The purpose of this paper is to show that arithmetically minimal systems of notations can be constructed which provide notations for all ramified analytical ordinals (all the ordinals in the minimum β-model for analysis). This is a much larger section of the second number class than the Church-Kleene constructive ordinals (although still only an initial segment of the ordinals). Arithmetic minimality means that if H is an “H-set” associated with an ordinal α in our system and H′ is an H-set associated with the same ordinal α in an arbitrary system of notations S, then H is arithmetical in H′. Thus the arithmetical degrees associated with ordinals in our system are as low as possible.

In order to clarify the structure of degrees of unsolvability and, more generally, to gain a deeper insight into the power set of the integers, coarser but neater classifications than the structure of Turing degrees have been sought. Several hierarchies of sets of integers have been studied, each of which organizes a certain class of sets (or their degrees of unsolvability) into a well-ordering of levels with increasing complexity of nonrecursiveness appearing at each new level. The best known of these hierarchies is the Kleene hierarchy of arithmetical sets.

An important problem, widely treated in the analysis of the structure of degree orderings, is that of partial order and lattice embeddings. Thus for example we have the results on embeddings of all countable partial orderings in the Turing degrees by Kleene and Post [3] and in the r.e. T-degrees by Sacks [10]. For lattice embeddings the work on T-degrees culminated in the characterization of countable initial segments by Lachlan and Lebeuf [4]. For the r.e. T-degrees there has been a continuing line of progress on this question. (See Soare [20] and Lerman, Shore, and Soare [8].) Similar projects have been undertaken for the T-degrees below 0′ (Kleene and Post [3], Lerman [6]) as well as for most other degree orderings. The results have been used not only to analyse individual orderings but also to distinguish between them (Shore [16], [19], [17]).

The situation for α-jecursive theory, the study of recursion in (admissible) ordinals, is similar to, though not as well developed as, that for Turing degrees. All afinite partial orderings have been embedded even in the α-r.e. degrees (see Lerman [5]). Lattice embedding results are somewhat fragmentary however. In terms of initial segments even the question of the existence of a minimal α-degree has not been settled for all admissibles. (See Shore [12] for a proof for Σ2-admissible ordinals, however.) Results on more complicated lattices have only reached to the finite distributive ones for Σ3-admissible ordinals (see Dorer [1]).

A model is called minimal if it does not contain a proper elementary submodel. A class of models is called Σ11(Π11 resp. elementary) if it is axiomatized by a sentence with σ in and some string of predicate symbols. All languages considered are assumed to be countable. For each model we shall define in a natural way its rank, denoted by rk (), which is an ordinal or ∞. Intuitively speaking, rk () is the least upper bound for the number of steps needed to define the elements of by first order formulas; e.g. we shall have rk((ω, <)) = 1 (each element is f.o. definable), rk ((Z, <)) = 2 (no element is f.o. definable, each element is f.o. definable using any other element as a parameter), rk ((Q, <) ) = ∞ (no element is f.o. definable by any number of steps). This notion of rank leads to a useful game theoretic characterization of minimal models which we apply to show that the Π11 class of minimal models is not Σ11.

This paper is a sequel to [3] and it contains, among other things, proofs of the results announced in the last section of that paper.

In §1, we use the general method of [3] together with reflection arguments to study the properties of forcing with Δ perfect trees, for certain Spector pointclasses Γ, obtaining as a main result the existence of a continuum of minimal Δ-degrees for such Γ's, under determinacy hypotheses. In particular, using PD, we prove the existence of continuum many minimal Δ½n+1-degrees, for all n.

Following an idea of Leo Harrington, we extend these results in §2 to show the existence of minimal strict upper bounds for sequences of Δ-degrees which are not too far apart. As a corollary, it is computed that the length of the natural hierarchy of Δ½n+1-degrees is equal to ω when n ≥ 1. (By results of Sacks and Richter the length of the natural hierarchy of -degrees is known to be equal to the first recursively inaccessible ordinal.)

We will follow in this paper standard notation and terminology, as in Moschovakis' book [7]. Letters e, i, j, k, l, m, n vary over the set of natural numbers ω, a, b, c over the Cantor space 2ω and α, β, γ, δ, … over the set of reals ωω. Finally ξ, η, κ, λ always denote ordinals.

The inverse of the distance between two structures ≢ of finite type τ is naturally measured by the smallest integer q such that a sentence of quantifier rank q − 1 is satisfied by but not by . In this way the space Strτ of structures of type τ is equipped with a pseudometric. The induced topology coincides with the elementary topology of Strτ. Using the rudiments of the theory of uniform spaces, in this elementary note we prove the convergence of every Cauchy net of structures, for any type τ.

Carnap (in [1], p. 566) has remarked that his measure function m* is fitting for finite languages using a fixed number of one-place predicates, i.e., for any sentence i, m*(i) is the same in all such restricted finite languages in which i occurs. The main purpose of this brief note is to show by means of a counter-example that m* does not have the intuitively desirable property of fittingness when we consider languages using two-place predicates.

We first state for finite languages a general theorem (related to the results in [2]) which guides the construction of large numbers of counter-examples.

Theorem. Let be the symmetric group of n letters. In a language which consists of a finite number of predicates and n individual names, the number of state descriptions in a structure description is equal to the index of some subgroup G in .

Proof. Let Σ be a structure description (of a language satisfying our hypothesis) and let S be an arbitrary state description in Σ. Let π be a permutation in , and let π*S be the state description which results from S by applying the permutation π to the individual names of . It is easily verified that for every π in π*S ∊ Σ. Also, for π, ψ in ,

We define the set G of permutations as follows :

.

The theory of computational complexity deals with those functions which can be computed subject to certain restrictions on the resources (for instance, time or memory) available for computation. Blum [5] gave an axiomatic characterization of some of the properties which should be possessed by a measure of computational complexity and established the existence of speed-upable functions—computable functions which fail to possess optimal programs in a particularly strong sense. Recursion theorists tend to like such functions, and people concerned with the specifics of real computing tend to consider such functions somewhat pathological. In Theorem 2 we show that such pathology is rampant: there is a great diversity of behavior among the collections of “run-times” of different functions which do not possess optimal programs, where such diversity is gauged by certain algebraic criteria which have computational significance. Roughly speaking, these algebraic criteria concern the ways in which various functions can be intermixed to satisfy requirements that certain functions can or cannot be computed more easily than certain other functions. (More detailed motivation for the relevance of these algebraic notions is given later.) Specifically, in Theorem 2 we generalize the embeddability theorem of McCreight and Meyer [12] (discussed below) by making speed-upable functions responsible for the embedding.

We study function fields over p-adically closed fields in the first-order language of fields. Using ideas of Duret [D], we show that the field of constants is definable, and that the genus is an elementary property.