Theorem 2.3 implies that $\bar {\mathsf {S}}$ is defined by a $\Sigma _n$ -formula in $\mathcal {M}$ . These definitions yield an interpretation, $\mathcal {I}$ , of an $\mathcal {L}^*_{\mathsf {S}}$ -structure $\mathfrak {A}_{\mathcal {N}}= \langle \mathcal {N}; \mathsf {Set}^{\mathcal {M}}, \mathcal {E}^{\mathcal {M}}, \bar {\mathsf {F}}^{\mathcal {M}}, \bar {\mathsf {S}}^{\mathcal {M}} \rangle $ , where $\mathcal {N}= \langle \mathsf {N}^{\mathcal {M}}, (\mathsf {E}^\prime )^{\mathcal {M}} \rangle $ . Table 1 extends the table on [Reference Barwise2, p. 373] and summarises the interpretation $\mathcal {I}$ :

Table 1

The interpretation $\mathcal {I}$ .

If $\phi $ is an $\mathcal {L}^*_{\mathsf {S}}$ -formula, then we write $\phi ^{\mathcal {I}}$ for the translation of $\phi $ into an $\mathcal {L}$ -formula described in Table 1. By ignoring the interpretation $\bar {\mathsf {S}}$ of $\mathsf {S}$ we obtain, instead, an interpretation, $\mathcal {I}^-$ , of an $\mathcal {L}^*$ -structure in $\mathcal {M}$ and we write $\mathfrak {A}_{\mathcal {N}}^-$ for this reduct. Note that the map $x \mapsto \langle 0, x \rangle $ defines an isomorphism between $\mathcal {M}$ and $\mathcal {N}= \langle \mathsf {N}^{\mathcal {M}}, (\mathsf {E}^\prime )^{\mathcal {M}} \rangle $ . Ressayre, refining [Reference Barwise2, Appendix Lemma 3.2], shows that if $\mathcal {M}$ satisfies $\mathsf {KP}+\Sigma _1\textsf {-Foundation}$ , then interpretation $\mathcal {I}^-$ yields a structure satisfying $\mathsf {KPU}_{\mathbb {C}\mathrm {ov}}$ .

Proof. We prove this result by induction on the complexity of $\phi $ . Above, we observed that $\mathsf {N}(x)$ , $x\mathsf {E}^\prime y$ , $x \mathcal {E} y$ and $y= \bar {\mathsf {F}}(x)$ can be written as $\Delta _0$ -formulae. And $\bar {\mathsf {S}}(x, y)$ can be written as a $\Sigma _n$ -formula. Now, $y \mathcal {E} \bar {F}(x)$ if and only if

$$ \begin{align*}\mathsf{fst}(y)=0 \land \mathsf{snd}(y) \in \mathsf{snd}(x),\end{align*} $$

which is $\Delta _0$ . Therefore, if $\phi (\vec {x})$ is a quantifier-free $\mathcal {L}_{\mathsf {S}}^*$ -formula, then $\phi ^{\mathcal {I}}(\vec {x})$ is equivalent to a $\Delta _{n+1}$ -formula in $\mathcal {M}$ . Now, suppose that $\phi (x_0, \ldots , x_{m-1})$ is in the form $(\exists y \in x_0) \psi (x_0, \ldots , x_{m-1}, y)$ where $\psi ^{\mathcal {I}}(x_0, \ldots , x_{m-1}, y)$ is equivalent to a $\Delta _{n+1}$ -formula in $\mathcal {M}$ . Therefore, $\phi ^{\mathcal {I}}(x_0, \ldots , x_{m-1})= (\exists y \mathcal {E} x_0) \psi ^{\mathcal {I}}(x_0, \ldots , x_{m-1}, y)$ , and $(\exists y \mathcal {E} x_0) \psi ^{\mathcal {I}}(x_0, \ldots , x_{m-1}, y)$ iff

$$ \begin{align*}(\exists y \in \mathsf{snd}(x_0)) \psi^{\mathcal{I}}(x_0, \ldots, x_{m-1}, y).\end{align*} $$

So, since $\mathcal {M}$ satisfies $\Pi _n\textsf {-Collection}$ , $\phi ^{\mathcal {I}}(x_0, \ldots , x_{m-1})$ is equivalent to a $\Delta _{n+1}$ -formula in $\mathcal {M}$ . Finally, suppose that $\phi (x_0, \ldots , x_{m-1})$ is in the form $(\exists y \in \mathsf {F}(x_0)) \psi (x_0, \ldots , x_{m-1}, y)$ where $\psi ^{\mathcal {I}}(x_0, \ldots , x_{m-1}, y)$ is equivalent to a $\Delta _{n+1}$ -formula in $\mathcal {M}$ . Therefore, $\phi ^{\mathcal {I}}(x_0, \ldots , x_{m-1})= (\exists y \mathcal {E} \bar {\mathsf {F}}(x_0)) \psi ^{\mathcal {I}}(x_0, \ldots , x_{m-1}, y)$ , and $(\exists y \mathcal {E} \bar {\mathsf {F}}(x_0)) \psi ^{\mathcal {I}}(x_0, \ldots , x_{m-1}, y)$ iff

$$ \begin{align*}\exists z (z= \bar{\mathsf{F}}(x_0) \land (\exists y \in \mathsf{snd}(z)) \psi^{\mathcal{I}}(x_0, \ldots, x_{m-1}, y))\end{align*} $$

$$ \begin{align*}\textrm{iff } \forall z(z= \bar{\mathsf{F}}(x_0) \Rightarrow (\exists y \in \mathsf{snd}(z)) \psi^{\mathcal{I}}(x_0, \ldots, x_{m-1}, y)).\end{align*} $$

Therefore, since $\mathcal {M}$ satisfies $\Pi _n$ -Collection, $\phi ^{\mathcal {I}}(x_0, \ldots , x_{m-1})$ is equivalent to a $\Delta _{n+1}$ -formula in $\mathcal {M}$ . The lemma now follows by induction.

