MATHEMATICS

A. D. TAIMANOV

A CHARACTERIZATION OF FINITELY AXIOMATIZABLE CLASSES OF MODELS

(Presented by Academician A. I. Mal'cev, 15 XII 1960)

In the note \((^1)\) a characterization was given of finitely axiomatizable classes of models for certain kinds of axioms. In the present note a characterization is given of finitely axiomatizable classes of models without restriction on the kinds of axioms.

A finite system of axioms of the narrow predicate calculus is equivalent to a single axiom of the form

\[ \forall x^1_1 x^1_2 \ldots x^1_{n_1} E x^2_1 \ldots x^2_{n_2} \ldots Q x^l_1 \ldots x^l_{n_l}\ \nu (x^1_1,\ldots,x^1_{n_1},\ldots,x^l_{n_l}) \tag{1} \]

or of the form

\[ E x^1_1 x^1_2 \ldots x^1_{n_1}\ \forall x^2_1 \ldots x^2_{n_2} \ldots Q x^l_1 \ldots x^l_{n_l}\ \nu (x^1_1,\ldots,x^1_{n_l},\ldots,x^l_{n_l}), \tag{2} \]

where \(Q=\forall\) or \(E\), depending on the parity of \(l\). Therefore it is enough to characterize the class of models described by an axiom of the form (1) or (2).

1. To characterize the class of models described by an axiom of the form (1), we introduce the following definition, which is a strengthening of the analogous definition from \((^1)\).

Definition 1. For a given tuple \((n_1,n_2,\ldots,n_l)\), a given model \(\mathfrak M\), and a submodel \(\mathfrak N_{n_1}\), \(\mathfrak N_{n_1}\in S_{n_1}(\mathfrak M)\), we define the notion of an \((n_1,n_2,\ldots,n_{l-1},n_l)\)-covering of the model \(\mathfrak M\) with respect to the submodel \(\mathfrak N_{n_1}=(a_1,a_2,\ldots,a_{n_1})\). For \(l=2\) the model \(\mathfrak M\) is called an \((n_1,n_2)\)-covering of the model \(\mathfrak R\) with respect to \(\mathfrak N_{n_1}\), if there exists an isomorphic mapping \(\varphi_1\) of the model \(\mathfrak N_{n_1}\) into \(\mathfrak M\) such that for every \(n_2\)-extension \(\mathfrak M_{n_1 n_2}\) (containing not more than \(n_1+n_2\) elements) of the model \(\varphi(\mathfrak N_{n_1})\) in \(\mathfrak M\) there exists an isomorphic mapping \(\varphi_2\) of the model \(\mathfrak M_{n_1 n_2}\) into \(\mathfrak R\), coinciding with \(\varphi_1^{-1}\) on \(\mathfrak N_{n_1}\). We shall write this briefly as follows:

\[ \mathfrak R \preccurlyeq_{(\mathfrak N_{n_1},\, n_1,\, n_2)} \mathfrak M. \]

The mappings \(\varphi_1,\varphi_2\) will be called admissible. Suppose that an \((n_1,n_2,\ldots,n_{l-1})\)-covering has been defined, and define an \((n_1,n_2,\ldots,n_l)\)-covering.

Consider two cases:

1) \(l-1=2s\). Then \(\mathfrak R \preccurlyeq_{(\mathfrak N_{n_1},\, n_1,\, n_2,\ldots,n_l)} \mathfrak M\) means that \(\mathfrak R \preccurlyeq_{\mathfrak N_{n_1},\, n_1,\ldots,n_{l-1}} \mathfrak M\), and for every sequence of admissible mappings \(\varphi_1,\varphi_2,\ldots,\varphi_{l-1}\), for every \(n_l\)-extension \(\mathfrak M_{n_1 n_2\ldots n_{l-1}n_l}\) of the model \(\mathfrak N_{n_1 n_2\ldots n_{l-1}}\) in \(\mathfrak M\), there exists an isomorphic mapping \(\varphi_l\) of the model \(\mathfrak N_{n_1 n_2\ldots n_{l-1}}\) into \(\mathfrak M\), coinciding with \(\varphi_{l-1}^{-1}\) on the model \(\mathfrak N_{n_1 n_2\ldots n_{l-1}}\).

