On first-order expressibility of satisfiability in submodels

TL;DR

This paper investigates when the property that a model has a submodel satisfying a certain sentence can be expressed in first-order logic or its extensions, providing characterizations for specific cases involving large cardinals.

Contribution

It offers syntactical and semantical characterizations of conditions under which submodel satisfaction can be expressed in $ ext{L}_{ ext{kappa,kappa}}$, especially for $ ext{kappa}= ext{omega}$ or large cardinals.

Findings

Characterizations of when $ heta( extphi)$ is in $ ext{L}_{ ext{kappa,kappa}}$

Results for $ ext{kappa}= ext{omega}$ and certain large cardinals

Conditions under which submodel satisfaction is first-order expressible

Abstract

Let $\kappa,\lambda$ be regular cardinals, $\lambda\le\kappa$, let $\varphi$ be a sentence of the language $\mathcal L_{\kappa,\lambda}$ in a given signature, and let $\vartheta(\varphi)$ express the fact that $\varphi$ holds in a submodel, i.e., any model $\mathfrak A$ in the signature satisfies $\vartheta(\varphi)$ if and only if some submodel $\mathfrak B$ of $\mathfrak A$ satisfies $\varphi$. It was shown in [1] that, whenever $\varphi$ is in $\mathcal L_{\kappa,\omega}$ in the signature having less than $\kappa$ functional symbols (and arbitrarily many predicate symbols), then $\vartheta(\varphi)$ is equivalent to a monadic existential sentence in the second-order language $\mathcal L^{2}_{\kappa,\omega}$, and that for any signature having at least one binary predicate symbol there exists $\varphi$ in $\mathcal L_{\omega,\omega}$ such that $\vartheta(\varphi)$ is not equivalent to any (first-order) sentence in $\mathcal L_{\infty,\omega}$. Nevertheless, in certain cases $\vartheta(\varphi)$ are first-order expressible. In this note, we provide several (syntactical and semantical) characterizations of the case when $\vartheta(\varphi)$ is in $\mathcal L_{\kappa,\kappa}$ and $\kappa$ is $\omega$ or a certain large cardinal.

Equations

$$\mathfrak{A}\models \vartheta (\phi )\text{ iff }\mathfrak{B}\models \phi \text{ for some submodel B of }\mathfrak{A}.$$

$$\exists x\;\bigl{(}X(x)\wedge\bigwedge\{\psi_{F}(X):F\text{ is a~{}functional symbol in }\tau\}\bigr{)}$$

$$\exists X\;\bigl{(}\psi(X)\wedge\varphi^{X}\bigr{)}$$

$$\displaystyle c^{\mathfrak{A}}_{\alpha}:=a,\quad\text{and}\quad c^{\mathfrak{B}}_{\alpha}:=\left\{\begin{array}[]{lll}a&\text{if }c_{\alpha}\text{ occurs in }\psi,\\ b&\text{otherwise.}\end{array}\right.$$

$$\mbox{\LARGE$\exists$}_{\alpha<\lambda}\,x_{\alpha}\;\varphi^{\,\{x_{\alpha}:\,\alpha<\lambda\}}$$

$$\mbox{\LARGE$\exists$}_{\beta<\gamma}\,y_{\beta}\;\bigl{(}\psi\wedge\bigwedge\nolimits_{\beta<\gamma}\bigvee\nolimits_{\!\!\!\alpha<\lambda}y_{\beta}=x_{\alpha}\bigr{)}.$$

$$\mbox{\LARGE$\exists$}_{\beta<\gamma}\,y_{\beta}\;\bigl{(}\psi\wedge\bigvee\nolimits_{\!\!\!f\in\lambda^{\gamma}}\bigwedge\nolimits_{\beta<\gamma}y_{\beta}=x_{f(\beta)}\bigr{)},$$

$${\bigvee }_{f\in {\lambda }^{\gamma }}\psi (y_{\beta }/x_{f(\beta )}{)}_{\beta <\gamma }$$

$$\mbox{\LARGE$\exists$}_{\alpha<\lambda}\,x_{\alpha}\;\varphi_{\beta}(x_{\alpha})_{\alpha<\lambda}$$