Proof. Let $\phi (x, \vec {z})$ be a $ \Delta _0(\mathcal {L}^*_{\mathsf {S}})$ -formula. Let $\vec {v}$ be a finite sequence of sets and/or urelements of $\mathfrak {A}_{\mathcal {N}}$ and a a set of $\mathfrak {A}_{\mathcal {N}}$ . Work inside $\mathcal {M}$ . Now, $a= \langle 1, a_0 \rangle $ . Let

$$\begin{align*}b_0=\{x \in a_0 \mid \phi^{\mathcal{I}}(x, \vec{v})\}, \end{align*}$$

which is a set by $\Delta _{n+1}\textsf {-Separation}$ . Let $b=\langle 1, b_0 \rangle $ . Therefore, for all x such that $\mathsf {Set}(x)$ ,

$$\begin{align*}x \mathcal{E} b \textrm{ if and only if } x \mathcal{E} a \land \phi^{\mathcal{I}}(x, \vec{v}). \end{align*}$$

This shows that $\mathfrak {A}_{\mathcal {N}}$ satisfies $\Delta _0(\mathcal {L}^*_{\mathsf {S}})\textsf {-Separation}$ .

Proof. Let $\phi (x, y, \vec {z})$ be a $\Delta _0(\mathcal {L}^*_{\mathsf {S}})$ -formula. Let $\vec {v}$ be a finite sequence of sets and/or urelements of $\mathfrak {A}_{\mathcal {N}}$ and let a be a set of $\mathfrak {A}_{\mathcal {N}}$ such that

$$\begin{align*}\mathfrak{A}_{\mathcal{N}} \models (\forall x \in a) \exists y \phi(x, y, \vec{v}). \end{align*}$$

Work inside $\mathcal {M}$ . Now, $a= \langle 1, a_0 \rangle $ . And,

$$\begin{align*}(\forall x \mathcal{E} a) \exists y ((\mathsf{N}(y) \lor \mathsf{Set}(y)) \land \phi^{\mathcal{I}}(x, y, \vec{v})). \end{align*}$$

So,

$$\begin{align*}(\forall x \in a_0) \exists y((\mathsf{N}(y) \lor \mathsf{Set}(y)) \land \phi^{\mathcal{I}}(x, y, \vec{v})). \end{align*}$$

Since $(\mathsf {N}(y) \lor \mathsf {Set}(y)) \land \phi ^{\mathcal {I}}(x, y, \vec {v})$ is equivalent to a $\Sigma _{n+1}$ -formula, we can use $\Pi _n\textsf {-Collection}$ to find $b_0$ such that

$$\begin{align*}(\forall x \in a_0) (\exists y \in b_0)((\mathsf{N}(y) \lor \mathsf{Set}(y)) \land \phi^{\mathcal{I}}(x, y, \vec{v})). \end{align*}$$

Let $b_1= \{y \in b_0 \mid \mathsf {N}(y) \lor \mathsf {Set}(y)\}$ , which is a set by $\Sigma _1\textsf {-Separation}$ . Let $b= \langle 1, b_1 \rangle $ . Therefore, $\mathsf {Set}(b)$ and

$$\begin{align*}(\forall x \mathcal{E} a)(\exists y \mathcal{E} b) \phi^{\mathcal{I}}(x, y, \vec{v}). \end{align*}$$

So,

$$\begin{align*}\mathfrak{A}_{\mathcal{N}} \models (\forall x \in a) (\exists y \in b) \phi(x, y, \vec{v}). \end{align*}$$

This shows that $\mathfrak {A}_{\mathcal {N}}$ satisfies $\Delta _0(\mathcal {L}^*_{\mathsf {S}})\textsf {-Collection}$ .

Proof. Let $\phi (x, \vec {z})$ be a $\Sigma _1(\mathcal {L}^*_{\mathsf {S}})$ -formula. Let $\vec {v}$ be a sequence of sets and/or urelements such that

$$\begin{align*}\{x \in \mathfrak{A}_{\mathcal{N}} \mid \mathfrak{A}_{\mathcal{N}} \models \phi(x, \vec{v}) \} \textrm{ is nonempty}. \end{align*}$$

Work inside $\mathcal {M}$ . Consider $\theta (\alpha , \vec {z})$ defined by

$$\begin{align*}(\alpha \textrm{ is an ordinal}) \land \exists x((\mathsf{N}(x) \lor \mathsf{Set}(x)) \land \rho(x)= \alpha \land \phi^{\mathcal{I}}(x, \vec{z})). \end{align*}$$

Note that $\theta (\alpha , \vec {z})$ is equivalent to a $\Sigma _{n+1}$ -formula and $\exists \alpha \theta (\alpha , \vec {v})$ . Therefore, using $\Sigma _{n+1}\textsf {-Foundation}$ , let $\beta $ be a $\in $ -least element of

$$\begin{align*}\{ \alpha \in M \mid \mathcal{M} \models \theta(\alpha, \vec{v})\}. \end{align*}$$

Let y be such that $(\mathsf {N}(y) \lor \mathsf {Set}(y))$ , $\rho (y)=\beta $ and $\phi ^{\mathcal {I}}(y, \vec {v})$ . Note that if $x \mathcal {E} y$ , then $\rho (x) < \rho (y)$ . Therefore y is an $\mathcal {E}$ -least element of

$$\begin{align*}\{x \in \mathfrak{A}_{\mathcal{N}} \mid \mathfrak{A}_{\mathcal{N}} \models \phi(x, \vec{v}) \}.\\[-34pt] \end{align*}$$

The results of [Reference Barwise2, Appendix Section 3] show that $\mathbb {C}\mathrm {ov}_{\mathcal {M}}$ is the $\mathcal {L}^*$ -reduct of the well-founded part of $\mathfrak {A}_{\mathcal {N}}$ .

