UDC 512.164

MATHEMATICS

G. V. CHUDNOVSKII

SOME RESULTS IN THE THEORY OF INFINITELY LONG EXPRESSIONS

(Presented by Academician P. S. Novikov on 15 VI 1967)

The paper considers properties of classes defined by systems of axioms of the language \(L_{\alpha\beta}\), and solves Scott’s problem.

Let \(\alpha, \beta\) be infinite regular cardinals, \(\alpha \geq \beta\). By the first-order language \(L_{\alpha\beta}\) we mean (see \((^{1,8})\)) the extension of the narrow predicate calculus obtained by allowing conjunctions and disjunctions of fewer than \(\alpha\) formulas and quantifications with variables whose number is less than \(\beta\).

1. A class of models \(K\) of a signature \(\sigma\) will be called \((\alpha,\beta)\)-\(F\)-compact (see \((^2)\)), where \(F\) is some set of axioms of the language \(L_{\alpha\beta}\), if for every system \(A\) of axioms from \(F\), satisfiability in \(K\) of every subsystem \(A_1\) of the system \(A\), \(A_1<\alpha\), entails satisfiability of \(A\) in \(K\). In the case when \(F\) is the set of all axioms of the language \(L_{\alpha\beta}\), we shall call \((\alpha,\beta)\)-\(F\)-compactness simply \((\alpha,\beta)\)-compactness. We shall call the language \(L_{\alpha\beta}\) \(F\)-compact (compact) if the class of all models of the signature of the language \(L_{\alpha\beta}\) is \((\alpha,\beta)\)-\(F\)-compact (\((\alpha,\beta)\)-compact).

The fundamental compactness theorem of Mal'cev in model theory shows the compactness of the language \(L_{\omega\omega}\). At the same time, results of Hanf, Tarski \((^4)\), Karp \((^8)\), and others show that compactness does not hold for many languages \(L_{\alpha\beta}\). In connection with these results it is of interest to find out whether classes axiomatizable in languages \(L_{\alpha\beta}\) will be \((\alpha,\beta)\)-\(F\)-compact for certain important sets \(F\). The cases considered are those in which \(F\) is the set of universal axioms or the set of positive axioms.

Theorem 1. Let \(\gamma\) be the cardinality of the signature of the language \(L_{\alpha\beta}\), and let \(2^\gamma\) be less than \(\alpha\). Then the language \(L_{\alpha\beta}\) is universally compact.

Let us note that the class of models of every infinitely long axiom is a projection of the class of models of some universal axiom. In addition, every positive axiom is satisfied on some one-element model. Hence we obtain:

Theorem 2. Every class of models defined by a single axiom of the language \(L_{\alpha\beta}\) is \((\alpha,\beta)\)-universally (and positively) compact.

We also note that Theorem 1 does not carry over to the class of models of an arbitrary signature.

Theorems 1 and 2 are used for the proof of Theorems 3 and 4.

1. Will an axiom of the logic \(L_{\omega_1\omega}\), whose class of models is closed under taking submodels, be equivalent to a universal axiom (or to a system of such axioms) of the logic \(L_{\omega_1\omega}\)? This problem was posed by D. Scott in \((^3)\). A positive answer to this question is given by the following theorem:

An axiom \(\psi\) of the logic \(L_{\omega_1\omega}\) is preserved under passage to submodels if and only if \(\psi\) is equivalent to a universal axiom of the logic \(L_{\omega_1\omega}\).

In \((^4)\) it was shown that in the logic \(L_{\alpha\alpha}\), where \(\alpha\) is nonmeasurable, there exists an axiom \(\varphi\) such that the class of models of \(\varphi\) is closed under taking submodels, but \(\varphi\) is not equivalent to any system of universal

* When this result had been obtained, the author learned of a brief communication by M. Malitz \((^7)\), who wrote that he had solved D. Scott’s problem, but did not give the method of solution.

axiom. The following theorem shows that in the case of a strictly inaccessible \(\theta\) there is no such axiom.

Theorem 3. An axiom \(\psi\) of the logic \(L_{\theta\theta}\) (where \(\theta\) is strictly inaccessible), defining a class \(K\), is equivalent to a universal axiom of the logic \(L_{\theta\theta}\) if and only if \(S(K)=K\).

We give a sketch of the proof of Theorem 3. We note that Theorem 4 is proved analogously.