$$\mbox{\LARGE$\exists$}_{\alpha<\lambda}\,x_{\alpha}\;\bigvee\nolimits_{\!\!\!\beta<\mu}\varphi_{\beta}(x_{\alpha})_{\alpha<\lambda}.$$

$$\displaystyle c_{a}(i):=\left\{\begin{array}[]{lll}a&\text{if }a\in i,\\ b_{i}&\text{otherwise.}\end{array}\right.$$

$$f(a):=[c_a{]}_D,$$

$$R^{\mathfrak{A}}(a_0,\dots ,a_{n-1})\text{ iff }{R}^{\mathfrak{B}}(f(a_0),\dots ,f(a_{n-1})).$$

$$R^{\mathfrak{A}}(a_0,\dots ,a_{n-1})$$

$$\displaystyle\;\text{ iff }\;\bigl{\{}i\in I:R^{\mathfrak{B}_{i}}(c_{a_{0}}(i),\ldots,c_{a_{n-1}}(i))\bigr{\}}\in D$$

$$f(F^{\mathfrak{A}}(a_0,\dots ,a_{n-1}))=F^{\mathfrak{B}}(f(a_0),\dots ,f(a_{n-1})).$$

$$f(F^{\mathfrak{A}}(a_0,\dots ,a_{n-1}))=[c_{F^{\mathfrak{A}}(a_0,\dots ,a_{n-1})}{]}_D$$

$$F^{\mathfrak{B}}(f(a_0),\dots ,f(a_{n-1}))=F^{\mathfrak{B}}([c_{a_0}{]}_D,\dots ,[c_{a_{n-1}}{]}_D)=[b{]}_D$$

$$\text{ where }b(i)=F^{{\mathfrak{B}}_i}(c_{a_0}(i),\dots ,c_{a_{n-1}}(i)).$$

Full text

Given a model-theoretic language $\mathcal{L}$ (in sense of [2]) and a sentence $\varphi$ in $\mathcal{L}$, let $\vartheta(\varphi)$ express the fact that $\varphi$ is satisfied in a submodel. Thus for any model $\mathfrak{A}$,

[TABLE]

We study when $\vartheta(\varphi)$, considered a priori as a meta-expression, is equivalent to a sentence in another (perhaps, the same) given model-theoretic language $\mathcal{L}^{\prime}$. Such questions naturally arise in studies of modal logics of submodels; if $\mathcal{L}$ is closed under $\vartheta$, then $\vartheta$ induces a modal operator on sentences (where a possibility of $\varphi$ means the satisfiability of $\varphi$ in a submodel), and the resulting modal logic can be regarded as a fragment of $\mathcal{L}$ with the submodel relation on a given class of models. These logics are an instance of modal logics of various model-theoretic relations, which were introduced and studied in [1]; another instance is modal logic of forcing (see, e.g., [3]). As another source of motivation for studies undertaken in this note, let us point out the paper [4] discussing reduction of higher-order logics to second-order one.

Here we concentrate on first-order languages $\mathcal{L}_{\kappa,\lambda}$. Recall that $\mathcal{L}_{\omega,\omega}$ is the usual first-order finitary language; $\mathcal{L}_{\kappa,\lambda}$ expands it by involving Boolean connectives of any arities $<\kappa$ and quantifiers over $<\lambda$ first-order variables, where $\lambda\leq\kappa$ are given regular cardinals; and $\mathcal{L}_{\infty,\lambda}$ is the union of $\mathcal{L}_{\kappa,\lambda}$ for all $\kappa$; see [2], [5], [6]. It was shown in [1] that, even for $\varphi$ in $\mathcal{L}_{\omega,\omega}$, it is possible that $\vartheta(\varphi)$ is not in $\mathcal{L}_{\infty,\omega}$; on the other hand, $\vartheta(\varphi)$ is equivalent to a second-order (in general, infinitary) sentence; these results are reproduced as Theorems 4 and 5 below. Nevertheless, in certain cases $\vartheta(\varphi)$ are first-order expressible. In this note, we provide several (syntactical as well as semantical) characterizations of the case when $\vartheta(\varphi)$ is equivalent to a sentence in $\mathcal{L}_{\kappa,\kappa}$ and $\kappa$ is either $\omega$ or a certain large (e.g., compact) cardinal.