We can extend this result to show that $\mathrm {WF}(\mathfrak {A}_{\mathcal {N}})$ is an n-admissible cover of $\mathcal {N}$ and, therefore, isomorphic to $\mathbb {C}\mathrm {ov}^n_{\mathcal {M}}$ .

Proof. Theorem 4.9, the fact that $\mathcal {M} \models \mathsf {KP}$ , and the fact that $\mathrm {WF}(\mathfrak {A}_{\mathcal {N}})$ is well-founded imply that $\mathrm {WF}(\mathfrak {A}_{\mathcal {N}})$ satisfies $\mathsf {KPU}^\prime _{\mathbb {C}\mathrm {ov}}+\mathcal {L}^*_{\mathsf {S}}\textsf {-Foundation}$ . The definition of $\bar {\mathsf {S}}$ in $\mathcal {M}$ ensures that $\mathrm {WF}(\mathfrak {A}_{\mathcal {N}})$ satisfies $n\textsf {-Sat}$ . If a is a set $\mathrm {WF}(\mathfrak {A}_{\mathcal {N}})$ and b is a set in $\mathfrak {A}_{\mathcal {N}}$ with $\mathfrak {A}_{\mathcal {N}} \models (b \subseteq a)$ , then $b \in \mathrm {WF}(\mathsf {Set}^{\mathcal {M}})$ . Therefore, since $\Delta _0(\mathcal {L}^*_{\mathsf {S}})$ -formulae are absolute between $\mathrm {WF}(\mathfrak {A}_{\mathcal {N}})$ and $\mathfrak {A}_{\mathcal {N}}$ , $\mathrm {WF}(\mathfrak {A}_{\mathcal {N}})$ satisfies $\Delta _0(\mathcal {L}^*_{\mathsf {S}})\textsf {-Separation}$ . To show that $\mathrm {WF}(\mathfrak {A}_{\mathcal {N}})$ satisfies $\Delta _0(\mathcal {L}^*_{\mathsf {S}})\textsf {-Collection}$ , let $\phi (x, y, \vec {z})$ be a $\Delta _0(\mathcal {L}^*_{\mathsf {S}})$ -formula. Let $\vec {v}$ be sets and/or urelements in $\mathrm {WF}(\mathfrak {A}_{\mathcal {N}})$ and let a be a set of $\mathrm {WF}(\mathfrak {A}_{\mathcal {N}})$ such that

$$\begin{align*}\mathrm{WF}(\mathfrak{A}_{\mathcal{N}}) \models (\forall x \in a)\exists y \phi(x, y, \vec{v}). \end{align*}$$

Consider the formula $\theta (\beta , \vec {z})$ defined by

$$\begin{align*}(\beta \textrm{ is an ordinal}) \land (\forall x \in a)(\exists \alpha \in \beta)\exists y (\rho(y)=\alpha \land \phi(x, y, \vec{z})). \end{align*}$$

Note that if $\beta $ is a nonstandard ordinal of $\mathfrak {A}_{\mathcal {N}}$ , then $\mathfrak {A}_{\mathcal {N}} \models \theta (\beta , \vec {v})$ . Using $\Delta _0(\mathcal {L}^*_{\mathsf {S}})\textsf {-Collection}$ , $\theta (\beta , \vec {z})$ is equivalent to a $\Sigma _1(\mathcal {L}^*_{\mathsf {S}})$ -formula in $\mathfrak {A}_{\mathcal {N}}$ . Therefore, by $\Sigma _1(\mathcal {L}^*_{\mathsf {S}})\textsf {-Foundation}$ in $\mathfrak {A}_{\mathcal {N}}$ , $\{ \beta \mid \mathfrak {A}_{\mathcal {N}} \models \theta (\beta , \vec {v})\}$ has a least element $\gamma $ . Note that $\gamma $ must be an ordinal in $\mathrm {WF}(\mathfrak {A}_{\mathcal {N}})$ . Consider the formula $\psi (x, y, \vec {z}, \gamma )$ defined by $\phi (x, y, \vec {z}) \land (\rho (y) < \gamma )$ . Then,

$$\begin{align*}\mathfrak{A}_{\mathcal{N}} \models (\forall x \in a) \exists y \psi(x, y, \vec{v}, \gamma). \end{align*}$$

By $\Delta _0(\mathcal {L}^*_{\mathsf {S}})\textsf {-Collection}$ in $\mathfrak {A}_{\mathcal {N}}$ , there is a set b of $\mathfrak {A}_{\mathcal {N}}$ such that

$$\begin{align*}\mathfrak{A}_{\mathcal{N}} \models (\forall x \in a) (\exists y \in b) \psi(x, y, \vec{v}, \gamma). \end{align*}$$

