Fact 2.1. (Downward Lowenheim-Skolem)

For every general structure $\mathcal {M}$ , there is a general structure $\mathcal {M}'\equiv \mathcal {M}$ such that $|M'|\le \aleph _0+|V|$ .

Fact 2.2. The union of an elementary chain of general structures $\langle \mathcal {M}_{\alpha }\colon \alpha <\beta \rangle $ is an elementary extension of each $\mathcal {M}_{\alpha }$ .

Fact 2.7. Let $\mathcal {M}_i$ be a general structure for each $i\in I$ , let $\mathcal {D}$ be an ultrafilter over I, and let $\mathcal {M}=\prod _{\mathcal {D}}\mathcal {M}_i$ be the ultraproduct. Then for each formula $\varphi $ and tuple $\vec {b}$ in the cartesian product $\prod _{i\in I} M_i$ ,

$$ \begin{align*}\varphi^{\mathcal{M}}(\vec{b}_{\mathcal{D}})=\lim\limits_{\mathcal{D}}\langle\varphi^{\mathcal{M}_i}(\vec{b}_i)\rangle_{i\in I}.\end{align*} $$

Fact 2.8. (Compactness)

If every finite subset of T has a general model, then T has a general model.

Proof. (i): Suppose not. Then for some $r\in (0,1]$ , is finitely satisfiable. By the Compactness Theorem, has a general model, so $T\not \models \{\varphi \}$ .

(ii): By (i), for each positive $n\in \mathbb {N}$ there is a finite subset $T_n$ of T such that . Then $T_{\infty }=\bigcup _n T_n$ is a countable subset of T, and $T_{\infty }\models \{\varphi \}$ . ⊣

Fact 2.13. (Uniqueness Theorem for Special Models)

If T is complete and $\mathcal {M}, \mathcal {N}$ are $\kappa $ -special models of T, then $\mathcal {M}$ and $\mathcal {N}$ are isomorphic.

Fact 2.14. (Existence Theorem for Special Models)

If $\kappa $ is a special cardinal and $\aleph _0+|V|<\kappa $ , then every reduced structure $\mathcal {M}$ such that $|M|\le \kappa $ has a $\kappa $ -special elementary extension.

Proof. Let V be the vocabulary of T. Then $|V|\le |T|+\aleph _0$ . Let U be the set of restricted continuous sentences $\theta $ in the vocabulary V such that $T\models \{\theta \}$ . Then $T\models U$ . Suppose $\mathcal {N}\models U$ but not $\mathcal {N}\models T$ . Then for some $\varphi \in T$ and some dyadic rational $r>0$ we have $\varphi ^{\mathcal {N}}\ge r$ . By Theorem 6.3 of [Reference Yaacov, Berenstein, Henson, Usvyatsov, Chatzidakis, Macpherson, Pillay and Wilkie18] (whose proof works for general structures as well as metric structures), there is a restricted continuous sentence $\theta $ in the vocabulary V such that $|\theta ^{\mathcal {M}}-\varphi ^{\mathcal {M}}|\le r/4$ for every general structure $\mathcal {M}$ . is also a restricted continuous sentence. Since $T\models \{\theta \}$ , we have , so . But since $\varphi ^{\mathcal {N}}\ge r$ , we have $\theta ^{\mathcal {N}}\ge r - r/4$ , so is not true in $\mathcal {N}$ , contradicting the assumption that $\mathcal {N}\models U$ . We conclude that $U\models T$ , so T is equivalent to U. ⊣

Fact 2.16. Let $(\mathcal {M},L)$ be a premetric structure with distinguished distance d. $\mathcal {M}$ is reduced if and only if for all $x,y\in M$ , if $d^{\mathcal {M}}(x,y)=0$ then $x=y$ .

Fact 2.18. (See [Reference CaicedoX. Caicedo and Iovino1], page 112.)

For each metric signature L over V, there is a theory $\operatorname {met}(L)$ whose general models are exactly the downgrades of premetric structures for L. Thus every premetric structure for L satisfies $\operatorname {met}(L)$ . Each sentence in $\operatorname {met}(L)$ consists of finitely many $\operatorname {sup} $ quantifiers followed by a quantifier-free formula.

Fact 2.20. Suppose V has countably many predicate symbols and T is a V-theory.

1. (i) (Theorem 3.3.4 of [Reference Keisler and Iovino12].) T has a premetric expansion.

2. (ii) (Proposition 4.4.1 of [Reference Keisler and Iovino12].) If $T_e$ a premetric expansion of T, $\mathcal {M}, \mathcal {N}\models T$ , and $\mathcal {M}\equiv \mathcal {N}$ , then $\mathcal {M}_e\equiv \mathcal {N}_e$ .

Fact 3.1. (Lemma 7.4.16 in [Reference Chang and Keisler2], with the order reversed.)

For every $g\colon I\to [0,1]$ ,

(1) $$ \begin{align} \operatorname {sup}\{\lim\limits_{\mathcal{D}}g\colon \mathcal{F}\subseteq\mathcal{D}\in\beta I\}=\inf\limits_{J\in\mathcal{F}}\operatorname {sup}_{i\in J}g(i). \end{align} $$

Proof. Let x denote the left side of (1), and y denote the right side of (1). For each $z\in (0,1)$ , let

$$ \begin{align*}K_z=\{i\in I\colon g(i)>z\}.\end{align*} $$

Suppose that $x>y$ , and take $x>z>y$ . Then there exists $\mathcal {D}\in \beta I$ such that $\mathcal {D}\supseteq \mathcal {F}$ and $\lim _{\mathcal {D}} g> z$ , and $J\in \mathcal {F}$ such that $z> \sup _{i\in J} g(i).$ Since $\lim _{\mathcal {D}} g> z$ we have $K_z\in \mathcal {D}$ . Since $J\in \mathcal {F}\subseteq \mathcal {D}$ we have $J\in \mathcal {D}$ . But then $K_z\cap J \in \mathcal {D}$ , so there exists $i\in J$ such that $z<g(i)$ , a contradiction.

Now suppose $x<y$ , and take $x<z<y$ . Since $z<y$ , for each $J\in \mathcal {F}$ there exists $i_J\in J$ such that $g(i_J)> z$ . Then $i_J\in J\cap K_z$ for each $J\in \mathcal {F}$ , so $\mathcal {F}\cup \{K_z\}$ generates a proper filter on I. Hence there is $\mathcal {D}\in \beta I$ for which $\mathcal {F}\subseteq \mathcal {D}$ and $K_z\in \mathcal {D}$ . Then $\{ i\in I\colon g(i)<z\}\notin \mathcal {D}$ , so we cannot have $\lim _{\mathcal {D}} g <z$ . Thus $x\ge \lim _{\mathcal {D}} g\ge z$ , and we again have a contradiction. ⊣