We start with some obvious observations, which confirm, in particular, that $\vartheta$ behaves like an S4 possibility operator.

• (i)

$\vartheta(\top)$* is equivalent to $\top$, and $\vartheta(\bot)$ is equivalent to $\bot$;*

• (ii)

$\varphi$* implies $\vartheta(\varphi)$;*

• (iii)

$\vartheta(\varphi)$* is equivalent to $\vartheta(\vartheta(\varphi))$;*

• (iv)

$\varphi\to\psi$* implies $\vartheta(\varphi)\to\vartheta(\psi)$;*

• (v)

$\vartheta(\bigwedge_{i\in I}\varphi_{i})$* implies $\bigwedge_{i\in I}\vartheta(\varphi_{i})$, and $\vartheta(\bigvee_{\!i\in I}\varphi_{i})$ is equivalent to $\bigvee_{\!i\in I}\vartheta(\varphi_{i})$;*

• (vi)

$\neg\,\vartheta(\varphi)$* implies $\neg\,\varphi$, and $\neg\,\varphi$ implies $\vartheta(\neg\,\varphi)$;*

• (vii)

$\vartheta(\neg\,\vartheta(\varphi))$* implies $\vartheta(\neg\,\varphi)$.*

Given cardinals $\kappa,\mu$ with $\mu\geq\kappa$, recall that $\kappa$ is $\mu$-compact iff the language $\mathcal{L}_{\kappa,\kappa}$ satisfies the $(\mu,\kappa)$-compactness, i.e., any theory in $\mathcal{L}_{\kappa,\kappa}$ of cardinality $\leq\mu$ has a model whenever each its subtheory of cardinality $<\kappa$ has a model. A cardinal $\kappa$ is weakly compact iff it is $\kappa$-compact, and strongly compact iff it is $\mu$-compact for all $\mu\geq\kappa$ (e.g., $\omega$ is strongly compact). By a compact cardinal $\kappa$ we shall mean strongly compact $\kappa$, and by an inaccessible, a strongly inaccessible, i.e., a regular $\kappa$ such that $2^{\lambda}<\kappa$ for all $\lambda<\kappa$. For more on these and other large cardinals and their connections with infinitary languages, we refer the reader to [5], [6].

• (i)

$\vartheta(\varphi)$* is equivalent to a sentence in $\mathcal{L}_{\kappa,\kappa}$;*

• (ii)

$\vartheta(\varphi)$* is equivalent to a sentence in $\varSigma^{0}_{1}(\mathcal{L}_{\kappa,\kappa})$, i.e., an existential sentence in $\mathcal{L}_{\kappa,\kappa}$.*

Two following results on expressibility of $\vartheta(\varphi)$ were essentially obtained in [1]. They show that the expressibility generally does not hold in first-order languages but is achieved in appropriate second-order ones.

Toward a contradiction, assume that there is $\kappa$ such that $\neg\,\vartheta(\varphi)$ is equivalent to some $\psi\in\mathcal{L}_{\kappa,\omega}$. It follows from Karp’s theorem (see, e.g., [8], Theorem 14.29) that there are models $\mathfrak{A}_{0}$ and $\mathfrak{B}_{0}$ of the subsignature $\tau_{0}=\{R\}$ such that $\mathfrak{A}_{0}$ is isomorphic to an ordinal while $\mathfrak{B}_{0}$ is not, and $\mathfrak{A}_{0}\equiv_{\mathcal{L}_{\kappa,\omega}}\!\mathfrak{B}_{0}.$ Expand $\mathfrak{A}_{0}$ and $\mathfrak{B}_{0}$ respectively to models $\mathfrak{A}$ and $\mathfrak{B}$ of $\tau$ by interpreting each predicate symbol other than $R$ by the empty set, each functional symbol of positive arity by the projection onto the first argument, and each constant symbol by the $R$-last element of the model (which w.l.g. can be assumed to exist). It is easy to see that in both $\mathfrak{A}$ and $\mathfrak{B}$ any formula of $\tau$ is equivalent to a formula of $\tau_{0}$; so we still have $\mathfrak{A}\equiv_{\mathcal{L}_{\kappa,\omega}}\!\mathfrak{B}.$ On the other hand, in both models every subset forms a submodel whenever it contains the $R$-last element of the whole model, whence it easily follows that $\mathfrak{A}\vDash\psi$ and $\mathfrak{B}\vDash\neg\,\psi$. A contradiction. ∎