Let $c= \{ y \in b \mid \rho (y)< \gamma \}$ , which is a set in $\mathfrak {A}_{\mathcal {N}}$ by $\Delta _1(\mathcal {L}^*_{\mathsf {S}})\textsf {-Separation}$ . Now, c is a set of $\mathrm {WF}(\mathfrak {A}_{\mathcal {N}})$ and

$$\begin{align*}\mathrm{WF}(\mathfrak{A}_{\mathcal{N}}) \models (\forall x \in a) (\exists y \in c) \phi(x, y, \vec{v}). \end{align*}$$

Therefore, $\mathrm {WF}(\mathfrak {A}_{\mathcal {N}})$ satisfies $\Delta _0(\mathcal {L}^*_{\mathsf {S}})\textsf {-Collection}$ , and so is an n-admissible set covering $\mathcal {N}$ . Since the $\mathcal {L}^*$ -reduct of $\mathrm {WF}(\mathfrak {A}_{\mathcal {N}})$ is isomorphic to $\mathbb {C}\mathrm {ov}_{\mathcal {M}}$ , $\mathrm {WF}(\mathfrak {A}_{\mathcal {N}})$ is isomorphic to $\mathbb {C}\mathrm {ov}^n_{\mathcal {M}}$ .

To summarise, we have proved the following.

Our analysis also yields the following version of [Reference Barwise2, Appendix Corollary 2.4], which plays an important role on compactness arguments.

In particular, we obtain:

Lemma 4.13. Let $\mathcal {M}= \langle M, \mathsf {E}^{\mathcal {M}} \rangle $ be such that $\mathcal {M}\models \mathsf {KP}+\Pi _n\textsf {-Collection}+\Sigma _{n+1}\textsf {-Foundation}$ . Let $T_0$ be an $\mathcal {L}_{\mathbb {C}\mathrm {ov}^n_{\mathcal {M}}}^{\mathtt {ee}}$ -theory. If $T_0 \in \mathbb {C}\mathrm {ov}^n_{\mathcal {M}}$ , then there exists $b \in M$ such that

$$\begin{align*}b^*= \{ a \in M \mid \bar{a} \textrm{ is mentioned in } T_0\}. \end{align*}$$

The next result connects definability in $\mathcal {M}$ with definability in $\mathbb {C}\mathrm {ov}^n_{\mathcal {M}}$ .

Lemma 4.14. Let $\mathcal {M}= \langle M, \mathsf {E}^{\mathcal {M}} \rangle $ be such that $\mathcal {M}\models \mathsf {KP}+\Pi _n\textsf {-Collection}+\Sigma _{n+1}\textsf {-Foundation}$ . Let $\phi (\vec {z})$ be a $\Sigma _{n+1}$ -formula. Then there exists a $\Sigma _1(\mathcal {L}^*_{\mathsf {S}})$ -formula $\hat {\phi }(\vec {z})$ such that for all $\vec {z} \in M$ ,

$$\begin{align*}\mathcal{M} \models \phi(\vec{z}) \textrm{ if and only if } \mathbb{C}\mathrm{ov}^n_{\mathcal{M}} \models \hat{\phi}(\vec{z}). \end{align*}$$

Proof. Let $\theta (x, \vec {z})$ be $\Pi _n$ such that $\phi (\vec {z})$ is $\exists x \theta (x, \vec {z})$ . Let $q \in \omega $ be such that $q= \ulcorner \neg \theta (\vec {z}) \urcorner $ . Let $z_0, \ldots , z_{m-1} \in M$ . Then

$$\begin{align*}\mathcal{M} \models \phi(z_0, \ldots, z_{m-1}) \textrm{ if and only if } \mathbb{C}\mathrm{ov}^n_{\mathcal{M}} \models \exists x \exists z(z= \langle x, z_0, \ldots, z_{m-1} \rangle \land \neg \mathsf{S}(q, z)). \end{align*}$$

Theorem 4.15. Let S be a recursively enumerable $\mathcal {L}$ -theory such that

$$\begin{align*}S \vdash \mathsf{KP}+\Pi_n\textsf{-Collection}+\Sigma_{n+1}\textsf{-Foundation}, \end{align*}$$

and let $\mathcal {M}= \langle M, \mathsf {E}^{\mathcal {M}} \rangle $ be a countable model of S. Then there exists an $\mathcal {L}$ -structure $\mathcal {N}= \langle N, \mathsf {E}^{\mathcal {N}} \rangle $ such that $\mathcal {M} \prec _{e, n} \mathcal {N} \models S$ and there exists $d \in N$ such that for all $x \in M$ , $\mathcal {N} \models (x \in d)$ .

Proof. Let T be the $\mathcal {L}^{\mathtt {ee}}_{\mathbb {C}\mathrm {ov}^n_{\mathcal {M}}}$ -theory that contains:

• • S;

• • for all $a, b \in M$ with $\mathcal {M} \models (a \in b)$ , $\bar {a} \in \bar {b}$ ;

• • for all $a \in M$ ,

  $$\begin{align*}\forall x \left(x \in a \iff \bigvee_{b \in a} (x=\bar{b}) \right); \end{align*}$$

• • for all $a \in M$ , $\bar {a} \in \mathbf {c}$ ;