Proof. Let $\mathcal {M}=\prod ^{\mathcal {F}}\mathcal {M}_i$ , and $\mathcal {M}^0=\prod ^{\mathcal {F}}\mathcal {M}^0_i$ . By definition, $\prod _{\mathcal {F}}\mathcal {M}_i$ is the reduction of $\mathcal {M}$ , and $\prod _{\mathcal {F}}\mathcal {M}^0_i$ is the reduction of $\mathcal {M}^0$ . It is clear that $\mathcal {M}^0$ is the $V^0$ -part of $\mathcal {M}$ . Therefore, by Remark 2.5 (ii), $\prod _{\mathcal {F}}\mathcal {M}^0_i$ is isomorphic to the $V^0$ -part of $\prod _{\mathcal {F}}\mathcal {M}_i$ . ⊣

Proof. The property

is equivalent to the countable set of primitive conditional formulas

Using a similar trick, the property

and the uniform continuity property

$$ \begin{align*}\max\limits_{k\le n} d(x_k,y_k)< \delta \Rightarrow |P(\vec x) - P(\vec y)|\le \varepsilon\end{align*} $$

can be expressed by countable sets of primitive conditional formulas. The sentences in $\operatorname {met}(L)$ can then be expressed by countable sets of sentences obtained by putting $\sup $ quantifiers in front of primitive conditional formulas. ⊣

Proof. If $\varphi $ is primitive conditional, then $B\varphi $ is equivalent to the primitive conditional formula obtained from Equation (2) by replacing each connective $C_k$ by $B\circ C_k$ . The result for arbitrary conditional formulas $\varphi $ follows by an easy induction on the complexity of $\varphi $ . ⊣

Fact 3.8. (Lemma 7.4.20 in [Reference Chang and Keisler2].)

Let $y_0,\ldots ,y_n\colon I\to [0,1]$ , and $C_0,\ldots ,C_n$ be increasing unary connectives. Suppose

$$ \begin{align*}J':= \{i\in I\colon \min(C_0(y_0(i)),C_1(1-y_1(i)),\ldots,C_n(1-y_n(i)))=0\}\in\mathcal{F}.\end{align*} $$

Then

$$ \begin{align*}\min(C_0(\limsup\limits_{\mathcal{F}}y_0),C_1(1-\limsup\limits_{\mathcal{F}}y_1),\ldots,C_n(1-\limsup\limits_{\mathcal{F}}y_n))=0.\end{align*} $$

Proof. We may assume that $C_k(0)=0$ for $1\le k\le n$ , because otherwise we may remove $C_{\ell }$ when $\ell $ is the least $\ell \ge 1$ such that $C_{\ell }(0)>0$ . For $1\le k\le n$ , let $z_k=\sup \{z\colon C_k(z)=0\}$ . Since each $C_k$ is increasing and continuous, for $1\le k\le n$ we have $C_k(z)=0$ if and only if $z\le z_k$ .

Suppose that $C_k(1-\limsup _{\mathcal {F}} y_k)>0$ for each $1\le k\le n$ . We prove that $C_0(\limsup _{\mathcal {F}}y_0)=0$ . Fix $1\le k\le n$ . Then $1-\limsup _{\mathcal {F}}y_k> z_k$ , so

$$ \begin{align*}\inf\limits_{J\in\mathcal{F}}\sup_{i\in J}y_k(i)=\limsup\limits_{\mathcal{F}} y_k< 1-z_k.\end{align*} $$

Hence there exists $J\in \mathcal {F}$ such that for every $i\in J$ and $1\le k\le n$ , $y_k(i)<1-z_k$ , so $1-y_k(i)> z_k$ and $C_k(1-y_k(i))>0$ . Then $J'\cap J\in \mathcal {F}$ , and therefore

$$ \begin{align*}J'\cap J\subseteq \{i\in I\colon C_0(y_0(i))=0\}\in\mathcal{F}.\end{align*} $$

Hence there is a set $J''\in \mathcal {F}$ such that $C_0(y_0(i))=0$ for all $i\in J''$ . Then there exists z such that $C_0(z)=0$ . Since $C_0$ is increasing and continuous, we have $\{z\colon C_0(z)=0\}=[0,z_0]$ for some $z_0\in [0,1]$ . So $y_0(i)\le z_0$ for all $i\in J''$ , and thus $\limsup _{\mathcal {F}} y_0\le z_0$ . It follows that $C_0(\limsup _{\mathcal {F}} y_0)=0$ , as required. ⊣

Proof. It suffices to prove the result when $T=\{\psi \}$ for a single conditional sentence $\psi $ . Consider an indexed family $\langle \mathcal {M}_i\colon i\in I\rangle $ of general structures, and let $\mathcal {N}=\prod _{\mathcal {F}}\mathcal {M}_i$ be the reduced product. First suppose that $\varphi $ is a primitive conditional formula as in (2) above, and let $\vec {a}$ be a tuple of elements of the cartesian product $\prod _{i\in I} \mathcal {M}_i$ . For $k\le n$ put $y_k(i)=\alpha _k^{\mathcal {M}_k}(\vec {a}(i))$ . Then Fact 3.8 shows that

$$ \begin{align*}\{i\in I\colon \varphi^{\mathcal{M}_i}(\vec{a}(i))=0\}\in\mathcal{F}\Rightarrow\varphi^{\mathcal{N}}(\vec{a}_{\mathcal{F}})=0.\end{align*} $$

By Lemma 3.7, for each $\varepsilon \in [0,1]$ , is equivalent to a primitive conditional formula. Moreover, for every $\mathcal {M}$ , tuple $\vec {a}$ in M, and $\varepsilon $ , $\varphi ^{\mathcal {M}}(\vec {a})\le \varepsilon $ if and only if Therefore for each $\varepsilon \in [0,1]$ ,

(3) $$ \begin{align} \{i\in I\colon \varphi^{\mathcal{M}_i}(\vec{a}(i))\le\varepsilon\}\in\mathcal{F}\Rightarrow\varphi^{\mathcal{N}}(\vec{a}_{\mathcal{F}})\le\varepsilon. \end{align} $$

We now complete the proof of the theorem by proving the following stronger statement: for every conditional formula $\psi $ and $\varepsilon \in [0,1]$ ,

(4) $$ \begin{align} \{i\in I\colon \psi^{\mathcal{M}_i}(\vec{a}(i))\le\varepsilon\}\in\mathcal{F}\Rightarrow\psi^{\mathcal{N}}(\vec{a}_{\mathcal{F}})\le\varepsilon. \end{align} $$

The base case of the induction is taken care of by Equation (3). For the inductive step, we only treat the $\inf $ case, the remaining cases being similar. Suppose (4) holds for $\theta (\vec {x},y)$ , and $\psi (\vec {x})=\inf _y\theta (\vec {x},y)$ . Assume that

$$ \begin{align*}\{i\in I\colon\psi^{\mathcal{M}_i}(\vec{a}(i))\le\varepsilon\}\in\mathcal{F}.\end{align*} $$

Then for each $\delta>0$ there exists $b\in \prod _{i\in I}M_i$ such that

$$ \begin{align*}\{i\in I\colon\theta^{\mathcal{M}_i}(\vec{a}(i),b(i))\le\varepsilon+\delta\}\in\mathcal{F}.\end{align*} $$

By inductive hypothesis, $\theta ^{\mathcal {N}}(\vec {a}_{\mathcal {F}},b_{\mathcal {F}})\le \varepsilon +\delta $ . $\psi ^{\mathcal {N}}(\vec {a}_{\mathcal {F}})\le \epsilon +\delta $ . Since this holds for all $\delta>0$ , $\psi ^{\mathcal {N}}(\vec {a}_{\mathcal {F}})\le \varepsilon $ , as required. ⊣

Proof. (i) and (ii): By Fact 2.18, Lemma 3.6, and Theorem 3.9.