Given a model-theoretic language $\mathcal{L}$, let $\mathcal{L}^{\alpha}$ denote the $\alpha$th-order extension of $\mathcal{L}$. A formula of $\mathcal{L}^{2}$ is monadic iff it involves only unary predicate variables, and existential second-order, respectively, universal second-order iff it involves only existential, respectively, universal quantifiers over second-order variables preceding a first-order formula (with arbitrary quantifiers). The monadic fragment of $\mathcal{L}^{2}$ consists of its monadic formulas; similarly for the existential and universal fragments of the language, which will be denoted by $\varSigma^{1}_{1}(\mathcal{L}^{2})$ and $\varPi^{1}_{1}(\mathcal{L}^{2})$, respectively.

• (i)

the monadic fragment of $\mathcal{L}^{2}_{\kappa,\lambda}$ for any $\lambda\leq\kappa$;

• (ii)

the existential fragment of $\mathcal{L}^{2}_{\kappa,\lambda}$ for any $\lambda\leq\kappa$;

• (iii)

$\mathcal{L}^{\alpha}_{\kappa,\lambda}$* for any $\lambda\leq\kappa$ and $\alpha\geq 2$.*

[TABLE]

stating that $X$ forms a submodel. Then $\vartheta(\varphi)$ is equivalent to the sentence

[TABLE]

where $\varphi^{X}$ is the relativization of $\varphi$ to $X$. ∎

As usual, a filter $D$ is $\kappa$-complete iff $\bigcap E\in D$ for all $E\in P_{\kappa}(D)$, where $P_{\kappa}(A)$ denotes the set of all subsets of $A$ which have cardinality $<\kappa$.

The next result on non-expressibility of $\vartheta(\varphi)$ shows that the restriction on the number of functional symbols in Theorem 5 is optimal.

[TABLE]

Since $|\psi|<\kappa$, there exists $\alpha<\kappa$ such that $c^{\mathfrak{B}}_{\alpha}=b$. So we have: $\mathfrak{A}\vDash\psi$ iff $\mathfrak{B}\vDash\psi$ (as $\mathfrak{A}$ and $\mathfrak{B}$ satisfy the same formulas involving only symbols from $\psi$), however, $\mathfrak{A}\vDash\vartheta(\varphi)$ and $\mathfrak{B}\vDash\neg\,\vartheta(\varphi)$ (as the singleton $\{a\}$ forms a submodel of $\mathfrak{A}$ while $\mathfrak{B}$ has no single-point submodels). ∎

Let $\vartheta_{\leq\lambda}(\varphi)$ denote that $\varphi$ is satisfied in a submodel generated by a set of cardinality $\leq\lambda$. Obviously, $\vartheta_{\leq\lambda}(\varphi)$ implies $\vartheta(\varphi)$. We are going to show that $\vartheta_{\leq\lambda}(\varphi)$ is an existential sentence in an appropriate first-order language. To simplify some formulations, we shall consider partial models in which their operations can be only partial. An atomic diagram of a partial model $\mathfrak{A}$ is defined in the same way as for usual models with total operations, i.e., it consists of all true in $\mathfrak{A}$ atomic and negated atomic sentences of the language expanded by constant symbols for all elements of $\mathfrak{A}$.

[TABLE]

where $\varphi^{\{x_{\alpha}:\,\alpha<\lambda\}}$ is the relativization of $\varphi$ to the set of (first-order) variables $x_{\alpha}$, $\alpha<\lambda$, which do not occur in $\varphi$. Let us verify that the relativization is equivalent to an open formula in $\mathcal{L}_{\kappa,\kappa}(\tau)$ (with parameters $x_{\alpha}$, $\alpha<\lambda$); it will clearly follow that \mbox{\Large\exists}_{\alpha<\lambda}\,x_{\alpha}\>\varphi^{\,\{x_{\alpha}:\,\alpha<\lambda\}} is equivalent to a $\varSigma^{0}_{1}(\mathcal{L}_{\kappa,\kappa})$-sentence.