• • for all $\Pi _n$ -formulae, $\phi (x_0, \ldots , x_{m-1})$ , and for all $a_0, \ldots , a_{m-1} \in M$ such that $\mathcal {M} \models \phi (a_0, \ldots , a_{m-1})$ ,

  $$\begin{align*}\phi(\bar{a}_0, \ldots, \bar{a}_{m-1}). \end{align*}$$

Since $\mathsf {S}$ is a satisfaction class for $\Sigma _n$ -formulae (and hence $\Pi _n$ -formula) of $\mathcal {M}$ in $\mathbb {C}\mathrm {ov}^n_{\mathcal {M}}$ , $T \subseteq \mathbb {C}\mathrm {ov}^n_{\mathcal {M}}$ is $\Sigma _1(\mathcal {L}^*_{\mathsf {S}})$ over $\mathbb {C}\mathrm {ov}^n_{\mathcal {M}}$ . Let $T_0 \subseteq T$ be such that $T_0 \in \mathbb {C}\mathrm {ov}^n_{\mathcal {M}}$ . Using Lemma 4.13, let $c \in M$ be such that

$$\begin{align*}c^*= \{ a \in M \mid \bar{a} \textrm{ is mentioned in } T_0\}. \end{align*}$$

Interpreting each $\bar {a}$ that is mentioned in $T_0$ by $a \in M$ and interpreting $\mathbf {c}$ by c, we expand $\mathcal {M}$ to a model of $T_0$ . Therefore, by the Barwise Compactness theorem, there exists $\mathcal {N} \models T$ . The $\mathcal {L}$ -reduct of $\mathcal {N}$ is the desired extension of $\mathcal {M}$ .

5 Well-founded models of collection

In this section we use Theorem 4.15 to show that for all $n \geq 1$ , $\mathsf {M}+\Pi _n\textsf {-Collection}+\Pi _{n+1}\textsf {-Foundation}$ proves $\Sigma _{n+1}\textsf {-Separation}$ . In particular, the theories $\mathsf {M}+\Pi _n\textsf {-Collection}$ and $\mathsf {M}+\textsf {Strong } \Pi _n\textsf {-Collection}$ have the same well-founded models.

In order to be able to apply Theorem 4.15 to countable models of $\mathsf {M}+\Pi _n\textsf {-Collection}+\Pi _{n+1}\textsf {-Foundation}$ , we first need to show that $\mathsf {M}+\Pi _n\textsf {-Collection}+\Pi _{n+1}\textsf {-Foundation}$ proves $\Sigma _{n+1}\textsf {-Foundation}$ . The proof presented here generalises the argument presented in [Reference Enayat and McKenzie5, Section 3] showing that $\mathsf {KP}^{\mathcal {P}}$ proves $\Sigma _1^{\mathcal {P}}\textsf {-Foundation}$ .

Definition 5.1. Let $\phi (x, y, \vec {z})$ be an $\mathcal {L}$ -formula. Define $\delta ^\phi (a, b, f)$ to be the $\mathcal {L}$ -formula:

$$\begin{align*}\begin{array}{c} (a \in \omega) \land (f \textrm{ is a function}) \land \mathsf{dom}(f)=a+1 \land f(0)= \{b\} \land\\ (\forall u \in \omega) \left(\!\!\!\begin{array}{c} (\forall x \in f(u))(\exists y \in f(u+1)) \phi(x, y, \vec{z})\\ (\forall y \in f(u+1))(\exists x \in f(u)) \phi(x, y, \vec{z}) \end{array}\!\!\!\right) \end{array}. \end{align*}$$

Define $\delta _\omega ^\phi (b, f, \vec {z})$ to be the $\mathcal {L}$ -formula:

$$\begin{align*}\begin{array}{c} (f \textrm{ is a function}) \land \mathsf{dom}(f)= \omega \land f(0)= \{b\} \land\\ (\forall u \in \omega) \left(\!\!\!\begin{array}{c} (\forall x \in f(u))(\exists y \in f(u+1)) \phi(x, y, \vec{z})\\ (\forall y \in f(u+1))(\exists x \in f(u)) \phi(x, y, \vec{z}) \end{array}\!\!\!\right) \end{array}. \end{align*}$$

Viewing $\vec {z}$ as parameters and letting $a \in \omega $ , $\delta ^\phi (a, b, f, \vec {z})$ says that f describes a family of directed paths of length $a+1$ starting at b through the directed graph defined by $\phi (x, y, \vec {z})$ . Similarly, viewing $\vec {z}$ as parameters, $\delta _\omega ^\phi (b, f, \vec {z})$ says that f describes a family of directed paths of length $\omega $ starting at b through the directed graph defined by $\phi (x, y, \vec {z})$ . Note that if $\phi (x, y, \vec {z})$ is $\Delta _0$ , then, in the theory $\mathsf {M}^-$ , both $\delta ^\phi (a, b, f \vec {z})$ and $\delta _\omega ^\phi (b, f, \vec {z})$ can be written as a $\Delta _0$ -formulae with parameter $\omega $ . Moreover, if $n \geq 1$ and $\phi (x, y, \vec {z})$ is a $\Sigma _n$ -formula ( $\Pi _n$ -formula), then, in the theory $\mathsf {M}^-+\Pi _{n-1}\textsf {-Collection}$ , both $\delta ^\phi (a, b, f \vec {z})$ and $\delta _\omega ^\phi (b, f, \vec {z})$ can be written as a $\Sigma _n$ -formulae ( $\Pi _n$ -formulae, respectively) with parameter $\omega $ .