(iii): For each $i\in I$ , let $\mathcal {M}^0_i$ be the reduction of the $V^0$ -part of $\mathcal {M}_i$ , and let $\mathcal {M}^0$ be the reduction of $\prod _{\mathcal {F}} \mathcal {M}^0_i$ . By Remarks 3.3 and 3.4, $\mathcal {M}^0$ is isomorphic to the $V^0$ -part of $\prod _{\mathcal {F}} \mathcal {M}_i$ . By hypothesis, each $\mathcal {M}^0_i$ is a metric structure. By (ii), $\mathcal {M}^0$ is a premetric structure, and its distinguished distance $d^{\mathcal {M}}$ is a metric. As in [Reference Lopes14], one can show that $d^{\mathcal {M}}$ is a complete metric by modifying the proof of Proposition 5.3 in [Reference Yaacov, Berenstein, Henson, Usvyatsov, Chatzidakis, Macpherson, Pillay and Wilkie18].⊣

Proof. For each i, $(\mathcal {M}_i,L)$ is a metric structure. By Corollary 3.10, $(\prod _{\mathcal {F}}\mathcal {N}_i,L)$ is a premetric structure, and $(\prod _{\mathcal {F}}\mathcal {M}_i,L)$ is a metric structure. For each $i\in I$ , $\mathcal {N}_i$ is a dense substructure of $\mathcal {M}_i$ . Therefore, for each $a\in \prod _{i\in I} M_i$ and $\varepsilon>0$ there exists $b\in \prod _{i\in I} N_i$ such that for all $i\in I$ . The formula is conditional, so by Theorem 3.9 we have . Therefore $\prod _{\mathcal {F}}\mathcal {N}_i$ is dense in $\prod _{\mathcal {F}}\mathcal {M}_i$ . By definition, all reduced products are reduced structures. It follows that $(\prod _{\mathcal {F}}\mathcal {M}_i,L)$ is the completion of $(\prod _{\mathcal {F}}\mathcal {N}_i,L)$ . ⊣

Fact 3.12. ([Reference Galvin5], Theorems 3.4, 5.2, and §6)

(i) A first-order sentence $\theta $ is preserved under reduced products if and only if $\theta $ is equivalent to a Horn sentence.

(ii) A first-order theory is preserved under reduced products if and only if it is equivalent to a set of Horn sentences.

Proof. (i) $\Rightarrow $ (ii) is clear from the definition. By Theorem 3.9, (ii) $\rightarrow $ (iii).

Assume (iii). Each first-order structure $\mathcal {M}$ without identity may be considered as a general structure in which every formula has truth values in $\{0,1\}$ . By Remark 2.3 in [Reference Lopes14], if each $\mathcal {M}_i$ is a first-order structure, then the reduced product $\prod _{\mathcal {F}} \mathcal {M}_i$ considered as a first-order structure is the same as the reduced product $\prod _{\mathcal {F}} \mathcal {M}_i$ considered as a general structure. Therefore $\theta $ is preserved under first-order reduced products, so by Fact 3.12, (iv) holds. ⊣

Proof. For example, let $P, Q$ be $0$ -ary predicates, and let $\theta $ be the first-order sentence $P \vee Q\vee \neg (P\vee Q)$ . Then $\theta $ is equivalent to the Horn sentence $P\vee \neg P$ , but we will show that $\theta ^c$ is not preserved under continuous reduced products. $\theta ^c$ is the continuous sentence

Let $I=\{1,2\}$ and let $\mathcal {F}$ be the filter $\mathcal {F}=\{I\}$ (so reduced products modulo $\mathcal {F}$ are direct products). Let $0<r\le 1/2$ . Let $\mathcal {M}_1$ be a general structure such that $\mathcal {M}_1\models P=r$ and $\mathcal {M}_1\models Q=0$ . Let $\mathcal {M}_2$ be a general structure such that $\mathcal {M}_2\models P=0$ and $\mathcal {M}_2\models Q=r$ . We leave it to the reader to check that $\mathcal {M}_1\models \theta =0$ and $\mathcal {M}_2\models \theta =0$ , but $\prod _{\mathcal {F}}\mathcal {M}_i\models \theta =r$ .⊣

Fact 3.15. (Proposition 3.6 in [Reference Ghasemi6].) If $\mathcal {F}$ is a proper filter over I, and $\mathcal {M}_i, \mathcal {N}_i$ are metric structures with $\mathcal {M}_i\equiv \mathcal {N}_i$ for each $i\in I$ , then $\prod _{\mathcal {F}}\mathcal {M}_i\equiv \prod _{\mathcal {F}}\mathcal {N}_i$ .

Proof. By Fact 2.20 (i), the empty theory T has a premetric expansion $T_e$ . For each $i\in I$ , $\mathcal {M}_{ie}$ and $\mathcal {N}_{ie}$ are premetric structures with signature $L_e$ . By Corollary 3.10, $\prod _{\mathcal {F}}(\mathcal {M}_{ie})$ and $\prod _{\mathcal {F}}(\mathcal {N}_{ie})$ are also premetric structure with signature $L_e$ . By Fact 2.20 (ii), $\mathcal {M}_{ie}\equiv \mathcal {N}_{ie}$ for each i. Then by Remark 3.16, $\prod _{\mathcal {F}}(\mathcal {M}_{ie})\equiv \prod _{\mathcal {F}}(\mathcal {N}_{ie})$ . Finally, taking the V-parts and using Remark 3.4, we have $\prod _{\mathcal {F}}\mathcal {M}_{i}\equiv \prod _{\mathcal {F}}\mathcal {N}_{i}$ . ⊣

Proof. The analogue of this lemma for classical model theory is Lemma 6.2.4 in [Reference Chang and Keisler3], and the reader can look at the proof in [Reference Chang and Keisler3] to fill in the details in this proof. We first construct a mapping h from $\prod _{i\in I} M_i$ onto $N'$ with the following property:

$\mathbf {P}$ : For every restricted conditional formula $\varphi (\vec {x})$ and tuple $\vec {a}$ in the domain of h, if $\varphi ^{\mathcal {M}_i}(\vec {a}(i))=0$ for most $i\in I$ , then $\varphi ^{\mathcal {N}'}(h(\vec {a}))=0$ .

The hypothesis $2^\lambda =\lambda ^+$ implies that $|\prod _{i\in I}M_i|\le \lambda ^+$ and $|N'|\le \lambda ^+$ . The mapping h is constructed by enumerating both $\prod _{i\in I}M_i$ and $N'$ by sequences of length $\lambda ^+$ , and then using a back-and-forth argument that is a transfinite induction with $\lambda ^+$ steps. To begin we note that the hypotheses of this lemma guarantee that the empty mapping has property $\mathbf {P}$ . At each stage of the induction, we assume that we have already constructed a mapping $h'$ from a subset of $\prod _{i\in I} M_i$ into $N'$ that has the property $\mathbf {P}$ . Let b be the next element in the enumeration of $\prod _{i\in I} M_i$ , and let e be the next element in the enumeration of $N'$ . We will use the fact that the set of restricted conditional formulas has cardinality $\le \lambda $ and is closed under $\max , \sup ,\inf $ , and that there are at most $\lambda $ previous stages.