Indeed, $\varphi^{\{x_{\alpha}:\,\alpha<\lambda\}}$ is obtained from $\varphi$ by successively replacing each subformula \mbox{\Large\exists}_{\beta<\gamma}\,y_{\beta}\>\psi with the $\mathcal{L}_{\kappa,\kappa}$-formula

[TABLE]

The latter formula is equivalent to the formula

[TABLE]

which is still in $\mathcal{L}_{\kappa,\kappa}$ since $|\lambda^{\gamma}|<\kappa$ due to the condition that $\kappa$ is inaccessible, and furthermore, to the open formula

[TABLE]

where $\psi(y_{\beta}/x_{f(\beta)})_{\beta<\gamma}$ is obtained from $\psi$ by substituting each variable $y_{\beta}$ with the variable $x_{f(\beta)}$. This eliminates all quantifiers in all subformulas of $\varphi^{\,\{x_{\alpha}:\,\alpha<\lambda\}}$, as required.

In the general case, the construction is slightly more complex. Let $\tau^{\prime}$ expand $\tau$ by $\lambda$ new constant symbols $c_{\alpha}$, $\alpha<\lambda$. We still have $|\tau^{\prime}|<\kappa$. Hence, since $\kappa$ is inaccessible, there exist only $<\kappa$ pairwise non-isomorphic partial models in $\tau^{\prime}$ satisfying $\varphi$ with the universe constisting of an interpretion of all closed terms; say, $\mathfrak{B}_{\beta}$, $\beta<\mu$, for some $\mu<\kappa$. Note that, though such partial models may have size $>\lambda$ (they interpret not only the $c_{\alpha}$ but all terms constructed from them), all they have size $\leq\nu$ for some fixed $\nu$ with $\lambda\leq\nu<\kappa$. For any $\beta<\mu$, let $\varDelta_{\beta}$ be the atomic diagram of $\mathfrak{B}_{\beta}$, and $\psi_{\beta}$ its conjunction $\bigwedge\varDelta_{\beta}$, which is still in $\mathcal{L}_{\kappa,\kappa}$ as $|\varDelta_{\beta}|<\kappa$. Let $x_{\alpha}$, $\alpha<\lambda$, be variables not occurring in $\varDelta_{\beta}$, and let $\varphi_{\beta}$ be the formula $\psi_{\beta}(c_{\alpha}/x_{\alpha})$ obtained from $\psi_{\beta}$ by replacing each constant symbol $c_{\alpha}$ with the variable $x_{\alpha}$. Then $\varphi_{\beta}$ is an open formula in $\tau$, and the $\varSigma^{0}_{1}(\mathcal{L}_{\kappa,\kappa})$-sentence

[TABLE]

characterizes the partial model $\mathfrak{B}_{\beta}$ up to isomorphism. It follows that $\vartheta_{\leq\lambda}(\varphi)$ is equivalent to the $\varSigma^{0}_{1}(\mathcal{L}_{\kappa,\kappa})$-sentence

[TABLE]

This completes the proof. ∎

A fragment of a model $\mathfrak{A}$ is an its partial submodel, i.e., a subset of the universe of $\mathfrak{A}$ together with the inherited structure. Thus for models in signatures without functional symbols, fragments are just submodels; while for models in signatures with functional symbols, operations on fragments can be partial. A fragment can be considered as a submodel of the corresponding model in the purely predicate language obtained from the original language by replacing each functional symbol of arity $\geq 1$ with a predicate symbol having the same interpretation. Clearly, for any fragment there exists the smallest submodel including it, the submodel generated by the fragment.

Let $\mathfrak{A}$ be a model and $I$ an ideal over $A$ (the universe of $\mathfrak{A}$) with $\bigcup I=A$. We shall say that the system $(\mathfrak{B}_{i})_{i\in I}$ of models (in the same signature) is coherent in $\mathfrak{A}$ iff for every $i\in I$, the set $i$ is included into $B_{i}$ (the universe of $\mathfrak{B}_{i}$) and the fragments of $\mathfrak{A}$ and of $\mathfrak{B}_{i}$ given by $i$ coincide.