2) \(l-1=2s+1\). Then \(\mathfrak R \preccurlyeq_{(\mathfrak N_{n_1},\, n_1,\, n_2,\ldots,n_l)} \mathfrak M\) means that \(\mathfrak R \preccurlyeq_{(\mathfrak N_{n_1},\, n_1,\ldots,n_{l-1})}\mathfrak M\), and for every sequence of admissible mappings \(\varphi_1,\varphi_2,\ldots,\varphi_{l-1}\), for every \(n_l\)-extension \(\mathfrak M_{n_1 n_2\ldots n_{l-1}n_l}\) of the model \(\mathfrak N_{n_1 n_2\ldots n_{l-1}}\) in \(\mathfrak M\), there exists an isomorphic mapping \(\varphi_l\) of the model \(\mathfrak M_{n_1 n_2\ldots n_{l-1}n_l}\) into \(\mathfrak R\), coinciding with \(\varphi_{l-1}^{-1}\) on the model \(\mathfrak N_{n_1 n_2\ldots n_{l-1}}\). Thus, for every tuple \((n_1,n_2,\ldots,n_l)\)

defined an \((n_1,n_2,\ldots,n_l)\)-covering of the model \(\mathfrak M\) with respect to \(\mathfrak M_{n_1}\), i.e.

\[ \mathfrak N \leqslant_{(\mathfrak M_{n_1},\, n_1,n_2,\ldots,n_l)} \mathfrak M . \tag{3} \]

Definition 2. If, for the given tuple \((n_1,n_2,\ldots,n_l)\) and for any submodel \(\mathfrak M_{n_1}\) from \(S_{n_1}(\mathfrak M)\), (1) holds, then we write

\[ \mathfrak N \leqslant_{(n_1,n_2,\ldots,n_l)} \mathfrak M . \tag{4} \]

If, for any tuple \((n_1,n_2,\ldots,n_l)\) of fixed rank \(l\), inequality (4) holds, then \(\mathfrak N \leqslant_{(l)} \mathfrak M\).

Let \(L\) be the class of all models of type \(\mathfrak M=\langle M,P\rangle\). For a given number \(n\) there exists in the class \(L\) a finite number \(\varphi_L(n)\) of pairwise nonisomorphic models containing no more than \(n\) elements. For given models \(\mathfrak M\) from \(L\) and a submodel \(\mathfrak M_n\) from \(S_n(\mathfrak M)\), there exist no more than \(\varphi_L(n+m)\) pairwise nonisomorphic \(m\)-extensions of the model \(\mathfrak M_n\) in the model \(\mathfrak M\).

Lemma 1. The class of all models \(\mathfrak M\) satisfying the condition \(\mathfrak N \leqslant_{(\mathfrak M_{n_1},\, n_1,\ldots,n_l)} \mathfrak M\) for given \(\mathfrak N,\mathfrak M_{n_1}, n_1,n_2,\ldots,n_l\) is described by an axiom of the form

\[ E x^1_1 \ldots x^1_{n_1}\, \forall x^2_1 \ldots x^2_{n_2}\, E x^3_1 \ldots x^3_{m_3}\, \forall x^4_1 \ldots x^4_{n_4}\, E x^5_1 \ldots x^5_{m_5}\,\ldots \]

\[ \ldots Q x^l_1 \ldots x^l_{m_l}\, v\bigl(x^1_1,\ldots,x^l_{n_l}\bigr), \]

where \(Q=E\) if \(l\) is odd, and \(Q=\forall\) if \(l\) is even; \(v(x^1_1,x^1_2,\ldots,x^l_{n_l})\) contains no quantifiers; the number \(m_i\) is not greater than

\[ n_i\varphi_L(n_1+n_2)\,\varphi_L(n_1+n_2+n_3)\cdots \varphi_L(n_1+n_2+\cdots+n_i),\quad i=3,5,7,\ldots . \]

This lemma is a strengthening of the analogous lemma from \((^1)\). Hence it follows:

Theorem 1. The following conditions are equivalent:

1) The class \(K\) is described by an axiom of the form (1).