Let \(M\) be a set of such universal axioms that \(\operatorname{Mod}(A)=\operatorname{Mod}(\psi)\) (see \((^3)\)). Suppose that for no \(A\in M\) does \(\operatorname{Mod}(A)\subseteq \operatorname{Mod}(\psi)\) hold. Then, by Theorem 2, \(\mathfrak M\in \operatorname{Mod}(\sim\psi\cup M)\) for some model \(\mathfrak M\). According to the Löwenheim—Skolem theorem for \(L_{\theta\theta}\), the model \(\mathfrak M\) has a submodel \(\mathfrak M_1\) of cardinality less than \(\theta\) such that \(\mathfrak M_1\in \operatorname{Mod}(\sim\psi)\). Using the method of diagrams, we establish that there exists an existential axiom \(\lambda\) (of the logic \(L_{\theta\theta}\)), true in \(\mathfrak M\), for which \(\operatorname{Mod}(\sim\lambda)\supseteq \operatorname{Mod}(\psi)\), which is impossible.

The following result also holds:

An axiom \(\psi\) of the logic \(L_{\omega_1\omega}\) \((L_{\theta\theta})\) is preserved under passage to extensions if and only if \(\psi\) is equivalent to an existential axiom of the logic \(L_{\omega_1\omega}\) \((L_{\theta\theta})\).

In the work of Lopez—Escobar \((^1)\) it was shown that: \((*)\) an axiom of the logic \(L_{\omega_1\omega}\) is preserved under passage to homomorphic images if and only if it is equivalent to a positive axiom of the logic \(L_{\omega_1\omega}\).

The result \((*)\) was proved by means of the interpolation theorem of the logic \(L_{\omega_1\omega}\). The interpolation theorem in the logic \(L_{\theta\theta}\), where \(\theta\) is strictly inaccessible, does not hold (see \((^5)\)). Nevertheless, the following holds.

Theorem 4. An axiom \(\psi\) of the logic \(L_{\omega_1\omega}\) \((L_{\theta\theta})\) is equivalent to an existential and positive axiom of the logic \(L_{\omega_1\omega}\) \((L_{\theta\theta})\) if and only if \(\psi\) holds in all homomorphic images of extensions of models of \(\psi\).

1. We shall call a class of models \(K\) \((\alpha,\beta)\)-\(F\)-axiomatizable, where \(F\) is some set of axioms of \(L_{\alpha\beta}\), if \(K\) consists of those and only those models in which a certain system of axioms from \(F\) holds. We shall call the language \(L_{\alpha\beta}\) local if, for every \((\alpha,\beta)\)-axiomatizable class \(K\), for every model \(\mathfrak N\), embeddability of each of its submodels \(\mathfrak N_1\) of cardinality less than \(\alpha\) in \(K\) entails embeddability of \(\mathfrak N\) in \(K\).

The well-known Henkin theorem shows the locality of the language \(L_{\omega\omega}\). At the same time, Tarski’s results \((^4)\) show that the languages \(L_{\alpha\alpha}\), where \(\alpha\) is singular, are not local. It is unknown whether the languages \(L_{\theta\theta}\), where \(\theta\) is strictly inaccessible, are local. One may hope that characteristics of \((\theta,\theta)\)-\(F\)-axiomatizable classes, even in terms of extensions of mappings, will help solve this problem.

Below we give characteristics of \((\alpha,\beta)\)-universally axiomatizable classes (where either \(\alpha=\beta\), or \(\mu<\alpha,\eta<\beta\Rightarrow \mu^\eta<\alpha\)), and characteristics of \((\theta,\theta)\)-axiomatizable classes, where \(\theta\) is strictly inaccessible*.

