UDC 519.50

MATHEMATICS

V. I. AMSTISLAVSKII

SET-THEORETIC OPERATIONS

AND RECURSIVE HIERARCHIES

(Presented by Academician A. I. Mal’tsev on 24 XI 1965)

A general definition is given of a recursive hierarchy (with \(\omega_1\) stages) in an arbitrary space. This definition may be regarded as an analogue of the definition of the classical hierarchies \((^6)\), which makes it possible to apply methods of the theory of operations on sets to establish a number of properties of recursive hierarchies. The hyperarithmetical \((^9)\) and \(\omega_1\)-stage \(\mathfrak S\)-hierarchies \((^8)\) turn out to be equivalent in the sense specified in Theorems 5 and 6 to the effective (eff.) \(B\)- and, respectively, eff. \(C\)-hierarchies, obtained in certain particular cases from our general definition**.

1. \(K\)-numberings. \(N=\{0,1,2,\ldots\}\); \(e,i,m,n\) are variables on \(N\). Subsets of \(N\) are called chains, and sets of chains are called bases. A collection \(\mathfrak M\) of arbitrary objects, supplied with some numbering \(\alpha\), is called a family and is denoted \(\langle\mathfrak M,\alpha\rangle\), or simply \(\alpha\). Let \(\langle\mathfrak M_1\alpha_1\rangle\) and \(\langle\mathfrak M_2\alpha_2\rangle\) be two arbitrary families; let \(P(x,y)\) be a predicate defined on \(\mathfrak M_1\times\mathfrak M_2\). We shall say: “for every \(x\;(\in\mathfrak M_1)\) one can find \(y\;(\in\mathfrak M_2)\) such that \(P(x,y)\),” meaning by this: “there exists a partial recursive function (p.r.f.) \(f(n)\) such that if \(n\) is an \(\alpha_1\)-number of some \(x\in\mathfrak M_1\), then \(f(n)\) is defined and is equal to the \(\alpha_2\)-number of such a \(y\in\mathfrak M_2\) that \(P(x,y)\).” The family of \(n\)-ary p.r.f.’s is assumed to be supplied with a Gödel numbering \((^1)\); for \(n=1\), the p.r.f. with number \(e\) is denoted by \(\langle e\rangle\); \(o\) is the one-place nowhere-defined p.r.f. The family of all recursively enumerable subsets of \(N\) is supplied with the numbering \(\tau\): \(\forall e.\tau e=\) the domain of definition of the p.r.f. \(\langle e\rangle\). For arbitrary families \(\alpha\) and \(\beta\), \(\alpha\leq [e]\beta\) (and \(\alpha\simeq [e]\beta\)) means that the numbering \(\alpha\) is reduced to the numbering \(\beta\) by means of the p.r.f. (recursive permutation) \(\langle e\rangle\), i.e. \(\alpha n=\beta\langle e\rangle n\); \(\alpha\leq\beta \Longleftrightarrow (\exists e)(\alpha\leq [e]\beta)\); and \(\alpha\simeq\beta \Longleftrightarrow (\exists e)(\alpha\simeq [e]\beta)\). If \(\{\alpha_n\}_{n\in A}\) and \(\{\beta_m\}_{m\in B}\) \((A,B\subseteq N)\) are two sequences of families and for any \(n\in A,\ m\in B\) one can find a number \(f(n,m)\) such that \(\alpha_n\leq [f(n,m)]\beta_m\), then we shall say that \(\alpha_n\leq\beta_m\) uniformly in \(n,m\). A simple numbering \(\alpha\) will be called a \(K\)-numbering if
\((\forall e_1,e_2)(\langle e_1\rangle=\langle e_2\rangle\Rightarrow \alpha e_1=\alpha e_2)\). If \(m_0\) is some number of the p.r.f. \(o\), then \(\alpha m_0\) will be denoted by \(o_\alpha\). For example, the family \(\tau\) is \(K\)-numbered. The numbering \(\langle e\rangle\) is effectively complete \((^5)\). Denote by \(e^*\) such a general recursive function (g.r.f.) that for every \(e\), \(\langle e^*\rangle\) is also a g.r.f. and for every \(n\)

\[ \langle\langle e^*\rangle(n)\rangle= \begin{cases} \langle\langle e\rangle(n)\rangle, & \text{if } \langle e\rangle(n) \text{ is defined},\\ o, & \text{otherwise}; \end{cases} \]

* \(\omega_1\) is the least nonconstructive ordinal number \((^{10})\).