The following generalises Rathjen’s $\Delta _0$ -weak dependent choices scheme from [Reference Rathjen14]:

• ( $\Delta _0\textrm {-}\mathsf {WDC}_\omega $ ) For all $\Delta _0$ -formulae, $\phi (x, y, \vec {z})$ ,

  $$\begin{align*}\forall \vec{z}(\forall x \exists y \phi(x, y, \vec{z}) \Rightarrow \forall w \exists f \delta_\omega^\phi(w, f, \vec{z})); \end{align*}$$

and for all $n \geq 1$ ,

• ( $\Delta _n\textrm {-}\mathsf {WDC}_\omega $ ) for all $\Pi _n$ -formulae, $\phi (x, y, \vec {z})$ , and for all $\Sigma _n$ -formulae, $\psi (x, y, \vec {z})$ ,

  $$\begin{align*}\forall \vec{z}(\forall x \forall y(\phi(x, y, \vec{z}) \iff \psi(x, y, \vec{z})) \Rightarrow (\forall x \exists y \phi(x, y, \vec{z}) \Rightarrow \forall w \exists f \delta_\omega^\phi(w, f, \vec{z}))). \end{align*}$$

The following is based on the proof of [Reference Rathjen14, Proposition 3.2].

Proof. Let T be the theory $\mathsf {KP}+\Pi _{n-1}\textsf {-Collection}+\Sigma _{n}\textsf {-Foundation}+\Delta _{n+1}\textrm {-} \mathsf {WDC}_\omega $ . Assume, for a contradiction, that $\mathcal {M}= \langle M, \in ^{\mathcal {M}} \rangle $ is such that $\mathcal {M} \models T$ and there is an instance of $\Sigma _{n+1}\textsf {-Foundation}$ that is false in $\mathcal {M}$ . Let $\phi (x, y, \vec {z})$ be a $\Pi _n$ -formula and let $\vec {a} \in M$ be such that

$$\begin{align*}\{x \mid \mathcal{M} \models \exists y \phi(x, y, \vec{a}) \} \end{align*}$$

is nonempty and has no $\in $ -minimal element. Let $b, d \in M$ be such that $\mathcal {M} \models \phi (b, d, \vec {a})$ . Now,

$$\begin{align*}\mathcal{M} \models \forall x \forall u \exists y \exists v (\phi(x, u, \vec{a}) \Rightarrow (y \in x) \land \phi(y, v, \vec{a})). \end{align*}$$

Therefore, $\mathcal {M} \models \forall x \exists y \theta (x, y, \vec {a})$ where $\theta (x, y, \vec {a})$ is the formula

$$\begin{align*}x= \langle x_0, x_1 \rangle \land y= \langle y_0, y_1 \rangle \land (\phi(x_0, x_1, \vec{a}) \Rightarrow (y_0 \in x_0) \land \phi(y_0, y_1, \vec{a})). \end{align*}$$

So, $\theta (x, y, \vec {a})$ is $\Delta _{n+1}^T$ . Work inside $\mathcal {M}$ . Using $\Delta _{n+1}\textrm {-}\mathsf {WDC}_\omega $ , let f be such that $\delta _\omega ^\theta (\langle b, d \rangle , f, \vec {a})$ . Note that $\Sigma _n\textsf {-Foundation}$ implies that for all $n \in \omega $ ,

1. (i) $f(n) \neq \emptyset $ ;

2. (ii) for all $x \in f(n)$ , $x= \langle x_0, x_1 \rangle $ and $\phi (x_0, x_1, \vec {a})$ .

Therefore, for all $n \in \omega $ ,

$$\begin{align*}\begin{array}{c} (\forall x \in f(n))(\exists y \in f(n+1))(x= \langle x_0, x_1 \rangle \land y= \langle y_0, y_1 \rangle \land y_0 \in x_0) \land\\ (\forall y \in f(n+1))(\exists x \in f(n))(x= \langle x_0, x_1 \rangle \land y= \langle y_0, y_1 \rangle \land y_0 \in x_0) \end{array}. \end{align*}$$

Let $B= \mathsf {TC}(\{b\})$ . $\textsf {Set-Foundation}$ implies that for all $n \in \omega $ ,

$$\begin{align*}(\forall x \in f(n))(x=\langle x_0, x_1 \rangle \land x_0 \in B). \end{align*}$$

Let

$$\begin{align*}A= \left\{ x \in B \left| (\exists n \in \omega)(\exists z \in f(n))\left(\exists y \in \bigcup z\right)(z= \langle x, y\rangle) \right. \right\}, \end{align*}$$

which is a set by $\Delta _0\textsf {-Separation}$ . Now, let $x \in A$ . Therefore, there exists $n \in \omega $ and $z \in f(n)$ such that $z= \langle x, x_0 \rangle $ . And, there exists $w \in f(n+1)$ such that $w= \langle y, y_0 \rangle $ and $y \in x$ . But $y \in A$ . So A is a set with no $\in $ -minimal element, which is the desired contradiction.

The following refinement of Definition 5.1 will allow us to show that for $n \geq 1$ , $\mathsf {M}+\Pi _n\textsf {-Collection}+\Pi _{n+1}\textsf {-Foundation}$ proves $\Delta _{n+1}\textrm {-}\mathsf {WDC}_\omega $ .