Let \(\alpha,\alpha_i\) be cardinals \((\alpha\geq\omega)\), \(\alpha>\alpha_i,\ i=1,\ldots;\ \mathfrak N,\mathfrak M\) models, \(\mathfrak N_1\subseteq S_{\alpha_1}(\mathfrak N)\). Then: 1) an isomorphic embedding of \(\mathfrak N_1\) into \(\mathfrak M\) will be called an \(\alpha\alpha(\alpha_1)\)-embedding of \(\mathfrak N\) into \(\mathfrak M\) relative to \(\mathfrak N_1\): \(\mathfrak N\leq_{\alpha\alpha}(\mathfrak N_1;\alpha_1)\mathfrak M\); 2) an isomorphism \(\varphi\) of the model \(\mathfrak N_1\) into \(\mathfrak M\) will be called an \(\alpha\alpha(\alpha_1,\alpha_2,\ldots,\alpha_{l+1})\)-embedding of \(\mathfrak N\) into \(\mathfrak M\) relative to \(\mathfrak N_1\): \(\mathfrak N\leq_{\alpha\alpha}(\mathfrak N_1;\alpha_1,\ldots,\alpha_{l+1})\mathfrak M\), if for every \(\alpha_2\)-extension \(\mathfrak M_2\) of the model \(\varphi(\mathfrak N_1)=\mathfrak M_1\) in \(\mathfrak M\) there is an \(\alpha_2\)-extension \(\mathfrak N_2\) of the model \(\mathfrak N_1\) in \(\mathfrak N\) and there exists an \(\alpha\alpha(\alpha_1+\alpha_2,\ldots,\alpha_{l+1})\)-embedding of \(\mathfrak M_2\) into \(\mathfrak N_2\) coinciding with \(\varphi^{-1}\) on \(\mathfrak M_1\).

We shall write \(\mathfrak N <^{\beta}_{\alpha}(\mathfrak N_1;\alpha_1)\mathfrak M\), \(\mathfrak N_1\in S_{\alpha_1}(\mathfrak N)\), if \(\mathfrak N_1 <^{\beta}_{\alpha}\mathfrak M\) in the sense of \((^1)\).

The following theorem generalizes the results of \((^{1,4})\).

Theorem 5. Let \(K,K_1\) be classes of arbitrary signature \(\sigma\), \(K\subseteq K_1\). Then condition 1) is equivalent to condition 2), and condition 3) to condition 4).

* Axioms transferred into \(L_{\theta\theta}\) are considered.

1) \(K\) is \((\alpha,\beta)\)-universally axiomatizable in the class \(K_1\), and for all \(\mu<\alpha\) and \(\eta<\beta\), \(\mu^\eta<\alpha\).

2) If, for a model \(\mathfrak A\in K_1\), for every system \(R\subseteq\sigma\), \(\overline{\overline R}<\alpha\), and every \(\mathfrak A_1\in S_{\alpha_1}(\mathfrak A)\), \(\overline{\overline{\mathfrak A_1}}=\alpha_1<\alpha\), there exists such an \(\mathfrak M^R\) in \(K_R\) (see \((^2)\)) that \(\mathfrak A_R\prec_{\alpha\beta}(\mathfrak A_{1R};\alpha_1)\mathfrak M^R\), then \(\mathfrak A\in K\) (and for all \(\mu<\alpha\), \(\eta<\beta\), \(\mu^\eta<\alpha\)).

3) \(K\) is \((\alpha,\alpha)\)-universally axiomatizable in \(K_1\).

4) If, for \(\mathfrak A\in K_1\), for every system \(R\subseteq\sigma\), \(\overline{\overline R}<\alpha\), and every \(\mathfrak A_1\in S_{\alpha_1}(\mathfrak A)\), \(\overline{\overline{\mathfrak A_1}}=\alpha_1<\alpha\), there exists \(\mathfrak M^R\) in \(K_R\) such that \(\mathfrak A_R\leq_{\alpha\alpha}(\mathfrak A_{1R};\alpha_1)\mathfrak M^R\), then \(\mathfrak A\in K\).

The proofs of the theorems are carried out by the method of \((^5)\).

Theorem 6. Let \(K,K_1\) be classes of models of signature \(\sigma\), \(\overline{\overline\sigma}<\theta\), \(K\subseteq K_1\). Then the following conditions are equivalent:

1) \(K\) is \((\theta,\theta)\)-\(\underbrace{\forall\exists\ldots}_{l}\)-axiomatizable in \(K_1\), where \(\theta\) is strongly inaccessible.

2) If, for a model \(\mathfrak A\in K_1\), any tuple \((a_1,\ldots,a_l)\) of length \(l\), \(a_i<\theta\), \(i=1,\ldots,l\), and any \(\mathfrak A_1\in S_{\alpha_1}(\mathfrak A)\) in \(K\), there is an \(\mathfrak M\) such that \(\mathfrak A\leq_{\theta\theta}(\mathfrak A_1;\alpha_1,\ldots,\alpha_l)\mathfrak M\), then \(\mathfrak A\in K\).

