Ever since Craig-Beth and Addison-Kleene proved their versions of the Lusin-Suslin theorem, work in model theory and recursion theory has demonstrated the value of classical descriptive set theory as a source of ideas and inspirations. During the sixties in particular, J.W. Addison refined the technique of “conjecture by analogy” and used it to generate a substantial number of results in both model theory and recursion theory (see, e.g., Addison [1], [2], [3]). During the past 15 years, techniques and results from recursion theory and model theory have played an important role in the development of descriptive set theory. (Moschovakis's book [6] is an excellent reference, particularly for the use of recursion-theoretic tools.) The use of “conjecture by analogy” as a means of transferring ideas from model theory and recursion theory to descriptive set theory has developed more slowly. Some notable recent examples of this phenomenon are in Vaught [9], where some results in invariant descriptive set theory reflecting and extending model-theoretic results are obtained and others are left as conjectures (including a version of the well-known conjecture on the number of countable models) and in Hrbacek and Simpson [4], where a notion analogous to that of Turing reducibility is used to study Borel isomorphism types. Moschovakis [6] describes in detail an effective descriptive set theory based in large part on classical recursion theory.

A LISP interpreter for the Zilog Z-80 CPU is described, together with the factors which influenced the design.

In [2], [3], J. Grassin characterized those classes of sets which are expressible as a Boolean combination of open sets in the Baire topology on the power set of the natural numbers. From this characterization he deduced that if is a Boolean combination of recursive open sets, then the minimum length for any Boolean combination representing can be realized using recursive open sets.

Let {Wk}k ≥ 0 be a standard enumeration of all recursively enumerable (r.e.) sets. If A is any class of r.e. sets, let θA denote the index set of A, i.e., θA = {k ∣ Wk ∈ A}. The one-one degrees of index sets form a partial order ℐ which is a proper subordering of the partial order of all one-one degrees. Denote by ⌀ the one-one degree of the empty set, and, if b is the one-one degree of θB, denote by the one-one degree of . Let . Let {Ym}m≥0 be the sequence of index sets of nonempty finite classes of finite sets (classified in [5] and independently, in [2]) and denote by am the one-one degree of Ym. As shown in [2], these degrees are complete at each level of the difference hierarchy generated by the r.e. sets. It was proved in [3] that, for each m ≥ 0, (a) am+1 and ām+1 are incomparable immediate successors of am and ām, and (b) . For m = 0, since Y0 = θ{⌀}, it follows from (a) that (c) . Hence it follows that (d) {⌀, , ao, ā0, a1, ā1 is an initial segment of ℐ.

Let {Wi} be a standard enumeration of all recursively enumerable (r.e.) sets, and for any class A of r.e. sets, let θA denote the index set of A = {n ∣ Wn ∈ A}. (Clearly, .) In [1], the index sets of nonempty finite classes of finite sets were classified under one-one reducibility into an increasing sequence {Ym}, 0 ≤ m < ∞. In this paper we examine further properties of this sequence within the partial ordering of one-one degrees of index sets. The main results are as follows: (1) For each m, Ym < Ym + 1 and < Ym + 1; (2) Ym is incomparable to ; (3) Ym + 1 and ; are immediate successors (among index sets) of Ym and m; (4) the pair (Ym + 1, ) is a “least upper bound” for the pair (Ym, ) in the sense that any successor of both Ym and is ≥ Ym + 1or ; (5) the pair (Ym, ) is a “greatest lower bound” for the pair (Ym + 1, ) in the sense that any predecessor of both Ym + 1 and is ≤ Ym or . Since and all Ym are in the bounded truth-table degree of K, this yields some local information about the one-one degrees of index sets which are “at the bottom” in the one-one ordering of index sets.

In [2] “recursive frames” were introduced as a means of extending relations R on the nonnegative integers to relations RΛ on the isols. In [1], this extension procedure was generalized by the introduction of “partial recursive frames”; the resulting extended relation on the isols was called RΛ. It was shown in [1] that the two extension procedures agree for recursive relations R, while RΛ ⊇ RΛ if R is . The case when R is , nonrecursive was left open. We show in this note that the extension procedures in fact agree for all relations R. In the following, the notation and terminology is that of [1] and [2]. Theorem. If R ⊆ XκQ is a recursively enumerable (r.e.) relation, then RΛ = RΛ. Proof. Clearly RΛ ⊆ RΛ, since every recursive frame is partial recursive. To prove RΛ ⊆ RΛ, we give a uniform effective method for expanding any partial recursiveR-frame F to a recursiveR-frame G such that F ⊆ G, so that So let F be a (nonempty) partial recursive R-frame, with CF(α) generated by . Let Rn denote the result of performing n steps in a fixed recursive enumeration of R. If g(α) is a partial recursive function, “g(α) is defined in n steps” means that in whichever coding of recursive computations is being used, a terminating computation for g with argument α has length ≤ n.

Let q0, q1,… be a standard enumeration of all partial recursive functions of one variable. For each i, let wi = range qi and for any recursively enumerable (r.e.) set α, let θα = {n | wn = α}. If A is a class of r.e. sets, let θA = the index set of A = {n | wn ∈ A}. It is the purpose of this paper to classify the possible recursive isomorphism types of index sets of finite classes of r.e. sets. The main theorem will also provide an answer to the question left open in [2] concerning the possible double isomorphism types of pairs (θα, θβ) where α ⊂ β.

The infinite-valued statement calculus to which this paper refers is that of Łukasiewicz [10], whose axiomatization was proved complete in [5]. In [9], Rutledge extended this system to include predicates and quantifiers2 and presented a deductively complete set of axioms for the monadic predicate calculus. This paper represents an attempt to axiomatize the full predicate calculus; for the proposed axiomatization, a property akin to but weaker than completeness is proved. An attempt to prove full completeness along similar lines failed; it has since been shown [11] that the set of valid formulas of the infinite-valued predicate calculus is not recursively enumerable. The method of this paper was suggested by Professor J. Barkley Rosser.