We first use the fact that $\mathcal {N}{'}$ is $\lambda ^+$ -saturated to find an element $c\in N'$ such that $h'\cup \{(b,c)\}$ has the property $\mathbf {P}$ . Let $\Gamma (y)$ be the set of all formulas of the form

such that $\varphi (\vec x,y)$ is restricted conditional, $\vec a$ is a tuple in the domain of $h'$ , r is a positive dyadic rational, and $\varphi ^{\mathcal {M}_i}(\vec a(i),b(i))=0$ for most $i\in I$ . For each finite subset $\Gamma _0\subseteq \Gamma $ , the formula

is restricted conditional. Moreover, $\psi ^{\mathcal {M}_i}(\vec a(i))=0$ for most $i\in I$ , so by property $\mathbf {P}$ for $h'$ , $\psi ^{\mathcal {N}{'}}(h'(\vec a))=0$ . It follows that $\Gamma (y)$ is finitely satisfiable in $\mathcal {N}{'}$ . The set $\Gamma (y)$ has cardinality $\le \lambda $ , and $\mathcal {N}{'}$ is $\lambda ^+$ -saturated, so $\Gamma (y)$ is satisfied by some element c in $\mathcal {N}{'}$ . Then the mapping $h'\cup \{(b,c)\}$ has property $\mathbf {P}$ .

We next use Lemma 6.1.6 in [Reference Chang and Keisler3] to find an element d of $\prod _{i\in I} M_i$ such that $h'\cup \{(b,c),(d,e)\}$ has property $\mathbf {P}$ . That lemma says that for any set Z of cardinality $|Z|\le \lambda $ and family $X=\langle X_{\zeta }\colon \zeta \in Z\rangle $ of sets of cardinality $\lambda $ , there is a family $Y=\langle Y_{\zeta }\colon \zeta \in Z\rangle $ of pairwise disjoint sets of cardinality $\lambda $ , called a refinement of X, such that $Y_{\zeta }\subseteq X_{\zeta }$ for each $\zeta \in Z$ .

If e is already in the range of $h'$ , we take d to be the first element such that $h'(d)=e$ . Otherwise, we argue as follows. Let Z be the set of $\varphi (\vec a)$ such that $\varphi (\vec x)=\sup _y\psi (\vec x,y)$ , where $\psi $ is restricted conditional, $\vec a$ is a tuple of elements of the domain of $h'\cup \{(b,c)\}$ , and $\varphi ^{\mathcal {N}{'}}(h(\vec a))>0$ . Then $\varphi (\vec x)$ is also restricted conditional. For each $\varphi (\vec a)\in Z$ , let $X_{\varphi (\vec a)}$ be the set of $i\in I$ such that $\varphi ^{\mathcal {M}_i}(\vec a(i))>0$ . Since $h'\cup \{(b,c)\}$ has property $\mathbf {P}$ , it follows that $X_{\zeta }$ has cardinality $\lambda $ for each $\zeta \in Z$ . Therefore X has a refinement Y. Since each $Y_{\zeta }\subseteq X_{\zeta }$ , we may build an element $d\in \prod _{i\in I} M_i$ such that for each $\varphi (\vec a)\in Z$ and $i\in Y_{\varphi (\vec a)}$ we have $\psi ^{\mathcal {M}_i}(\vec a(i),d(i))>0$ . Since each $Y_{\zeta }$ has cardinality $\lambda $ , the function $h'\cup \{(b,c),(d,e)\}$ has property $\mathbf {P}$ . This completes the induction.

Now let h be a mapping from $\prod _{i\in I} M_i$ onto $N'$ with property $\mathbf {P}$ . Let $\mathcal {E}$ be the set of all $J\subseteq I$ such that for some atomic formula $\alpha (\vec {x})$ , some tuple $\vec {a}$ in $\prod _{i\in I} M_i$ , and some $\varepsilon \in [0,1)$ ,

$$ \begin{align*}\alpha^{\mathcal{N}'}(h(\vec{a}))<1-\varepsilon\mbox{ and } J=\{i\in I\colon \alpha^{\mathcal{M}_i}(\vec{a}(i))<1-\varepsilon\}.\end{align*} $$

Proof of Claim 1. Let $J_0,\ldots ,J_n\in \mathcal {E}$ and $K=J_0\cap \cdots \cap J_n$ . For each $k\le n$ , take an atomic formula $\alpha _k$ , tuple $\vec {a}_k$ in $\prod _{i\in I} M_i$ , and a dyadic rational $\varepsilon _k\in [0,1)$ such that

$$ \begin{align*}\alpha_k^{\mathcal{N}'}(h(\vec{a}_k))<1-\varepsilon_k\mbox{ and } J_k=\{i\in I\colon \alpha_k^{\mathcal{M}_i}(\vec{a}_k(i))<1-\varepsilon_k\}.\end{align*} $$

Let

$$ \begin{align*}\psi=\max(\alpha_0\dotplus \varepsilon_0,\ldots,\alpha_n\dotplus \varepsilon_n)\end{align*} $$

and $\vec {b}=(\vec {a}_0,\ldots ,\vec {a}_n)$ . Then

(5) $$ \begin{align} \psi^{\mathcal{N}'}(h(\vec{b}))<1\mbox{ and }K=\{i\in I\colon\psi^{\mathcal{M}_i}(\vec{b}(i))<1\}. \end{align} $$

Assume, towards a contradiction, that $|K|<\lambda $ and let $\varphi =1-\psi $ . $\varphi $ is equivalent to the restricted primitive conditional formula

$$ \begin{align*}\min(1-(\alpha_0\dotplus\varepsilon_0),\ldots,1-(\alpha_n\dotplus\varepsilon_n)).\end{align*} $$

For most $i\in I$ we have $i\notin K$ , so $\psi ^{\mathcal {M}_i}(\vec {b}(i))=1$ and hence $\varphi ^{\mathcal {M}_i}(\vec {b}(i))=0.$ Then by property $\mathbf {P}$ , we have $\varphi ^{\mathcal {N}'}(h(\vec {b}))=0.$ But this means that $\psi ^{\mathcal {N}'}(h(\vec {b}))=1,$ contradicting (5). Therefore $|K|=\lambda $ and Claim 1 is proved.

By Claim 1, $\mathcal {E}$ generates a proper filter $\mathcal {F}$ over I.