It is not difficult to show that the class of models \(\mathfrak M\) satisfying the condition
\(\mathfrak A\leq_{\theta\theta}(\mathfrak A_1;\alpha_1,\ldots,\alpha_l)\mathfrak M\) is described by an axiom of the form \(\exists\forall\ldots\) of the logic \(L_{\theta\theta}\).

Hence it follows that the model \(\mathfrak A\) will be an \(F^{\theta\theta}_{A_l}\)-touching point of the class \(K\) (see \((^6)\)) if and only if \(\mathfrak A\) satisfies condition 2), which proves theorem 6.

Let us note that a model will be an \(F^{\theta\theta}_{A}\)-touching point of the class \(K\) if and only if \(\mathfrak A\) is an \(F^{\theta\theta}_{A_l}\)-touching point (or an \(F^{\theta\theta}_{E_l}\)-touching point) for any \(l\in\omega\). Hence we obtain:

Theorem 7. Let \(K\) be a class of signature \(\sigma\), \(\overline{\overline\sigma}<\theta\), where \(\theta\) is strongly inaccessible. Then the following conditions are equivalent:

1) \(K\) is \((\theta,\theta)\)-axiomatizable.

2) If, for a model \(\mathfrak A\), any tuple \((\alpha_1,\ldots,\alpha_l)\), and any \(\mathfrak A_1\in S_{\alpha_1}(\mathfrak A)\) in \(K\), there is an \(\mathfrak M\) such that \(\mathfrak A\leq_{\theta\theta}(\mathfrak A_1;\alpha_1,\ldots,\alpha_l)\mathfrak M\), then \(\mathfrak A\in K\).

3) If, for a model \(\mathfrak A\), any tuple \((\alpha_1,\ldots,\alpha_l)\), there is in \(K\) an \(\mathfrak M\) such that \(\mathfrak M\leq_{\theta\theta}(\alpha_1,\ldots,\alpha_l)\mathfrak A\), then \(\mathfrak A\in K\).

In addition, using the results of \((^6)\), one can obtain a large series of characteristics of \((\theta,\theta)\)-axiomatizable classes.

Proposition 1. Let \(\mathfrak A,\mathfrak M\) be models. Then:

1) \(\mathfrak A\) and \(\mathfrak M\) are \(L_{\alpha\alpha}\)-universally equivalent if and only if
\(\mathfrak A\leq_{\alpha\alpha}[1]\mathfrak M\) and \(\mathfrak M\leq_{\alpha\alpha}[1]\mathfrak A\).

2) \(\mathfrak A\) and \(\mathfrak M\) are \(L_{\theta\theta}\)-elementarily equivalent if and only if
\(\mathfrak A\leq_{\theta\theta}[l]\mathfrak M\) and \(\mathfrak M\leq_{\theta\theta}[l]\mathfrak A\) for any \(l\in\omega\) (where \(\theta\) is strongly inaccessible).

Here \(\mathfrak A\leq_{\alpha\alpha}[l]\mathfrak M\) denotes that for any sequence \((\alpha_1,\ldots,\alpha_l)\) and any \(\mathfrak A_1\in S_{\alpha_1}(\mathfrak A)\),
\(\mathfrak A\leq_{\alpha\alpha}(\mathfrak A_1;\alpha_1,\ldots,\alpha_l)\mathfrak M\).

From proposition 1 it follows that

\(L_{\alpha\alpha}\)-universal and \(L_{\theta\theta}\)-elementary equivalences are preserved under direct products (cf. \((^1)\)).

The author expresses deep gratitude to S. R. Kogalovskii for valuable advice and comments relating to this work.

Received
1 VI 1967

CITED LITERATURE

\(^1\) E. G. K. Lopez-Escobar, Bull. Acad. polon. sci., ser. mat., 13, No. 6, 383 (1965).
\(^2\) A. I. Mal’tsev, Tr. IV Matem. s”ezda, 1, L., 1963.
\(^3\) The Theory of Models (Proc. of the 1963 Intern. Symposium at Berkeley), Amsterdam, 1965, p. 329 (158).
\(^4\) A. Tarski, Colloq. math., 6, 171 (1958).
\(^5\) A. D. Taimanov, Algebra i logika, 1, No. 4 (1962).
\(^6\) S. R. Kogalovskii, DAN, 136, No. 6 (1961).
\(^7\) J. Mallitz, Not. Am. Math. Soc., 13, 128 (1966).
\(^8\) C. Karp, Languages with Expressions of Infinite Length. Amsterdam, 1964.