As usual, an upper cone of a partially ordered set $(P,\leq)$ is an $C\subseteq P$ which is upward closed, i.e., such that $b\in C$ whenever $a\leq b$ for some $a\in C$. Clearly, the set of upper cones of $P$ generates a filter over $P$ whenever $P$ is directed.

[TABLE]

Now define $f:A\to B$ by letting for all $a\in A$,

[TABLE]

and check that $f$ is an isomorphic embedding of $\mathfrak{A}$ into $\mathfrak{B}$.

Let $R$ be an $n$-ary predicate symbol in our signature. We must check that for all $a_{0},\ldots,a_{n-1}$ in $A$,

[TABLE]

Since $\bigcup I=A$, for any $k<n$ there is $i_{k}\in I$ with $a_{k}\in B_{i_{k}}$, and since $I$ is an ideal, $\bigcup_{k<n}i_{k}\in I$. Moreover, since $i\subseteq\mathfrak{B}_{i}$ for all $i\in I$, whenever $\bigcup_{k<n}i_{k}\subseteq i$ then $\{a_{k}\}_{k<n}\subseteq B_{i}$ and $a_{k}=c_{a_{k}}(i)$, and so, since the fragments of $\mathfrak{A}$ and $\mathfrak{B}_{i}$ given by $i$ coincide, $R^{\mathfrak{A}}(a_{0},\ldots,a_{n-1})$ is equivalent to $R^{\mathfrak{B}_{i}}(c_{a_{0}}(i),\ldots,c_{a_{n-1}}(i))$. Thus we have:

[TABLE]

where one implication in the second equivalence holds since $D$ extends the filter generated by upper cones of $I$ while the converse implication holds since the property inherits upward. Finally, the latter assertion is equivalent to $R^{\mathfrak{B}}([c_{a_{0}}]_{D},\ldots,[c_{a_{n-1}}]_{D})$ by definition of reduced products, and thus to $R^{\mathfrak{B}}(f(a_{0}),\ldots,f(a_{n-1}))$, as required.

Let now $F$ be an $n$-ary functional symbol in the signature. We must check that for all $a_{0},\ldots,a_{n-1}$ in $A$,

[TABLE]

Indeed,

[TABLE]

while

[TABLE]

Again, if for $k<n$, $i_{k}\in I$ is such that $a_{k}\in B_{i_{k}}$, and also $i_{n}\in I$ is such that $F^{\mathfrak{A}}(a_{0},\ldots,a_{n-1})\in B_{i_{n}}$, whenever $\bigcup_{k\leq n}i_{k}\subseteq i$ then $c_{F^{\mathfrak{A}}(a_{0},\ldots,a_{n-1})}(i)=F^{\mathfrak{A}}(a_{0},\ldots,a_{n-1})=F^{\mathfrak{B}_{i}}(a_{0},\ldots,a_{n-1})$ and also $F^{\mathfrak{B}_{i}}(c_{a_{0}}(i),\ldots,c_{a_{n-1}}(i))=F^{\mathfrak{B}_{i}}(a_{0},\ldots,a_{n-1}).$ It follows $[c_{F^{\mathfrak{A}}(a_{0},\ldots,a_{n-1})}]_{D}=[b]_{D},$ as required.

The proof is complete. ∎

The theorem below is the main result of this note; it extends Corollary 3 by providing new characterizations – syntactical in item (iii) and semantical in items (iv) and (v) – of the case when $\vartheta(\varphi)$ is equivalent to a first-order formula.

• (i)

$\vartheta(\varphi)$* is equivalent to a $\varSigma^{0}_{1}(\mathcal{L}_{\kappa,\kappa})$-sentence;*

• (ii)

$\vartheta(\varphi)$* is equivalent to an $\mathcal{L}_{\kappa,\kappa}$-sentence;*

• (iii)

$\vartheta(\varphi)$* is equivalent to a $\varPi^{1}_{1}(\mathcal{L}^{2}_{\kappa,\kappa})$-sentence;*

• (iv)

any model satisfying $\varphi$ has a fragment of cardinality $<\kappa$ such that each model having the fragment satisfies $\vartheta(\varphi)$;

• (v)

there exists $\lambda<\kappa$ such that any model satisfying $\varphi$ has a fragment of cardinality $\leq\lambda$ and such that each model having the fragment satisfies $\vartheta(\varphi)$.