Claim 2. For each atomic formula $\alpha (\vec {x})$ and tuple $\vec {a}$ in $\prod _{i\in I}M_i$ ,

$$ \begin{align*}\alpha^{\mathcal{N}'}(h(\vec{a}))=\limsup\limits_{\mathcal{F}}\alpha^{\mathcal{M}_i}(\vec{a}(i)).\end{align*} $$

Proof of Claim 2. Let $r_N=\alpha ^{\mathcal {N}'}(h(\vec {a}))$ , and

$$ \begin{align*}r_M=\limsup\limits_{\mathcal{F}}\alpha^{\mathcal{M}_i}(\vec{a}(i))=\inf\limits_{J\in\mathcal{F}}\sup_{i\in J} \alpha^{\mathcal{M}_i}(\vec{a}(i)).\end{align*} $$

For every $\varepsilon \in [0,1]$ such that $r_N<1-\varepsilon $ , we have

$$ \begin{align*}J_{\varepsilon} :=\{i\in I\colon\alpha^{\mathcal{M}_i}(\vec{a}(i))<1-\varepsilon\}\in\mathcal{E}\subseteq\mathcal{F},\end{align*} $$

so

$$ \begin{align*}r_M\le\sup_{i\in J_{\varepsilon}} \alpha^{\mathcal{M}_i}(\vec{a}(i))\le 1-\varepsilon.\end{align*} $$

Since this holds for all $\varepsilon>0$ , we have $r_M\le r_N$ .

Now suppose $r_M<r_N$ , and take $\delta \in (r_M,r_N)$ . Since $r_M<\delta $ , there is a set $J\in \mathcal {F}$ such that $\sup _{i\in J}\alpha ^{\mathcal {M}_i}(\vec {a}(i))\le \delta $ . Hence

(6)

There are sets $J_0,\ldots ,J_n\in \mathcal {E}$ such that $J_0\cap \cdots \cap J_n\subseteq J$ . Take $\alpha _k, \vec {a}_k,$ and $\varepsilon _k$ such that

(7) $$ \begin{align} \alpha_k^{\mathcal{N}'}(h(\vec{a}_k))<1-\varepsilon_k\mbox{ and } J_k=\{i\in I\colon \alpha_k^{\mathcal{M}_i}(\vec{a}_k(i))<1-\varepsilon_k\}. \end{align} $$

Since $J_0\cap \cdots \cap J_n\subseteq J$ we have

$$ \begin{align*}I=J\cup(I\setminus J_0)\cup\cdots\cup(I\setminus J_n).\end{align*} $$

For each $k\le n$ ,

$$ \begin{align*}J_k=\{i\in I\colon 1-(\alpha^{\mathcal{M}_i}(\vec{a}_k(i))\dotplus\varepsilon_k)>0\},\end{align*} $$

so

(8) $$ \begin{align} (\forall i\in I\setminus J_k)\ 1-(\alpha^{\mathcal{M}_i}(\vec{a}_k(i))\dotplus\varepsilon_k)=0. \end{align} $$

Let $\varphi $ be the restricted primitive conditional formula

and $\vec {b}=(\vec {a},\vec {a}_0,\ldots ,\vec {a}_n)$ . By (6) and (8),

$$ \begin{align*}(\forall i\in I) \varphi^{\mathcal{M}_i}(\vec{b}(i))=0,\end{align*} $$

so by property $\mathbf {P}$ ,

$$ \begin{align*}\varphi^{\mathcal{N}'}(h(\vec{b}))=0.\end{align*} $$

Therefore either

$$ \begin{align*}r_N=\alpha^{\mathcal{N}'}(h(\vec{a}))\le\delta,\end{align*} $$

or

$$ \begin{align*}\alpha^{\mathcal{N}'}_k(h(\vec{a}_k))\ge 1-\varepsilon_k \mbox{ for some } k\le n.\end{align*} $$

But this contradicts (7) and the fact that $\delta \in (r_M,r_N)$ . Therefore $r_M=r_N$ , and Claim 2 is proved.

Now let $\mathcal {M}'$ be the pre-reduced product of $\langle \mathcal {M}_i\colon i\in I\rangle $ modulo $\mathcal {F}$ . Then $\prod _{\mathcal {F}}\mathcal {M}_i$ is the reduction of $\mathcal {M}'$ , and h is a mapping from $M'$ onto $N'$ . By Claim 2 and Remark 3.3, h is an embedding from $\mathcal {M}'$ onto $\mathcal {N}'$ , so $\mathcal {M}'\cong \mathcal {N}'$ . Therefore $\mathcal {N}'\cong \prod _{\mathcal {F}}\mathcal {M}_i$ , and the proof is complete. ⊣

Proof. (ii) $\Rightarrow $ (i) is trivial. Even without the hypothesis that $2^\lambda =\lambda ^+$ , the implication (i) $\Rightarrow $ (iii) follows from Theorem 3.9.

(iii) $\Rightarrow $ (ii): Assume (iii). Let U be the set of all restricted conditional sentences $\varphi $ such that every general model of $S\cup T$ is a general model of U. Then every general model of $S\cup T$ is a general model of $S\cup U$ . We will show that every general model $\mathcal {N}$ of $S\cup U$ is a general model of $S\cup T$ . It will then follow that $S\cup T$ and $S\cup U$ have the same general models, and (i) follows.

By the GCH at $\lambda $ , $2^\lambda $ is a special cardinal. By Facts 2.1 and 2.14, we may assume that $\mathcal {N}$ is a special model such that N is either finite or of cardinality $2^\lambda $ . For each conditional sentence $\varphi $ that is not true in $\mathcal {N}$ , we have $\varphi \notin U$ , so there exists a model of $S\cup T$ in which $\varphi $ is not true. Let I be a set of cardinality $\lambda $ . Since there are $\aleph _0+|V|\le \lambda $ restricted sentences, there is an indexed family $\langle \mathcal {M}_i\colon i\in I\rangle $ of general models of $S\cup T$ with $|M_i|\leq 2^\lambda $ such that every restricted conditional sentence that is true in $\mathcal {M}_i$ for most $i\in I$ is true $\mathcal {N}$ . By Lemma 4.1, there exists a filter $\mathcal {F}$ over I such that $\mathcal {N}\cong \prod _{\mathcal {F}}\mathcal {M}_i$ . Then by (iii), $\mathcal {N}$ is a model of T, and (ii) follows. ⊣

Proof. Since S is a metric theory with signature L, $S\models \operatorname {met}(L)$ , and similarly for T. As in the proof of Theorem 4.2, the implication (ii) $\Rightarrow $ (i) is trivial, ant the implication (i) $\Rightarrow $ (iii) follows from Theorem 3.9 (without the hypothesis $2^\lambda =\lambda ^+$ ).

We next assume (iii) and prove Condition (iii) of Theorem 4.2. To distinguish premetric structures from their downgrades, we will use the full notation $(\mathcal {N},L)$ for a premetric structure. Let $\langle \mathcal {N}_i\colon i\in I\rangle $ be an indexed family of general models of $S\cup T$ such that $\prod _{\mathcal {F}}\mathcal {N}_i$ is a general model of S. Since $S\models \operatorname {met}(L)$ , each $(\mathcal {N}_i,L)$ is a premetric model of S. By Corollary 3.10, $(\prod _{\mathcal {F}}\mathcal {N}_i,L)$ is a premetric structure, and hence a premetric model of S. For each i, let $(\mathcal {M}_i,L)$ be the completion of $(\mathcal {N}_i,L)$ . By Corollary 3.11, $(\prod _{\mathcal {F}}\mathcal {M}_i,L)$ is the completion of $(\prod _{\mathcal {F}}\mathcal {N}_i,L)$ . Therefore $\prod _{\mathcal {F}}\mathcal {M}_i\equiv \prod _{\mathcal {F}}\mathcal {N}_i$ , so $(\prod _{\mathcal {F}}\mathcal {M}_i,L)$ is a metric model of S. By (iii) above, $(\prod _{\mathcal {F}}\mathcal {M}_i,L)$ is a metric model of T. It follows that $\prod _{\mathcal {F}}\mathcal {N}_i$ is a general model of T. This proves Condition (iii) of Theorem 4.2.