2) There exists a number \(N\) such that, if for a model \(\mathfrak A\) and any of its submodels \(\mathfrak A'\) from \(S_N(\mathfrak A)\) (containing no more than \(N\) elements) there is found in \(K\) a model \(\mathfrak B_{\mathfrak A'}\) satisfying the condition

\[ \mathfrak A \leqslant_{(\mathfrak A',\, \underbrace{N,N,\ldots,N}_{l})} \mathfrak B_{\mathfrak A'}, \]

then \(\mathfrak A\in K\).

1. An axiom of the form (2) may be regarded as an axiom of the form (1), where the number of variables \(x^1\) is equal to zero. Therefore the following holds:

Theorem 2. The following conditions are equivalent:

1) The class \(K\) is described by an axiom of the form (2).

2) There exists a number \(N\) such that, if for a model \(\mathfrak A\) one can indicate a model \(\mathfrak B_{\mathfrak A}\) from \(K\) satisfying the condition

\[ \mathfrak B_{\mathfrak A} \leqslant_{(\underbrace{N,N,\ldots,N}_{l})} \mathfrak A , \tag{5} \]

then \(\mathfrak A\in K\).

From these theorems follow the characterizations of finitely axiomatizable classes of models given in \((^{1,2})\).

1. In order to obtain a strengthening of the Łoś–Tarski theorem, we introduce the following definitions:

Definition 3. A submodel \(\mathfrak N\) of a model \(\mathfrak M\) is called an \((N,l)\)-submodel if \(\mathfrak N \leqslant_{(\underbrace{N,N,\ldots,N}_{l})}\mathfrak M\). The class of all \((N,l)\)-submodels of the model \(\mathfrak M\) will be denoted by \(S_{(N,l)}(\mathfrak M)\). Then

\[ \mathfrak N\in S_{(N,l)}(K)\equiv (E\mathfrak M)\bigl(\mathfrak M\in K\ \&\ \mathfrak N\in S_{(N,l)}(\mathfrak M)\bigr). \]

Definition 4. A class of models \(K\) is called an \((N,l)\)-class if it has the following property: if, for a model \(\mathfrak A\), any submodel \(\mathfrak A_N\)

from $S_N(\mathfrak A)$ in the class $K$ there is a model
$\mathfrak M_{\mathfrak A,\mathfrak A_1,\underbrace{N,N,\ldots,N}_{l}}$ such that

\[ \mathfrak A \leqslant_{\langle \mathfrak A_N,\underbrace{N,N,\ldots,N}_{l}\rangle} \mathfrak M_{\mathfrak A,\mathfrak A_N,\underbrace{N,\ldots,N}_{l}}, \]

then there exists a model $\mathfrak M$ in $K$ containing $\mathfrak A$ and

\[ \mathfrak A \leqslant \langle \underbrace{N,N,\ldots,N}_{l}\rangle \mathfrak M, \]

i.e. $\mathfrak A \in S_{(N,l)}(\mathfrak M)$.

Theorem 3. The following conditions are equivalent:
1) The class $K$ is defined by an axiom of the form (1).
2) There exist numbers $N,l$ such that $K$ is an $(N,l)$-class and $S_{N,l}(K)\subset K$.

Proof. If the class $K$ is described by an axiom of the form (1), then $K$ is an $(N,l)$-class for some pair of numbers $N,l$.

This assertion is an analogue of Henkin’s theorem.

Theorem 4. If $K$ is an $(N,l)$-class for some pair of numbers $N,l$, then $S_{(N,l)}(K)$ is described by an axiom of the form (1).

Mathematical Institute
of the Siberian Branch of the Academy of Sciences of the USSR

Received
10 XII 1960

REFERENCES

¹ A. D. Taimanov, Izv. AN SSSR, Ser. Mat., 25 (1961). ² R. Vaught, Proc. Acad. van Wetensch., A 57, No. 5 (1954). ³ R. Vaught, J. Symbolic Logis, 16, 589 (1954).