(iii)$\to$(iv). Assume that (iv) does not hold. Then there is $\mathfrak{A}$ such that $\mathfrak{A}\vDash\varphi$, and for every set $i\subseteq A$ of size $<\kappa$, there exists a model $\mathfrak{B}_{i}$ such that $i\subseteq B_{i}$, the fragments of $\mathfrak{A}$ and of $\mathfrak{B}_{i}$ given by $i$ coincide, and $\mathfrak{B}_{i}$ has no submodels satisfying $\varphi$, thus $\mathfrak{B}_{i}\vDash\neg\,\vartheta(\varphi)$. Let $\mathfrak{B}:=\prod_{D}\mathfrak{B}_{i}$ where $D$ is a $\kappa$-complete ultrafilter over $P_{\kappa}(A)$ which is fine, i.e., extends the filter generated by the sets $\{i\in P_{\kappa}(A):a\in i\}$ for all $a\in A$ (recall that the existence of such an ultrafilter follows from the compactness of $\kappa$; see, e.g., [6], Corollary 22.18). By Lemma 9, $\mathfrak{A}$ isomorphically embeds into $\mathfrak{B}$; therefore, $\mathfrak{B}\vDash\vartheta(\varphi)$. Let us show that (iii) fails.

Indeed, if $\vartheta(\varphi)$ is equivalent to a $\varPi^{1}_{1}(\mathcal{L}^{2}_{\kappa,\kappa})$-formula, then $\neg\,\vartheta(\varphi)$ is equivalent to a $\varSigma^{1}_{1}(\mathcal{L}^{2}_{\kappa,\kappa})$-formula, and hence, is preserved under ultraproducts by $\kappa$-complete ultrafilters (as was pointed out in the proof of Corollary 6), whence we get also $\mathfrak{B}\vDash\neg\,\vartheta(\varphi)$; a contradiction.

(iv)$\to$(v). Assume that (v) does not hold. Let us again use an ultraproduct argument: for any $\alpha<\kappa$ pick a model $\mathfrak{A}_{\alpha}$ which satisfies $\varphi$ and does not have fragments of size $\leq\lambda$ generating submodels satisfying $\varphi$, pick any $\kappa$-complete ultrafilter $D$ over $\kappa$, and consider $\mathfrak{A}:=\prod_{D}\mathfrak{A}_{\alpha}$. Clearly, $\mathfrak{A}$ satisfies $\varphi$. Let us show that $\mathfrak{A}$ does not have fragments of size $<\kappa$ generating submodels satisfying $\varphi$, thus proving that (iv) fails.

Indeed, if there is $\lambda<\kappa$ such that $\mathfrak{A}$ has some $\lambda$-generated submodel satisfying $\varphi$, i.e., $\mathfrak{A}\vDash\vartheta_{\leq\lambda}(\varphi)$, then this fact is expressed by a first-order (and even existential) sentence by Lemma 8, and hence, should hold in $\mathfrak{A}_{\alpha}$ for $D$-almost all $\alpha$, which is, however, not true.

(v)$\to$(i). Assume (v). Then $\vartheta(\varphi)$ is equivalent to $\vartheta_{\leq\lambda}(\varphi)$, which is equivalent to a $\varSigma^{0}_{1}(\mathcal{L}_{\kappa,\kappa})$-sentence by Lemma 8, thus proving (i).

The theorem is proved. ∎

• (i)

$\vartheta(\varphi)$* is equivalent to a sentence in $\mathcal{L}_{\kappa,\kappa}$ (or $\varSigma^{0}_{1}(\mathcal{L}_{\kappa,\kappa})$, or $\varPi^{1}_{1}(\mathcal{L}^{2}_{\kappa,\kappa})$);*

• (ii)

any model satisfying $\varphi$ has a submodel of cardinality $<\kappa$ satisfying $\varphi$ (or $\vartheta(\varphi)$);

• (iii)

there exists $\lambda<\kappa$ such that any model satisfying $\varphi$ has a submodel of cardinality $\leq\lambda$ satisfying $\varphi$ (or $\vartheta(\varphi)$).