** This clarifies the connection between \(\mathfrak S\)- and eff. \(C\)-hierarchies \((^8)\).

*** For the concepts and notation used below see \((^1,^4)\) or \((^5,^6)\).

then if \(\alpha\) is a \(K\)-enumeration, then

\[ \alpha\langle e^*\rangle(n)= \begin{cases} \alpha\langle e\rangle(n), & \text{if } \langle e\rangle(n) \text{ is defined},\\ o_\alpha & \text{otherwise.} \end{cases} \tag{1} \]

2. Standard operators. Let \(E\) be an arbitrary set, called a space; let \(\Psi_M\) be a set-theoretic operation (s.t.o.) with base \(M\). A function which assigns to each simply enumerated family \(\alpha\) of subsets of the space \(E\) a family \(\beta=H_M\alpha\), defined as follows: \(\forall e.\,\beta e=\Psi_M\{\alpha\langle e^*\rangle(n)\}\), will be called a standard operator with base \(M\). From (1) it is clear that if \(\alpha\) is a \(K\)-enumerated family, then \(H_M\alpha\) is the family of all sets obtained as a result of the s.t.o. \(\Psi_M\) applied to all possible enumerated sequences of sets of the family \(\alpha\). A composite standard operator with base is a function which assigns to a sequence \(\{\alpha_n\}_{n\in N}\) of simply enumerated families the family \(\beta=H_M\{\alpha_n\}\), where \(\forall e.\,\beta e=\Psi_M\{\alpha_n\langle e^*\rangle(n)\}\). If all

\[ \alpha_n=\alpha,\quad \text{then } H_M\{\alpha_n\}=H_M\alpha. \]

In what follows, all \(K\)-enumerated families are families of subsets of the space \(E\).

Theorem 1. If all \(\alpha_n\), \(n\in N\), are \(K\)-enumerated families, then \(H_M\{\alpha_n\}\) is also a \(K\)-enumerated family.

Theorem 2. If for each \(n\) the families \(\beta_n\) are simple, and \(\alpha_n\) are \(K\)-enumerated, then for any general recursive function \(\langle e\rangle\) such that \(\alpha_n \leqslant [\langle e\rangle(n)]\beta_n\), one can find a number \(h(e)\) such that

\[ H_M\{\beta_n\}\leqslant [h(e)]H_M\{\alpha_n\}. \]

Let a general recursive function \(\sigma(m,n)\) map \(N^2\) one-to-one onto \(N\), and let \((M\mid L_i)\) be the base of such an s.t.o. that, for any sequence \(\{a_i\}_{i\in N}\) of families of sets,

\[ \Psi_M\{\Psi_{L_i}\{a_{in}\}\}=\Psi_{M\mid L}\{\beta m\}, \]

where \(\beta\sigma(i,n)=a_{in}\).

Theorem 3. There exists a number \(e_0\) such that, for any \(K\)-enumerated family \(\alpha\) and any bases \(M,L_0,\ldots,L_i,\ldots\),

\[ H_M\{H_{L_i}\alpha\}\cong [e_0]H_{(M\mid L_i)}\alpha. \]

3. Definition of recursive hierarchies. Let \((O,<_{0})\) be Kleene’s system of notations for constructive ordinal numbers \((^{10})\). Let \(a'=2^a\), if \(a>0\), and let the partial recursive function \([a]_i\) be such that if \(a\in O\) and \(a=2^{(a)_0}\), then for every \(i\), \([a]_i\) is defined,

\[ [a]_i <_{0} [a]_{i+1} <_{0} a \quad\text{and}\quad \lim_i |[a]_i|=|a|. \]

Let \(\mathscr{H}r(n)\) be such a general recursive predicate that

\[ (n\in O \ \text{and}\ \mathscr{H}r(n)) \Longleftrightarrow (|n| \text{ is a limit nonzero ordinal number}). \]

We shall say that a base \(M\) has property \((A)\) if \(N\in M\) and all chains belonging to \(M\) are infinite. By \(M^c\) is denoted the base of the s.t.o. complementary to \(\Psi_M\).