Finally by Theorem 4.2, Condition (ii) of Theorem 4.2, which is the same as Condition (ii) above, holds. ⊣

Question 4.4. Can the conclusions of Theorem 4.2 and Corollary 4.3 be proved in ZFC (without the continuum hypothesis)?

Proof. The lemma follows from the observation that $P^{\mathcal {M}}(\vec x)\le r$ if and only if for all rational $s\in (r,1)$ we have $P^{\mathcal {M}}(\vec x)\le s$ . ⊣

Proof. (i) is clear from the definitions.

(ii) Suppose $\mathcal {M}\models T$ and $\theta \in T_{\downarrow }$ . By Lemma 5.5, $\mathcal {M}=(\mathcal {M}_{\downarrow })_{\uparrow }\models T$ , so $\mathcal {M}_{\downarrow }\models \theta $ . Therefore $\mathcal {M}_{\downarrow }\models T_{\downarrow }$ .

(iii) The property of being increasing is expressed by a set of $V_{\downarrow }$ -sentences. ⊣

Proof. It is enough to prove the result in the case that V contains a predicate symbol d for the discrete metric, because then the general case follows by removing d from the vocabulary and taking the reduction of both sides. In that case, for each n-ary $P\in V$ , rational $r\in [0,1)$ , and $\vec x\in (\prod K_i)^n$ , the following are equivalent.

• • .

• • .

• • .

• • .

• • $(\forall s\in (r,1))(\exists J\in \mathcal {F})(\forall i\in J){\mathcal {K}}_i\models P_{\le s}(\vec x_i)$ .

• • $(\forall s\in (r,1))\{i\colon {\mathcal {K}}_i\models P_{\le s}(\vec x_i)\}\in \mathcal {F}$ .

• • $(\forall s\in (r,1))\prod _{\mathcal {F}}({\mathcal {K}}_i)\models P_{\le s}(\vec x_{\mathcal {F}})$ .

• • .

The second statement follows from the first statement and Lemma 5.5. ⊣

Proof. (i) is clear. We prove (ii). By the Keisler-Shelah Theorem (Theorem 6.1.15 in [Reference Chang and Keisler3]), there are ultrafilters $\mathcal {D}$ over H and $\mathcal {E}$ over I such that $\mathcal {K}^{H}/{\mathcal {D}}\cong {\mathcal {K}{'}}^{J}/{\mathcal {E}}$ . Then by (i) we have $(\mathcal {K}^{H}/{\mathcal {D}})_{\uparrow }\cong ({\mathcal {K}{'}}^{J}/{\mathcal {E}})_{\uparrow }$ . By Lemma 5.8, $({\mathcal {K}}_{\uparrow })^{H}/{\mathcal {D}}\cong ({\mathcal {K}{'}}_{\uparrow })^{J}/{\mathcal {E}}$ . It follows that ${\mathcal {K}}_{\uparrow }\equiv {\mathcal {K}{'}}_{\uparrow }$ . ⊣

Proof. It follows immediately from the definition of $T_{\downarrow }$ that if ${\mathcal {K}}_{\uparrow }\models T$ , then $\mathcal {K}\models T_{\downarrow }$ . For the other direction, suppose that $\mathcal {K}\models T_{\downarrow }$ . Then there is a set I, an ultrafilter $\mathcal {D}$ on I, and increasing structures ${\mathcal {K}}_i$ , $i\in I$ , such that ${\mathcal {K}}_{i\uparrow } \models T$ for each $i\in I$ , and $\mathcal {K}\equiv \prod _{\mathcal {D}}{\mathcal {K}}_i$ . By Lemmas 5.9 and 5.8, we have

$$ \begin{align*}{\mathcal{K}}_{\uparrow} \equiv \left(\prod\limits_{\mathcal{D}}{\mathcal{K}}_i\right)_{\uparrow}=\prod\limits_{\mathcal{D}}({\mathcal{K}}_{i\uparrow}),\end{align*} $$

from which it follows that ${\mathcal {K}}_{\uparrow } \models T$ .⊣

Proof. Let $\mathcal {F}$ be a proper filter over a set I.

(i) $\Rightarrow $ (ii): Assume (i). For each $i\in I$ , let ${\mathcal {K}}_i\models S_{\downarrow }$ . By Lemma 5.10, ${\mathcal {K}}_{i\uparrow }\models S$ , so $\prod _{\mathcal {F}}({\mathcal {K}}_{i\uparrow })\models T$ . But $\prod _{\mathcal {K}}({\mathcal {K}}_{i\uparrow })=(\prod _{\mathcal {F}}({\mathcal {K}}_i))_{\uparrow }$ by Lemma 5.8. Therefore $(\prod _{\mathcal {F}}({\mathcal {K}}_i))_{\uparrow }\models T$ , so $\prod _{\mathcal {F}}({\mathcal {K}}_i)\models T_{\downarrow }$ by Lemma 5.10. This proves (ii).

(ii) $\Rightarrow $ (i): Assume (ii). Let $\mathcal {M}_i\models S$ for each $i\in I$ . We have $\mathcal {M}_{i\downarrow \uparrow }=\mathcal {M}_i$ by Lemma 5.5, so $\mathcal {M}_{i\downarrow }\models S_{\downarrow }$ . Then $\prod _{\mathcal {F}}(\mathcal {M}_{i\downarrow })\models T_{\downarrow }$ . By Lemma 5.10, $(\prod _{\mathcal {F}}(\mathcal {M}_{i\downarrow }))_{\uparrow }\models T$ . By the second statement of Lemma 5.8, we have $ \prod _{\mathcal {F}}(\mathcal {M}_i)\models T$ , and (i) holds. ⊣

Proof. Take $S=T$ in Lemma 5.11. ⊣

Fact 5.13. Every sentence of set theory of the form $Q_1 Q_2\theta $ , where $Q_1$ is a sequence of second order universal quantifiers over subsets of $\operatorname {HF}$ and first-order quantifiers restricted to $\operatorname {HF}$ , $Q_2$ is a sequence of second order existential quantifiers over subsets of $\operatorname {HF}$ and first-order quantifiers restricted to $\operatorname {HF}$ , and $\theta $ is quantifier-free, is equivalent to a $\Pi ^1_2$ sentence.

Fact 5.14. Every $\Pi ^1_2$ sentence over $\operatorname {HF}$ that is provable from ZFC + CH is provable from ZFC.

