UDC 519.52

MATHEMATICS

V. A. LYUBETSKII

FROM THE EXISTENCE OF A NONMEASURABLE SET OF TYPE \(A_2\) THERE FOLLOWS THE EXISTENCE OF AN UNCOUNTABLE SET CONTAINING NO PERFECT SUBSET OF TYPE \(CA\)

(Presented by Academician P. S. Novikov, April 27, 1970)

The following problems of descriptive set theory are known: is assertion (I) true: “there exists an uncountable set of type \(CA\) containing no perfect subset”;

is assertion (II) true: “there exists a set of type \(B_2\) that is not Lebesgue measurable.”

The formulation of these problems was refined in terms of axiomatic set theory. Namely, the question was posed as to the derivability of assertions (I) and (II), or of their negations, in one or another axiomatic theory—for example, in Zermelo–Fraenkel theory (denoted by \(ZF\)) or in Gödel–Bernays theory (denoted by \(GB\) or \(\Sigma\)). We note that, by virtue of general metatheorems of axiomatic set theory (see \((^1)\), p. 149), derivability in \(ZF\) of assertions (I), (II), or of their negations is equivalent to the derivability of these same propositions in \(GB\).

P. S. Novikov \((^2)\) showed that the negations of propositions (I) and (II) cannot be derived in \(GB\). In an abstract \((^3)\), Solovay announced, without giving a proof, the existence of a model of \(ZF\) in which, in particular, assertions (I) and (II) are false. The author of the present note also constructed a model of \(ZF\) in which, in particular, assertion (I) is false, and reported this result at the All-Union Symposium on Mathematical Logic (Alma-Ata, June 2–7, 1969; see theorem 3a in the abstracts of that symposium \((^4)\)). At the symposium the following theorem was also reported (not included, however, in the abstracts \((^4)\)):

Theorem 1. There exists a model of the theory \(ZF\) in which (II) is false and (I) is true.

After a number of very diverse propositions that proved to be undecidable by the means of ordinary set theory (for example, propositions (I) and (II), the proposition that (II) follows from (I), etc.), it seems somewhat surprising that one classical question of descriptive set theory can be solved within the framework of the usual naive set theory without any additional assumptions.

Theorem 2. It is derivable in \(ZF\) that from the existence of a nonmeasurable set of type \(A_2\) there follows the existence of an uncountable set of type \(CA\) containing no perfect subset.

Using the following criterion (see \((^{4,6})\)):
\(ZF \vdash\) “there exists \(u\bigl(L^{+}(u)\)—uncountable \(\equiv (I)\bigr)\),” Theorem 2 is easy to derive from the following more general assertion.

Theorem \(2'\).

\[ ZF \vdash \neg u \neg \varphi\bigl(\varphi \in \Sigma^1_2(u) f \{x \mid \operatorname{Icm}\varphi(x)\}\text{—nonmeasurable}\bigr) \equiv R(u)\text{—not of measure }0). \]

In the formulation of Theorem \(2'\) the following abbreviations are used:
\(\varphi \in \Sigma^1_2(u) \cup \Pi^1_2(u)\) means that the formula \(\varphi\) contains only two quantifiers over real numbers (one existential quantifier and one quantifier ...

generality) and any number of quantifiers over natural numbers, prefixed to a predicate decidable relative to the real number \(u\) (for the exact definition see \((^5)\)); \(L^+(u)\) is a term which, from the real number \(u\), yields the set of all real numbers constructive (in the sense of Gödel) relative to \(u\). \(\operatorname{Ist}\varphi\) is the formula \(ZF\), having the following meaning: if the formula \(\varphi\) contains only quantifiers over sets of real or natural numbers, then \(\operatorname{Ist}\varphi\) means that the formula \(\varphi\) is true. It is easy to see that for every \(\varphi\), \(\varphi\in \Sigma^1_2(u)\cup \Pi^1_2(u)\), the set \(\{x\mid \operatorname{Ist}\varphi(x)\}\) is of type \(A_2\) or \(CA_2\). Conversely, every set of real numbers of type \(A_2\) or \(CA_2\) can be described in this sense by a formula \(\varphi\), \(\varphi\in \Sigma^1_2(u)\cup \Pi^1_2(u)\). \(\overline{R}(u)\) is a term which, from the number \(u\), yields the set of real numbers non-random relative to \(u\).

When the problems formulated at the beginning of the present note were posed, the assertions themselves (I) and (II) were understood by some authors (and, first of all, by N. N. Luzin: see \((^7)\), pp. 553–556, 565) differently from assertions of pure existence. What was meant was “effective existence,” i.e., whether one can effectively specify this or that object, even if one allows that such an object exists.

