UDC 519.40

MATHEMATICS

S. R. KOGALOVSKII

SOME REMARKS ON LOGIC OF HIGHER ORDERS

(Presented by Academician P. S. Novikov, 13 IV 1967)

In this note the results of ((^{1,2})) are strengthened.

We shall say that an ordinal (\alpha) is defined by a formula (\sigma) (of finite or infinite order) if (\sigma) is true on all sets well ordered by type (\alpha), and only on them.

By induction we define the notion of a generally definable ordinal:

1. An ordinal definable by a formula of finite order is a generally definable ordinal.

2. If an ordinal (\alpha) is defined by a formula such that the types of all variables occurring in it are generally definable ordinals, then (\alpha) is a generally definable ordinal.

We shall call a formula generally definable if the types of all variables occurring in it are generally definable.

Theorem 1. For every generally definable formula (\sigma) one can effectively construct a second-order formula (\Phi(\sigma)) such that the truth (satisfiability, categoricity) of (\sigma) is equivalent to the truth (satisfiability, categoricity) of (\Phi(\sigma)).

This theorem strengthens Theorem 5 of ((^{2})) and, consequently, Hintikka’s result ((^{3})) on the reduction of type theory. From this theorem and a well-known result of A. Tarski ((^{4})) it follows that

Theorem 2. For every effective enumeration of second-order formulas, the set of numbers of true second-order formulas is not definable in arithmetic by any generally definable formula.

Let, as in ((^{1,2})), (k(\sigma)) be the least cardinal in the spectrum of the formula (\sigma), and let (k_0) be the upper bound of the set ({k(\sigma)}), where ({\sigma}) is the collection of all satisfiable second-order formulas. It is unknown whether (k_0) coincides with the upper bound of the set ({k(\tau)}), where ({\tau}) is the set of all satisfiable formulas whose orders are less than (k_0). In the case of coincidence, one can, assuming the axiom of choice, prove that for every ordinal (\alpha < k_0) the set of numbers of true second-order formulas is not definable in arithmetic by any formula of order (\alpha). At the same time, the set of numbers of true second-order formulas is definable in arithmetic by a formula of order ((k_0 + 1)).

Let us express the result established by Theorem 2 in another form. We shall call a structure (\mathfrak{S}) generally definable if there exists a generally definable formula true on all structures isomorphic to (\mathfrak{S}), and only on them. We shall call a structure arithmetized if in it there is defined a sequence, of type (\omega), of elements (\Delta_0, \Delta_1, \ldots). The set of natural numbers (N) will be called definable (generally definable) in an arithmetized structure (\mathfrak{S}) if there exists a formula of finite order (a generally definable formula) (\sigma) in the signature (\mathfrak{S}) such that (n \in N) is equivalent to the truth of (\sigma(\Delta_n)) for every natural (n). For every effective enumeration of second-order formulas, the set of numbers of true second-order formulas is not definable in any arithmetized structure definable by a second-order formula

order. If some set of natural numbers is generally definable in a generally definable arithmetized structure, then it is definable by a second-order formula in some structure definable by a second-order formula. Hence it follows that the set of numbers of true second-order formulas will not be generally definable in any generally definable arithmetized structure. Hence, and from the known theorem of Zykov ($^{5}$), it follows:

Theorem 3. For every effective enumeration of second-order formulas, the set of numbers of true second-order formulas of the form

[
(\exists X)(Y)\gamma,
]

where $X$ is a binary and $Y$ a unary predicate variable, and $\gamma$ contains no quantifiers over predicate variables, is not generally definable in any generally definable arithmetized structure.

We shall call the genus of a generally definable formula the set of types of all variables occurring in it. We shall call the genus of a system $\Sigma$ of generally definable formulas the union of the set of genera of the formulas from $\Sigma$.

The following theorem, proved with the aid of Theorem 1, generalizes Theorem 1 from ($^{2}$).

Theorem 4. Let $\Sigma$ be a system of generally definable formulas with a finite set of nonlogical constants such that the set of degrees of formulas defining ordinals from the genus $\Sigma$ is bounded by a generally definable ordinal, and the set of numbers of formulas from $\Sigma$ (for some effective enumeration of the set of generally definable formulas) is defined in arithmetic by a generally definable formula. Then the system $\Sigma$ is equivalent to some generally definable formula $\sigma$.

Without giving the exact formulation, in view of its cumbersomeness, let us only note that the relation between the genus $\Sigma$ and the genus $\sigma$ is analogous to the relation between the degrees of a definable system of formulas and an equivalent formula, established by Theorem IV of ($^{1}$).

We shall say that a set of ordinals $\mathfrak{D}$ is defined by a formula $\sigma$ if $\sigma$ is true on all sets from $\mathfrak{D}$ that are fully ordered by type, and only on them. We shall call a set of ordinals generally definable if it is defined by some generally definable formula. With the aid of Theorems 1 and 4 one proves:

Theorem 5. For every generally definable formula $\sigma$ and every generally definable set $\mathfrak{D}$ of generally definable ordinals, one can effectively construct second-order formulas $\Phi(\sigma)$ and $\Psi(\sigma)$ such that:

1) the equivalence of $\sigma$ to some generally definable formula whose genus is included in $\mathfrak{D}$ is equivalent to the truth of $\Phi(\sigma)$;

2) the equivalence of $\sigma$ to some system of generally definable formulas whose genus is included in $\mathfrak{D}$ is equivalent to the truth of $\Psi(\sigma)$.

This theorem generalizes and strengthens Theorem VI of ($^{1}$). It implies, in particular, that for every generally definable formula $\sigma$ and every natural number $n$ one can effectively construct second-order formulas $\Phi(\sigma)$ and $\Psi(\sigma)$ such that the equivalence of $\sigma$ to some formula of the $n$-th order is equivalent to the truth of $\Phi(\sigma)$, and the equivalence of $\sigma$ to some system of formulas of the $n$-th order is equivalent to the truth of $\Psi(\sigma)$.

The method of proof of Theorem 1 (see also the proof of Theorem 1.1 from ($^{1}$)) shows that the problem of characterizing the spectra of generally definable formulas reduces to the problem of characterizing the spectra of second-order formulas. The complexity of the latter problem is indicated, in part—

...that in the sentence of the strict axiom of infinity there occurs a formula of the 2nd order whose spectrum has the greatest cardinal, greater than (k_0). With the aid of Theorems 1 and 4 one proves

Theorem 6. For every recursively definable set (\mathfrak{D}) of recursively definable ordinals there exists a formula of the 2nd order whose spectrum is not predicatively definable in the sense of ((2)) by any formula whose genus is included in (\mathfrak{D}).

Ivanovo Textile Institute
named after I. V. Frunze

Received
3 III 1967

REFERENCES

1. S. R. Kogalovskii, Izv. Vyssh. Uchebn. Zaved., Mathematics, No. 1, 50, 89 (1966).
2. S. R. Kogalovskii, DAN, 171, No. 6, 1272 (1966).
3. K. J. J. Hintikka, Acta Philos. Fennica, No. 8, 566 (1955).
4. A. Tarski, A. Mostowski, R. Robinson, Undecidable Theories, Amsterdam, 1953.
5. A. A. Zykov, Izv. AN SSSR, ser. matem., 17, No. 3, 63 (1953).
6. S. R. Kogalovskii, Abstracts of reports at the International Mathematical Congress in Moscow, 1966, Information Bulletin, No. 6.