Proof. First assume the continuum hypothesis. By Fact 3.12, $T_{\downarrow }$ is preserved under reduced products. By Corollary 5.12, T is preserved under reduced products. Since $V\subseteq \operatorname {HF}$ , V is countable. By the continuum hypothesis and Theorem 4.2, T is equivalent to a set of restricted conditional sentences.

By Fact 5.14, to show that this lemma is provable in ZFC, it suffices to show that the statement of this lemma is a $\Pi ^1_2$ sentence. To do that, we will freely use Fact 5.13, which allows us to ignore quantifiers over elements of $\operatorname {HF}$ .

It is elementary that the statements “ $\theta $ is a first-order $V_{\downarrow }$ -sentence”, “ $\theta $ is a Horn sentence”, “ $\varphi $ is a strict V-sentence”, and “ $\varphi $ is a strict conditional sentence”, are $\Delta ^1_1$ . It follows that the first hypothesis of this lemma, “T is a set of strict continuous sentences in a vocabulary $V\subseteq \operatorname {HF}$ ”, is $\Delta ^1_1$ .

Every countable $V_{\downarrow }$ -structure is isomorphic to a $V_{\downarrow }$ -structure that is a subset of $\operatorname {HF}$ and similarly for general structures. In the following we let $\mathcal {K}$ be a variable that ranges over $V_{\downarrow }$ -structures that are subsets of $\operatorname {HF}$ with universe K, and let $\mathcal {M}, \mathcal {N}$ be variables ranging over V-structures that are subsets of $\operatorname {HF}$ . By unravelling the definition of satisfaction in [Reference Chang and Keisler3] and [Reference Yaacov, Berenstein, Henson, Usvyatsov, Chatzidakis, Macpherson, Pillay and Wilkie18], one can show that the statements “ $\mathcal {K}\models \theta $ ”, “ $\mathcal {K}$ is increasing”, “ ${\mathcal {K}}_{\uparrow }\models \varphi $ ”, “ ${\mathcal {K}}_{\uparrow }\models T$ ”, “ $\mathcal {N}\models \varphi $ ”, and “ $\mathcal {N}\models T$ ”, are $\Delta ^1_1$ .

To illustrate, we show that the statement “ ${\mathcal {K}}_{\uparrow }\models \varphi $ ” is $\Delta ^1_1$ . The statement “ $\psi $ is a strict V-sentence with parameters in $\mathcal {K}$ and r is a dyadic rational in $[0,1)$ ” is a $\Delta ^1_1$ formula $\delta (\mathcal {K},\psi ,r)$ . Let us say that D is an $\uparrow $ -diagram of $\mathcal {K}$ if D is the set of all pairs $(\psi ,r)$ such that $\delta (\mathcal {K},\psi ,r)$ and . Then D is an $\uparrow $ -diagram of $\mathcal {K}$ if and only if each of the following $\Delta ^1_1$ sentences with the parameters $\mathcal {K}, D$ hold.

• • $(\forall \psi )(\forall r)[(\psi ,r)\in D\Rightarrow \delta (\mathcal {K},\psi ,r)] .$

• • $(\forall \mbox { atomic }P(\vec \tau ))(\forall r)[(P(\vec \tau ),r)\in D\Leftrightarrow \mathcal {K} \models P_{\le r}( \vec \tau )]$ .

• • $(\forall \psi _1)(\forall \psi _2)(\forall r)[(\max (\psi _1,\psi _2),r)\in D\Leftrightarrow [(\psi _1,r)\in D\wedge (\psi _2,r)\in D]].$

• • Similar rules for connectives .

• • $(0,r)\in D\wedge (1,r)\notin D$ .

• • $(\forall \psi (x))(\forall r)[(\sup _x\psi (x),r)\in D\Leftrightarrow (\forall a\in K)(\psi (a),r)\in D]$ .

• • A similar rule for $\inf _x$ .

Then ${\mathcal {K}}_{\uparrow }\models \varphi $ if and only if $(\varphi ,0)\in D$ for every $\uparrow $ -diagram D of $\mathcal {K}$ , and also if and only if $(\varphi ,0)\in D$ for some $\uparrow $ -diagram D of $\mathcal {K}$ . This shows that the statement “ ${\mathcal {K}}_{\uparrow }\models \varphi $ ” is $\Delta ^1_1$ .

By Lemma 5.10, the statement “ $\mathcal {K}\models T_{\downarrow }$ ” is also $\Delta ^1_1$ . It follows that the second hypothesis of this lemma, that “ $T_{\downarrow }$ is equivalent to a set of Horn sentences”, is equivalent in ZFC to the following $\Sigma ^1_2$ sentence with the parameter T:

$$ \begin{align*}(\exists U\subseteq\operatorname{HF})(\forall\mathcal{K})[(\forall \theta)(\theta\in U\Rightarrow \theta \mbox{ is Horn}) \wedge (\mathcal{K}\models T_{\downarrow} \Leftrightarrow \mathcal{K}\models U)].\end{align*} $$

The statement “ $T\models \varphi $ ” is $\Pi ^1_1$ , because it is equivalent in ZFC to

$$ \begin{align*}(\forall \mathcal{N})[\mathcal{N}\models T\Rightarrow \mathcal{N}\models\varphi].\end{align*} $$

The conclusion of this lemma says that for every $\mathcal {M}$ , if every strict conditional consequence of T holds in $\mathcal {M}$ , then $\mathcal {M}\models T$ , that is,

$$ \begin{align*}(\forall\mathcal{M})[(\forall\varphi)(\varphi \mbox{ strict conditional and } T \models\varphi)\Rightarrow\mathcal{M}\models T].\end{align*} $$

This is a $\Pi ^1_2$ sentence with a parameter for T.

So the whole statement of this lemma has the form

$$ \begin{align*}(\forall T)[\alpha(T)\Rightarrow \beta(T)],\end{align*} $$

where $\alpha (T)$ is a $\Sigma ^1_2$ sentence, and $\beta (T)$ is a $\Pi ^1_2$ sentence. It follows that the statement of this lemma is a $\Pi ^1_2$ sentence. ⊣

Proof. (ii) $\Rightarrow $ (i) is trivial, and (i) $\Rightarrow $ (iii) follows from Theorem 3.9

(iii) $\Rightarrow $ (ii): Assume (iii). By Lemma 2.15, we may assume without generality that T is a set of strict continuous sentences, and that every consequence of T that is a strict continuous sentence in vocabulary V belongs to T. For each vocabulary $V'\subseteq V$ , let $[V']$ be the set of strict continuous sentences in vocabulary $V'$ , and let $T'=T\cap [V']$ .

Claim 1. For every countable vocabulary $V'\subseteq V$ , $T'$ is preserved under reduced products.