Definition 5.2. Let $\phi (x, y, \vec {z})$ be an $\mathcal {L}$ -formula. Define $\eta ^\phi (a, b, f, \vec {z})$ to the $\mathcal {L}$ -formula:

$$\begin{align*}\begin{array}{c} \delta^\phi(a, b, f, \vec{z}) \land\\ (\forall u \in a) \exists \alpha \exists X \left(\!\!\!\begin{array}{c} (\alpha \textrm{ is an ordinal}) \land (X= V_\alpha) \land\\ (\forall x \in f(u+1))(x \in X)\\ (\forall y \in X)(\forall x \in f(u))(\phi(x, y, \vec{z}) \Rightarrow y \in f(u+1)) \land\\ (\forall \beta \in \alpha)(\forall Y \in X)\left(\!\!\! \begin{array}{c} Y= V_\beta \Rightarrow\\ (\exists x \in f(u))(\forall y \in Y) \neg \phi(x, y, \vec{z}) \end{array}\!\!\!\right) \end{array}\!\!\!\right). \end{array} \end{align*}$$

The formula $\eta ^\phi (a, b, f, \vec {z})$ says that f is a function with domain $a+1$ and for all $u \in a$ , $f(u+1)$ is the set of $y \in V_\alpha $ such that there exists $x \in f(u)$ with $\phi (x, y, \vec {z})$ and $\alpha $ is least such that for all $x \in f(u)$ , there exists $y \in V_\alpha $ such that $\phi (x, y, \vec {z})$ . In the theory $\mathsf {M}+ \Pi _1\textsf {-Collection}+\Pi _2\textsf {-Foundation}$ , ‘ $X=V_\alpha $ ’ can be expressed as both a $\Sigma _2$ -formula and a $\Pi _2$ -formula. If $n \geq 1$ and, for given parameters $\vec {c}$ , $\phi (x, y, \vec {c})$ is equivalent to both a $\Sigma _{n+1}$ -formula and a $\Pi _{n+1}$ -formula, then, in the theory $\mathsf {M}+ \Pi _n\textsf {-Collection}+\Pi _2\textsf {-Foundation}$ , $\eta ^\phi (a, b, f, \vec {z})$ is equivalent to a $\Sigma _{n+1}$ -formula.

Proof. Work in the theory $\mathsf {M}+\Pi _n\textsf {-Collection}+\Pi _{n+1}\textsf {-Foundation}$ . Let $\phi (x, y, \vec {z})$ be a $\Pi _{n+1}$ -formula. Let $\vec {a}$ , b be sets and let $\theta (x, y, \vec {z})$ be a $\Sigma _{n+1}$ -formula such that

$$\begin{align*}\forall x \forall y(\phi(x, y, \vec{a}) \iff \theta(x, y, \vec{a})). \end{align*}$$

We begin by claiming that for all $m \in \omega $ , $\exists f \eta ^\phi (m, b, f, \vec {a})$ . Assume, for a contradiction, that this does not hold. Using $\Pi _{n+1}\textsf {-Foundation}$ , let $k \in \omega $ be least such that $\neg \exists f \eta ^\phi (k, b, f, \vec {a})$ . Since $k \neq 0$ , there exists a function g with $\mathsf {dom}(g)=k$ and $\eta ^\phi (k-1, b, g, \vec {a})$ . Consider the class

$$\begin{align*}A& = \{ \alpha \in \mathsf{Ord} \mid \forall X(X= V_\alpha \Rightarrow (\forall x \in g(k-1)) (\exists y \in X) \phi(x, y, \vec{a}))\}\\&= \{ \alpha \in \mathsf{Ord} \mid \exists X(X=V_\alpha \land (\forall x \in g(k-1))(\exists y \in X) \theta(x, y, \vec{a})) \}. \end{align*}$$

Applying $\Sigma _{n+1}\textsf {-Collection}$ to the formula $\theta (x, y, \vec {a})$ shows that A is nonempty. Moreover, $\Delta _{n+1}\textsf {-Foundation}$ ensures that there is a least element $\beta \in A$ . Now, let

$$\begin{align*}C= \{y \in V_\beta \mid (\exists x \in g(k-1)) \phi(x, y, \vec{a})\}, \end{align*}$$

which is a set by $\Delta _{n+1}\textsf {-Separation}$ . Let $f= g \cup \{\langle k, C \rangle \}$ . Then f is such that $\eta ^\phi (k, b, f, \vec {a})$ , which contradicts our assumption that no such f exists. Therefore, for all $m \in \omega $ , $\exists f \eta ^\phi (m, b, f, \vec {a})$ . Using $\Sigma _{n+1}\textsf {-Collection}$ , let D be such that $(\forall m \in \omega )(\exists f \in D) \eta ^\phi (m, b, f, \vec {a})$ . Note that for all $m \in \omega $ and for all functions f and g, if $\eta ^\phi (m, b, f, \vec {a})$ and $\eta ^\phi (m, b, g, \vec {a})$ , then $f=g$ . Now, let

$$\begin{align*}h= \{\langle m, X \rangle \in \omega \times \mathsf{TC}(D) \mid (\exists f \in D)(\eta^\phi(m, b, f, \vec{a}) \land f(m)= X)\}. \end{align*}$$

Since

$$\begin{align*}h= \{\langle m, X \rangle \in \omega \times \mathsf{TC}(D) \mid (\forall f \in D)(\eta^\phi(m, b, f, \vec{a}) \Rightarrow f(m)=X) \}, \end{align*}$$

h is a set by $\Delta _{n+1}\textsf {-Separation}$ . Now, h is the function required by $\Delta _{n+1}\textsf {-}\mathsf {WDC}_\omega $ .

Note $\Pi _{n+1}\textsf {-Foundation}$ is only used in the proof of Theorem 5.2 to find the least element of a $\Pi _{n+1}$ -definable subclass of naturals numbers. Therefore, the proof of Theorem 5.2 also yields the following result.

Theorem 5.3. Let $n \in \omega $ with $n \geq 1$ . Let $\mathcal {M}$ be an $\omega $ -standard model of $\mathsf {M}+\Pi _n\textsf {-Collection}+\Pi _2\textsf {-Foundation}$ . Then

$$\begin{align*}\mathcal{M} \models \Delta_{n+1}\textsf{-}\mathsf{WDC}_\omega. \end{align*}$$

Note that $\Pi _2\textsf {-Foundation}$ coupled with $\Pi _1\textsf {-Collection}$ ensures that the function $\alpha \mapsto V_\alpha $ is total.