Basic definition. Let the base \(M\) or \(M^c\) have property \((A)\), and let \(\lambda\) be a \(K\)-enumerated family of subsets of the space \(E\), with \(\varnothing\) and \(E\) belonging to \(\lambda\). The sequence \(\{\lambda_a\}_{a\in O}\) of families of subsets of the space \(E\), defined as follows:

1) \(\lambda_1=\lambda\);

2)

\[ \lambda_{a'}= \begin{cases} H_M\lambda_a, & \text{if } \mathscr{H}r(a'),\\ H_{M^c}\lambda_a & \text{otherwise;} \end{cases} \]

3) if \(a\ne 2^{(a)_0}\), then

\[ \lambda_a=H_{M^c}\{\lambda_{[a]_i}\} \]

is called a recursive hierarchy (r.h.) with base \(M\) and initial family \(\lambda\), or, more briefly, an \(\langle M,\lambda\rangle\)-hierarchy.

It follows from Theorem 1 that each \(\lambda_a,\ a\in O\), is a \(K\)-numbered family. By \(\bar\alpha\) we shall denote the family of complements of the sets of the family \(\alpha\): \(\bar\alpha n=E\setminus \alpha n\). Theorems 4–6 are proved by effective induction over \((O,<_{0})\) \((^{11})\).

Theorem 4. For any \(a,b\in O\), \(a<_{0}b \Rightarrow \lambda_a \leq \lambda_b\) uniformly in \(ab\); if, moreover, \(\bar\lambda \leqslant' H_M\lambda\), then \(a<_{0}b \Rightarrow \lambda_a \leqslant \lambda_b\) uniformly in \(a,b\).

4. Relation with the hyperarithmetical and \(\mathfrak S\)-hierarchies. Here \(E=N\). Let, for \(a\in O\), the predicates \(H_a(x)\) and \(\mathfrak S_a(x)\) be defined as in \((^{8,9})\); put
\[ h_a n=x(\exists y)T_1^{H_a}(n,x,y),\qquad \mathfrak h_a n=x(\forall f)(\exists y)T_1^{\mathfrak S_{a'}f}(n,x,y). \]

Theorem 5. Let \(M\) be a base of the t.m.o. of intersection, and let \(\{\tau_a\}_{a\in O}\) be an r.u. with base \(M\) and initial family \(\tau\). Then for every \(a\in O\)
\[ \tau_a \simeq \begin{cases} h_{a'}, & \text{if } \mathfrak Hr(a'),\\ \bar h_{a'}, & \text{otherwise} \end{cases} \]
uniformly in \(a\).

Theorem 6. Let \(M\) be a complete base of the \(A\)-operation and let \(\lambda=H_M\mathfrak ct\). Then the \(\langle M,\lambda\rangle\)-hierarchy is such that, for \(a\in O\),
\[ \lambda_a \simeq \begin{cases} \mathfrak h_{a'}, & \text{if } \mathfrak Hr(a'),\\ \bar{\mathfrak h}_{a'}, & \text{otherwise} \end{cases} \]
uniformly in \(a\).

5. Properties of recursive hierarchies. We denote the totality of all chains by \(Y\) and shall regard it as a \(T_0\)-space with the following topology \((^7)\): for any finite chain \(\xi'\), the set \(\sigma_{\xi'}\) of all chains containing \(\xi'\) as a subset is called elementarily open; an open set is any union of elementarily open sets. Let \(\alpha=\{\alpha n\}\), \(\alpha n\subseteq E,\ n\in N\); the function \(F:E\to Y\), defined as follows:
\[ \forall x\in E\ .\ F(x)=n\quad (x\in \alpha n), \]
will be called the component function corresponding to the sequence \(\alpha\). From the definition of a t.m.o. it is clear that \(\Psi_M\{\alpha n\}=F^{-1}(M)\), where \(F\) is the component function corresponding to \(\alpha\). The class of continuous mappings of \(Y\) into itself coincides with the class of component functions corresponding to all possible sequences of open sets. An effectively open set will mean any enumerable union of elementarily open sets; denote by \(\gamma\) the numbering of the family of effectively open sets induced by the numbering \(\tau\). A component function \(F:Y\to Y\) corresponding to some \(\gamma\)-enumerable sequence of effectively open sets will be called a computable operation. This definition is equivalent to the definition of a computable operation \((^7)\).

