UDC 519.40

S. R. KOGALOVSKII

ON LOGIC OF HIGHER LEVELS

(Presented by Academician P. S. Novikov, 19 II 1966)

In this note the results from (¹), concerning logic of higher levels, are strengthened. Speaking of the models of one formula or another, we shall mean its standard models.

In (¹) the following was proved.

Theorem I. If a class of infinite structures is defined by a set of formulas of the \(k\)-th level, definable by a formula of the \(n\)-th level, then it is definable by a formula of the \([\max(n,k)+1]\)-st level.

With the help of Theorem I the following theorem can be proved, strengthening Theorem V from (¹).

Theorem 1. For every natural \(n\) there exists a closed (i.e., containing no free variables) formula of the \((n+1)\)-st level which is not equivalent to any set of formulas of lower levels*.

By the spectrum of a formula we shall mean the class of cardinalities of its models. The least of the cardinalities belonging to the spectrum of a satisfiable formula \(\sigma\) will be denoted by \(k(\sigma)\). A cardinal \(\mathfrak n\) will be called definable if there exists a formula all of whose models have cardinality \(\mathfrak n\). Obviously, the cardinal \(\mathfrak n\) is definable if \(\mathfrak n=k(\sigma)\) for some formula \(\sigma\).

It can be shown that the least of the cardinals not definable by any set of closed formulas of the \(n\)-th level is defined by a formula of the \((n+1)\)-st level. This proves Theorem 1.

From Theorem I and from the fact that the notion of satisfiability for formulas of the \(n\)-th level is defined by a recursive set of formulas of the \(n\)-th level, there follows the well-known result of Tarski (²): the semantics of the language of the \(n\)-th level is expressible in the language of the \((n+1)\)-st level.

Let \(F\) be the set of formulas of the \(n\)-th level (from \(\mathfrak S\)) that have no free variables other than object variables. Denote by \(E_n\) the formula in \(F\) with Gödel number \(n\), and by \(E_n(n)\) the formula obtained from \(E_n\) by replacing all occurrences of free variables by \(n\) (more precisely, by the term \(\Delta_n\), interpreted as the natural number \(n\)). By \(D(n)\) denote the Gödel number of the formula \(E_n(n)\). If, however, \(n\) is not the Gödel number of any formula in \(F\), we put \(D(n)=0\). Obviously, \(D\) is a recursive function and, hence, there exists an elementary formula \(\Phi\) such that the truth of \(\Phi(n,v)\) is equivalent to \(v=D(n)\).

The proof of the following theorem, established in (²), is almost a word-for-word repetition of Theorem 1, Ch. II, from (³).

Theorem 2. The set \(V\) of Gödel numbers of true formulas from \(F\) is not definable by a formula of the \(n\)-th level.

Suppose, contrary to the assertion, that there exists a formula \(\Psi\) of the \(n\)-th level such that \(n\in V\) is equivalent to the truth of \(\Psi(n)\). Let \(m\) be the Gödel number of the formula (of the \(n\)-th level) \((\forall v)(\Phi(u,v)\to \neg \Psi(v))\). Hence,
\[ E_m(m)=(\forall v)(\Phi(m,v)\to \neg \Psi(v)). \]
If \(E_m(m)\) is true, then \(\neg\Psi(D(m))\) is true. If \(E_m(m)\) is false, then the number \(E_m(m)\) does not belong to \(V\), and hence \(\neg\Psi(D(m))\) is true. Thus, \(\neg\Psi(D(m))\) is true. But

* Two sets of formulas are called equivalent if every model of one of them is a model of the other.

then the formula \(E_m(m)\) is true. Hence, \(D(m)\in V\), and therefore \(\Psi(D(m))\) is true. We have arrived at a contradiction.

In \((^1)\) the following was proved (without using the axiom of choice).

Theorem. There exists an algorithm which, for every formula (of any finite order) \(\sigma\), makes it possible to construct a formula \(\Phi(\sigma)\) of second order such that the satisfiability (truth, categoricity) of \(\sigma\) is equivalent to the satisfiability (truth, categoricity) of \(\Phi(\sigma)\).

Using this theorem and Theorem 2, it is not difficult to show that for every \(n\) there exists a second-order formula \(\Phi\) such that the set of numbers of all formulas \(\Psi\) for which \(\Phi\to\Psi\) is true is not definable by a formula of \(n\)-th order.

Hence one derives (without using the axiom of choice)

Theorem 3. The set of Gödel numbers of true formulas of the extended predicate calculus is not definable in arithmetic.

In \((^1)\) this theorem was proved with the aid of the axiom of choice. The use of the generalized continuum hypothesis proves

Theorem 4. For every cardinal \(\mathfrak n\), the set \(F_{\mathfrak n}\) of Gödel numbers of formulas of the extended predicate calculus that are true on sets of cardinality \(\mathfrak n\) is definable if and only if \(\mathfrak n<\aleph_\omega\). Moreover, for every natural \(n\), the set \(F_{\aleph_n}\) is definable by a formula of \((n+1)\)-st order.