Combining Theorem 5.1 with Theorems 5.2 and 5.3 yields:

Corollary 5.5. Let $n \in \omega $ with $n \geq 2$ . Let $\mathcal {M}$ be an $\omega $ -standard model of $\mathsf {M}+\Pi _n\textsf {-Collection}$ . Then

$$\begin{align*}\mathcal{M} \models \Sigma_{n+1}\textsf{-Foundation}. \end{align*}$$

The proof of [Reference Enayat and McKenzie5, Theorem 3.11] shows how the use of the cumulative hierarchy can be avoided in the argument used in the proof of Theorem 5.2. The following is [Reference Enayat and McKenzie5, Corollary 3.12] combined with [Reference Mathias13, Proposition Scheme 6.12] and provides a version of Corollary 5.5 when $n=1$ .

Theorem 5.6. Let $\mathcal {M}$ be an $\omega $ -standard model of $\mathsf {MOST}+\Pi _1\textsf {-Collection}$ . Then

$$\begin{align*}\mathcal{M} \models \Sigma_{2}\textsf{-Foundation}. \end{align*}$$

Equipped with these results, we are now able to show that, in the theory $\mathsf {M}+\Pi _n\textsf {-Collection}$ , $\Pi _{n+1}\textsf {-Foundation}$ implies $\Sigma _{n+1}\textsf {-Separation}$ .

Proof. Assume that $\mathcal {M} \prec _{e, 1} \mathcal {N}$ . Let $x \in M$ and let $y \in N$ with $\mathcal {N} \models (y \subseteq x)$ . We need to show that $y \in M$ . Let $a \in M$ be such that $\mathcal {M} \models (a= \mathcal {P}(x))$ . Therefore, $\mathcal {M} \models \theta (x, a)$ where $\theta (x, a)$ is the $\Pi _1$ -formula

$$\begin{align*}\forall z(z \subseteq x \iff z \in a). \end{align*}$$

So, $\mathcal {N} \models \theta (x, a)$ . Therefore, $\mathcal {N} \models (y \in a)$ and so $y \in M$ .

As alluded to in [Reference Mathias13, Remark 3.21], the theory $\mathsf {KP}+\Sigma _1\textsf {-Separation}$ is capable of endowing any well-founded partial order with a ranking function.

Proof. Work in the theory $\mathsf {KP}+\Sigma _1\textsf {-Separation}$ . Let X be a set and $R \subseteq X \times X$ be such that $\langle X, R \rangle $ is a well-founded strict partial order. Let $\theta (x, g, X, R)$ be the conjunction of the following clauses:

1. (i) g is a function;

2. (ii) $\mathsf {rng}(g)$ is a set of ordinals;

3. (iii) $\mathsf {dom}(g)= \{y \in X \mid \langle y, x \rangle \in R \lor y= x\}$ ;

4. (iv) $(\forall y, z \in \mathsf {dom}(g))(\langle y, z \rangle \in R \Rightarrow g(y) < g(z))$ ;

5. (v) $(\forall y \in \mathsf {dom}(g))(\forall \alpha \in g(y))(\exists z \in X)(\langle z, y \rangle \in R \land g(z) \geq \alpha )$ .

Note that $\theta (x, g, X, R)$ can be written as a $\Delta _0$ -formula. Moreover, for all $x \in X$ and functions $g_0$ and $g_1$ , if $\theta (x, g_0, X, R)$ and $\theta (x, g_1, X, R)$ , then $g_0=g_1$ . And, if $x, y \in X$ with $\langle x, y \rangle \in R$ and $g_0$ and $g_1$ are functions with $\theta (y, g_0, X, R)$ and $\theta (x, g_1, X, R)$ , then $g_0= g_1 \upharpoonright \mathsf {dom}(g_0)$ . Now, consider

$$\begin{align*}A= \{ x \in X \mid \neg \exists g \theta(x, g, X, R)\}, \end{align*}$$

which is a set by $\Pi _1\textsf {-Separation}$ . Assume, for a contradiction, that $A \neq \emptyset $ . Let $x_0 \in A$ be R-minimal. Let $B= \{y \in X \mid \langle y, x_0 \rangle \in R\}$ . Using $\Delta _0\textsf {-Collection}$ , let $C_0$ be such that $(\forall y \in B)(\exists g \in C_0) \theta (y, g, X, R)$ . Let

$$\begin{align*}D= \{ g \in C_0 \mid (\exists y \in B) \theta(y, g, X, R)\}. \end{align*}$$

Let

$$\begin{align*}\beta= \sup\{g(y)+1 \mid y \in B \textrm{ and } g \in D \textrm{ with } y \in \mathsf{dom}(g)\}. \end{align*}$$

Then $f= \bigcup D \cup \{ \langle x_0, \beta \rangle \}$ is such that $\theta (x_0, f, X, R)$ , which contradicts the fact that $x_0 \in A$ . Therefore, $A= \emptyset $ . Using $\Delta _0\textsf {-Collection}$ , let $C_1$ be such that $(\forall x \in X)(\exists g \in C_1) \theta (x, g, X, R)$ . Let

$$\begin{align*}F= \{ g \in C_1 \mid (\exists x \in X) \theta(x, g, X, R)\}. \end{align*}$$

Then $h= \bigcup F$ is the function we require.