Theorem 3. There is no term \(T\) with free variables \(x_1\ldots x_n\), for which in \(ZF+(I)\) the assertion is provable: “\(\exists a_1\ldots \exists a_n\) (\(a_1\ldots a_n\) are ordinals and \(T(a_1\ldots a_n)\) is an \(A\)-set whose complement is uncountable and contains no perfect kernel).”

We give a sketch of the proof of Theorem 3. First of all, from a term \(T_1\) of type \(\Pi^1_1\) one can construct a term \(T\) which specifies a sieve (see \((^7)\), p. 162) not precisely sifting the set \(\{x\mid T_1(x)\}\). Suppose now the contrary: let the term \(T(a_1\ldots a_n)\), where \(a_1\ldots a_n\) are arbitrary ordinals, specify an uncountable \(CA\)-set containing no perfect kernel; more precisely

\[ \underset{ZF}{\vdash}\ \exists a_1\ldots \exists a_n\ (a_1\ldots a_n \text{ are ordinals } \& T(a_1\ldots a_n) \text{ is} \]

an uncountable \(CA\)-set without a perfect kernel). In particular, it is derivable in \(ZF\) that the set \(T(a_1\ldots a_n)\) is the union of an uncountable number of nonintersecting constituents, each of which is countable. P. S. Novikov indicated an effective method for choosing a point from a \(CA\)-set specified by a sieve (see \((^7)\), p. 617). With the aid of this method, from the term \(T(a_1\ldots a_n)\) one can construct another term \(T'(a_1\ldots a_n)\) possessing the following property: for every ordinal \(\alpha\) such that the \(\alpha\)-th constituent of the sieve \(T(a_1\ldots a_n)\) is nonempty, the term \(T'(a_1\ldots a_n\alpha)\) is a real number from this constituent. The term \(T'(a_1\ldots a_n\alpha)\) therefore specifies an uncountable well-ordered sequence of distinct real numbers, and this circumstance is derivable in \(ZF\), i.e., is true in every model of \(ZF\). Consider the model of \(ZF\) described in the book \((^1)\), p. 267, taking as the \(\alpha\) occurring there the first uncountable ordinal \(\omega_1^M\) of the minimal model \(M\).

We shall denote this model by \(N_h\). It can be shown that in \(N_h\) proposition (I) is valid. The term \(T'\), relativized to \(N_h\), gives an uncountable sequence of real numbers in \(N_h\), definable in \(N_h\) by this term \((T')_N\). On the other hand, using the notion of permutation introduced by A. Lévy in \((^8)\) in connection with another model, one can show that every sequence of real numbers definable in \(N_h\) is countable (in \(N_h\)). This gives a contradiction.

Corollary. There is no term \(T\) without free variables for which in \(ZF+(I)\) the assertion is provable: “\(T\) is an \(A\)-set whose complement is uncountable and contains no perfect kernel.”

Theorem 4. There is no term \(T\) with free variables \(x_1\ldots x_n\), for which in \(ZF+(I)+(II)\) the assertion is provable: “\(\exists a_1\ldots \exists a_n\) (\(a_1\ldots a_n\) are ordinals and \(T(a_1\ldots a_n)\) is a nonmeasurable set).”

Corollary. There is no term \(T\), without free variables, for which the assertion “\(T\) is a nonmeasurable set” is proved in \(ZF + (I) + (II)\).

I wish to thank my supervisor V. A. Uspenskii, who set before me a number of interesting problems (answers to some of them are given here) and constantly helped me in my work, and A. G. Dragalin for repeated discussions of these theorems and for advice on improving their proofs.

Moscow State University
named after M. V. Lomonosov

Received
18 III 1970

REFERENCES

1. P. J. Cohen, Set Theory and the Continuum Hypothesis, 1969.
2. P. S. Novikov, Tr. Mat. Inst. im. V. A. Steklova, 38, 279 (1951).
3. R. Solovay, Abstract 65T–62, Notices Am. Math. Soc., 12, 217 (1965).
4. A. G. Dragalin, V. A. Lyubetskii, Abstracts of reports, All-Union Symposium on Mathematical Logic, Alma-Ata, 1969.
5. J. W. Addison, Fund. Math., 46, 337 (1959).
6. V. A. Lyubetskii, DAN, 182, No. 4, 758 (1968).
7. N. I. Luzin, Collected Works, 2, 1958.
8. A. Levy, Proc. of the 1964 Intern. Congr., Amsterdam, 1966, p. 127.