At the same time it is easily proved that the set of numbers of all closed formulas of finite orders that are true on all finite sets is the complement of a recursively enumerable set.

It follows from this, in particular, that for the decidability of the theory of the class \(K_{\mathrm{fin}}\), consisting of the finite models of a class \(K\) defined by a single axiom, it is necessary and sufficient that this theory be recursively axiomatizable. (It also follows from this that if the theory of a class \(K\) defined by an elementary axiom is undecidable, then the theory of the class \(K_\infty\), consisting of the infinite models from \(K\), is also undecidable.)

We shall call an ordinal \(\alpha\) definable if there exists a formula of finite order that is true only on sets well ordered in type \(\alpha\). If every ordinal \(\beta\leqslant\alpha\) is definable, then \(\alpha\) will be called a strictly definable ordinal.

Analyzing the proof of Theorem II (contained in \((^1)\)), one may observe that the following stronger statement is valid.

Theorem 5. For every strictly definable ordinal \(\alpha\) there exists an algorithm which, from every formula \(\sigma\) of \(\alpha\)-th order, constructs a formula \(\Phi(\sigma)\) of second order such that the satisfiability (truth, categoricity) of \(\sigma\) is equivalent to the satisfiability (truth, categoricity) of \(\Phi(\sigma)\), and \(k(\sigma)<k(\Phi(\sigma))\) under the assumption of the axiom of choice.

Let \(\Phi\) be the set of all satisfiable formulas of second order, and let \(k_0\) be the least upper bound of the set \(\{k(\varphi)\}_{\varphi\in\Phi}\). From Theorem 5 it follows, under the assumption of the axiom of choice, that for every strictly definable ordinal \(\alpha\) and every formula \(\sigma\) of \(\alpha\)-th order one has \(k(\sigma)<k_0\).

Let \(\Psi\) be the set of formulas of second order satisfiable on infinite domains. For every formula \(\varphi\) from \(\Psi\) denote by \(\varphi^*\) the formula
\[ (\exists X)(\widetilde{\varphi}\wedge\rho), \]
where \(\widetilde{\varphi}\) is obtained from \(\varphi\) by relativizing the individual variables to \(X\), and \(\rho\) asserts that \(X\) is infinite. Suppose that the set of Gödel numbers of formulas from \(\Psi\) is definable by a formula of \(\alpha\)-th order for some strictly definable ordinal \(\alpha\). Then the set of Gödel numbers of formulas from the set \(\{\varphi^*\}_{\varphi\in\Psi}\) is also definable by a formula of \(\alpha\)-th order. The class defined by the latter set is defined by the formula \(\sigma^*\) of \((\alpha+1)\)-st order. This set has no models of cardinalities less than \(k_0\).

* See \((^1)\), Theorem III.

On the other hand, it must be that \(k(\sigma)<k_0\). The contradiction obtained proves the validity (under the assumption of the axiom of choice) of the following theorem, which strengthens Theorem 3.

Theorem 6. For no strictly definable ordinal \(\alpha\) is the set of Gödel numbers of true formulas of the extended predicate calculus definable by any formula of level \(\alpha\).

Let us denote by \(A\) the set of closed formulas of the second level having the form \((\exists X)(Y)\mathfrak A(X,Y,x_1,\ldots,x_n)\), where \(X\) is a four-place and \(Y\) a one-place predicate variable, and \(\mathfrak A\) is a formula containing no quantifiers over predicate variables. From the fact that for every formula \(\sigma\) of the second level one can effectively construct a formula of the form \(A\), whose truth is equivalent to the truth of \(\sigma\), there follows the stronger assertion: for no strictly definable ordinal \(\alpha\) is the set of Gödel numbers (for any Gödel numbering) of true formulas of the form \(A\) definable in arithmetic of level \(\alpha\).

We shall say that a class of cardinals \(C\) is predicatively defined by a sentence \(\sigma\) if it consists of cardinals \(\aleph_\alpha\) such that well-ordered sets of ordinals \(\alpha\) satisfy \(\sigma\). If a class \(C\) is predicatively defined by a sentence of the \(n\)-th level, then it is the spectrum of some sentence of the \(n\)-th level. At the same time, with the help of the theorems given above, for Gödel set theory \(\Sigma^*\) one can prove that there exists a sentence of the second level whose spectrum is a set not predicatively definable by any sentence of level \(\beta\), whatever the strictly definable ordinal \(\beta\) may be.

Ivanovo Textile Institute
named after M. V. Frunze

Received
2 II 1966

REFERENCES

1. S. R. Kogalovsky, Izv. Vyssh. Uchebn. Zaved., Matematika, No. 1 (50), 89 (1966).
2. A. Tarski, Studia Philos., 1, 261 (1936).
3. A. Tarski, A. Mostowski, R. Robinson, Undecidable Theories, Amsterdam, 1953.