Let a signature $\tau$ consist of a single unary functional symbol $F$. Let $\varphi$ be the $\mathcal{L}_{\omega,\omega}$-sentence $\exists x\>F(x)\neq x$ in $\tau$. Then $\vartheta(\varphi)$ is equivalent to $\varphi$ itself. Clearly, the cardinality $\lambda<\omega$ of a finite fragment determining satisfiability of $\vartheta(\varphi)$ in submodels extending it, stated in Theorem 10 (v), is $1$; note that there are models of $\varphi$ without finite submodels at all (e.g., so is the free $1$-generated unar $(\omega,S)$ where $S$ is the successor operation).

More generally, let $\tau$ consist of $\kappa$ unary functional symbols $F_{\alpha}$, $\alpha<\kappa$, and let $\varphi$ be the $\mathcal{L}_{\kappa^{+},\omega}$-sentence $\exists x\bigvee_{\!\alpha<\kappa}F_{\alpha}(x)\neq x$ in $\tau$. Then again, $\vartheta(\varphi)$ is equivalent to $\varphi$, $\lambda$ is $1$, and there are models of $\varphi$ without submodels of size $<\kappa$ (e.g., any free $\tau$-unoid).

Let $\varphi$ be the $\mathcal{L}_{\omega,\omega}$-sentence $\exists x\,\forall y\>R(x,y)$ in $\tau$. Then $\vartheta(\varphi)$ is equivalent to $\exists x\,R(x,x)$, and $\lambda$ from Corollary 11 (iii) is $1$. Observe that $\varphi$ is in $\varSigma^{0}_{2}\,\setminus\,\varPi^{0}_{2}$. Indeed, assume that $\varphi$ is in $\varPi^{0}_{2}$. Recall that $\varPi^{0}_{2}$-formulas are characterized as those that are preserved under chain unions (see [7], Theorem 3.2.2). Let $\mathfrak{A}_{n}$ be $(n+1,\geq)$, and let $\mathfrak{A}$ be the union of the chain of the $\mathfrak{A}_{n}$, $n\in\omega$. Then $\mathfrak{A}_{n}\vDash\varphi$ for all $n\in\omega$ but $\mathfrak{A}\vDash\neg\,\varphi$; a contradiction.

Let also $\psi$ be the $\mathcal{L}_{\omega,\omega}$-sentence $\forall x\,\exists y\>R(x,y)$ in $\tau$. Then $\vartheta(\psi)$ is not equivalent to any $\mathcal{L}_{\omega,\omega}$-sentence. Indeed, the model $(\omega,<)$ satisfies $\psi$ but includes no finite submodels satisfying $\psi$; apply Corollary 11 (iii). Similar arguments show that $\psi$ is in $\varPi^{0}_{2}\,\setminus\,\varSigma^{0}_{2}$ and that $\vartheta(\neg\,\psi)$ is $\exists x\,\neg\,R(x,x)$ while $\vartheta(\neg\,\varphi)$ is not equivalent to any $\mathcal{L}_{\omega,\omega}$-sentence.

First, in Theorem 10 (and Corollary 11) it suffices to assume that $\kappa$ is inaccessible. Moreover, for $\kappa>\omega$, if a sentence $\varphi$ in $\mathcal{L}_{\kappa,\kappa}$ holds in a model then it holds in its submodel of size $<\kappa$ (see, e.g., [2], Corollary 3.1.3), hence $\vartheta(\varphi)$ is equivalent to $\vartheta_{\leq\lambda}(\varphi)$ for some $\lambda<\kappa$; then items (i)–(v) of Theorem 10 follow due to Lemma 8 (even in a nicer form with submodels instead of fragments, like as in Corollary 11).

Further, the obtained results can be relativized to theories $T$ in a given language. Another natural generalization concerns the question when $\vartheta(T)$ is equivalent to some theory in the same language.