Proof of Claim 1

Let $\{ {\mathcal {M}{'}}_i\colon i\in I\}$ be a family of general models of $T\cap [V']$ , let $\mathcal {F}$ be a proper filter over I, and let $\mathcal {M}{'}=\prod _{\mathcal {F}}({\mathcal {M}{'}}_i)$ . For each i, let $U_i$ be the set of strict continuous sentences

Suppose that $U_i$ is not finitely satisfiable. Then there is a strict sentence $\varphi \in Th({\mathcal {M}{'}}_i)$ and a dyadic rational $r>0$ such that

. The sentence

belongs to $[V']$ , and also belongs to T because it is a consequence of T. But then ${\mathcal {M}{'}}_i\models \varphi = 0$ and

, which is a contradiction. Therefore $U_i$ is finitely satisfiable. By the Compactness Theorem, $U_i$ has a general model $\mathcal {N}_i$ . Then $\mathcal {N}_i{{\kern-1.5pt}\models{\kern-1.5pt}} T$ for each i, so by (iii) we have $\mathcal {N}:=\prod _{\mathcal {F}}(\mathcal {N}_i)\models T$ . Let ${\mathcal {N}{'}}_i$ be the $V{'}$ -part of $\mathcal {N}_i$ . By Remark 3.4, $\mathcal {N}{'}{{\kern-1.2pt}:={\kern-1.2pt}}\prod _{\mathcal {F}}({\mathcal {N}{'}}_i)$ is the $V'$ -part of $\mathcal {N}$ , so $\mathcal {N}{'}{{\kern-1.2pt}\models{\kern-1.2pt}} T{'}$ . Since $\mathcal {N}_i{{\kern-1.2pt}\models{\kern-1.2pt}} U_i$ , we have ${\mathcal {N}{'}}_i\equiv {\mathcal {M}{'}}_i$ . Then by Proposition 3.17, $\mathcal {N}{'}\equiv \mathcal {M}'$ . Therefore $\mathcal {M}{'}\models T'$ , and Claim 1 is proved.

Claim 2. For every countable $V'\subseteq V$ , $T'$ is equivalent to a set of strict conditional sentences.

Proof of Claim 2

By Claim 1, $T'$ is preserved under reduced products. We may take $V{'}$ , $T{'}$ , ${V{'}}_{\downarrow }$ , and ${T{'}}_{\downarrow }$ to be subsets of $\operatorname {HF}$ . By Lemma 5.11, ${T{'}}_{\downarrow }$ is preserved under reduced products. By Fact 3.12 (ii), ${T{'}}_{\downarrow }$ is equivalent to a set of Horn sentences with vocabulary $V_{\downarrow }$ . By Lemma 5.15, $T'$ is equivalent to a set of strict conditional sentences in the vocabulary $V'$ , so Claim 2 is proved.

It follows at once from Claim 2 that Condition (i) holds. ⊣

Proof. It is clear that (ii) implies (i).

Assume (i). Then $\{\varphi \}$ is equivalent to the countable set $\{\psi _n\colon n>0\}$ of conditional sentences. Then by Theorem 5.1, (iii) holds.

Now assume (iii). By Theorem 5.1, $\{\varphi \}$ is equivalent to a set U of restricted conditional sentences. By Corollary 2.9, for each $n>0$ there is a finite $U_0\subseteq U$ such that . Let $\psi _n=\max (U_0)$ . Then $\psi _n$ is a restricted conditional sentence, $\varphi \models \psi _n$ and , so (ii) holds. ⊣

Proof. (i) $\Rightarrow $ (ii) follows from Theorem 3.9 and Corollary 3.10. (ii) $\Rightarrow $ (iii) follows from Corollary 3.10.

(iii) $\Rightarrow $ (ii): Assume (iii), and let $(\prod _{\mathcal {F}}\mathcal {N}_i,L)$ be a reduced product of premetric models of T. For each $i\in I$ , let $(\mathcal {M}_i,L)$ be the completion of $(\mathcal {N}_i,L)$ , which is a metric model of T. By (iii) above, $(\prod _{\mathcal {F}} \mathcal {M}_i,L)$ is a metric model of T. By Corollary 3.11, $(\prod _{\mathcal {F}} \mathcal {M}_i,L)$ is the completion of $(\prod _{\mathcal {F}} \mathcal {N}_i,L)$ . Therefore $(\prod _{\mathcal {F}} \mathcal {N}_i,L)$ is a premetric model of T, and (ii) holds.

(ii) $\Rightarrow $ (i): Assume (ii). We first show that T is preserved under reduced products of general structures. Suppose $\prod _{\mathcal {F}} \mathcal {N}_i$ is a reduced product of general models of T. Since $T\models \operatorname {met}(L)$ , $(\mathcal {N}_i,L)$ is a premetric model of T for each $i\in I$ . Therefore, by (ii), $(\prod _{\mathcal {F}}\mathcal {N}_i,L)$ is a premetric model of T, and hence $\prod _{\mathcal {F}}\mathcal {N}_i$ is a general model of T. Thus T is preserved under reduced products of general structures. So by Theorem 5.1, (i) holds. ⊣

Question 5.19. Suppose $S, T$ are continuous theories, and every reduced product of general models of $ S$ is a general model of T. Is there a set U of conditional sentences such that $S\models U$ and $U\models T$ ?

Proof. In the case of metric structures, this is Proposition 7.15 of [Reference Yaacov, Berenstein, Henson, Usvyatsov, Chatzidakis, Macpherson, Pillay and Wilkie18]. However, the proof given there also works in the case of general structures. ⊣

Proof. It is clear that (2) implies (1). Conversely, suppose that (1) holds. Then by Corollary 5.17, there are conditional sentences $\psi _n$ such that $\sigma \models \psi _n$ and . Set $\psi :=\sum _n 2^{-n}\psi _n$ , a definable predicate. Note that $\varphi \models \psi $ and $\psi \models \varphi $ . By Lemma 5.20, there are increasing continuous functions $\alpha _n:[0,1]\to [0,1]$ and $\beta :[0,1]\to [0,1]$ with $\alpha _n(0)=\beta (0)=0$ and such that $\psi _n^{\mathcal {M}}\leq \alpha _n(\varphi ^{\mathcal {M}})$ and $\varphi ^{\mathcal {M}}\leq \beta (\psi ^{\mathcal {M}})$ for all general structures $\mathcal {M}$ . Set $\alpha :=\sum _n 2^{-n}\alpha _n$ , so that $\alpha :[0,1]\to [0,1]$ is an increasing continuous function with $\alpha (0)=0$ . Finally, set $\gamma :=\beta \circ \alpha $ .

Suppose now that we have a set I, general structures $\mathcal {M}_i$ for each $i\in I$ , and a filter $\mathcal {F}$ on I, and set $\mathcal {M}:=\prod _{\mathcal {F}} \mathcal {M}_i$ . Let $r:=\limsup _{\mathcal {F}} \varphi ^{\mathcal {M}_i}$ and take $s>r$ . Take $J\in \mathcal {F}$ such that $\sup _{i\in J}\varphi ^{\mathcal {M}_i}\leq s$ . It follows that $\psi _n^{\mathcal {M}_i}\leq \alpha _n(s)$ for all $i\in J$ . By (4) in the proof of Theorem 3.9, it follows that $\psi _n^{\mathcal {M}}\leq \alpha _n(s)$ and thus $\psi ^{\mathcal {M}}\leq \alpha (s)$ and $\varphi ^{\mathcal {M}}\leq \beta (\psi ^{\mathcal {M}})\leq \beta (\alpha (s))=\gamma (s)$ . Since $s>r$ was arbitrary, we have that $\varphi ^{\mathcal {M}}\leq \gamma (r)$ , as desired. ⊣