Theorem 7 (fixed point theorem). If the function \(F:Y\to Y\) is continuous, then there exists a chain \(\xi_0\) such that \(F(\xi_0)=\xi_0\). If, moreover, \(F\) is a computable operation, then \(\xi_0\) is a recursively enumerable subset of \(N\).

Let \(\langle\mathfrak M,\alpha\rangle\) be a family of bases; a base \(L\in\mathfrak M\) will be called universal in the family \(\langle\mathfrak M,\alpha\rangle\) if, for any base \(M\in\mathfrak M\), one can find a computable operation \(F\) such that \(M=F^{-1}(L)\).

Theorem 8. Let \(\{\lambda_a\}_{a\in O}\) be an r.u. with base \(M\) and initial family \(\lambda\) in the space \(E\), and let \(\{\gamma_a\}_{a\in O}\) be an r.u. with the same base \(M\) and initial family \(\gamma\) of effectively open sets in \(Y\). Then for every \(a\in O\), in the family \(\gamma_a\) one can find a base \(L_a\) such that: 1) \(L_a\) is universal in \(\gamma_a\); 2) if \(\gamma_a \leq H_M\gamma\), then for every \(b<_{0}a\), \(L_a\) does not belong to \(\gamma_b\); 3) \(\lambda_a\simeq H_{L_a}\lambda\) uniformly in \(a\).

Remark. 2) follows from (1) and Theorem 9; thus, the “non-emptiness” of the families of \(\langle M,\gamma\rangle\)-hierarchies is established by means of the fixed point theorem instead of the traditional diagonal method.

A simply numbered family \(\alpha\) is called e.f.f. closed with respect to enumerable sums if for every \(e\) one can find a number \(s(e)\) such that
\[ \bigcup_{n\in \tau e} \alpha n=\alpha s(e). \]
Analogously one defines e.f.f. closedness with respect to pairwise intersections.

Theorem 9 (an analogue of Kolmogorov’s “empty classes” theorem \({}^{(2)}\)).
Let \(\langle M,\lambda\rangle\)-hierarchy be such that \(\lambda\) is an e.f.f. closed with respect to enumerable sums and pairwise intersections, and in \(\lambda\) there exists an enumerable sequence of nonempty, pairwise disjoint sets. Then, for any \(a\in O\), in the family \(\lambda_a\) one can find a set that belongs to no \(\lambda_b\) with \(b<_O a\).

The hypotheses of Theorem 9 are satisfied, for example, when \(E=N\) and \(\lambda=\tau\), or \(E\) is Baire space and \(\lambda\) is a family of e.f.f. open sets \({}^{(3)}\).

We shall say that a sequence \(\{\alpha_a\}_{a\in O}\) of families is cofinal in a sequence \(\{\beta_b\}_{b\in O}\) if, for every \(a\in O\), one can find a number \(h(a)\) such that \(\alpha_a\leqslant \beta_{h(a)}\) uniformly in \(a\).

Theorem 10. Consider two r.s. in a space \(E\), having bases \(L\) and \(M\), and one and the same initial family \(\lambda\). If the base \(L\) belongs to one of the families of the \(\langle M,\gamma\rangle\)-hierarchy in the space \(Y\), then the \(\langle L,\lambda\rangle\)-hierarchy is cofinal in the \(\langle M,\lambda\rangle\)-hierarchy.

Institute of Cybernetics
Academy of Sciences of the Azerbaijan SSR

Received
18 XI 1965

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