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.

The relation M 1 ⊰ M 2, where M 1 and M 2 are elementary submodels of a given model M, defines a partial ordering ⊰(M). A natural question is: What are the general properties of this ordering and what special properties can it have for given M? Here we give an example of a prime model M for which ⊰(M) is a dense linear order, in particular ⊰(M) is isomorphic to the natural order of (− ∞, ∞], the reals with a last element (Theorem 1). In fact, this is the only way for ⊰(M) to be a linear order when M is prime (Theorem 3).

We then remark that if N is a model of an ℵ0-categorical theory then ⊰(N) cannot be a linear order, but we give an example where ⊰(N) is a dense partial order, N a model of an ℵ0-categorical theory.

Independently, and about the same time, Benda [1] found nonprime models M where ⊰(M) is isomorphic to any one of a large class of orders, including (− ∞, ∞]. His methods are very similar to ours.

Notation. We assume the reader is familar with Vaught [2] where he will find the definitions and standard results pertaining to prime models, homogeneous models, and ℵ0-categorical theories.

This note presents a practical algorithm for carrying out abstraction on combinatory terms. The well-known abstraction algorithms [1, pp. 188ff.] defining abstracts in terms of the combinator sets { S, K }, { B, C, K, W }, etc. operate on one variable at a time, and result in rather long abstracts when several variables are involved. These algorithms are not practical for the applications of the combinatory logic to the theory of computing which make much use of multi-variable abstraction (e.g., [2], [3]). The present algorithm performs the abstraction with respect to all specified variables in a single step, and yields abstracts in a concise form (with sizes proportional to those of given combinatory terms).

In a paper on combinatorial properties and large cardinals [2], Jech extended several combinatorial properties of a cardinal κ to analogous properties of the set of all subsets of λ of cardinality less than κ, denoted by “p κλ”, where λ is any cardinal ≤κ. We shall consider in this paper one of these properties which is historically rooted in a theorem of Ramsey [10] and in work of Rowbottom [12].

As in [2], define [p κλ]2 = {{x, y}: x, y ∈ p κλ and x ≠ y}. An unbounded subset A of p κλ is homogeneous for a function F: [p κλ]2 → 2 if there is a k < 2 so that for all x, y ∈ A with either x ⊊ y or y ⊊ x, F({x, y}) = k. A two-valued measure ü on p κλ is fine if it is κ-complete and if for all α < λ, ü({x ∈ p κλ: α ∈ x}) = 1, and ü is normal if, in addition, for every function f: p κλ → λsuch that ü({x ∈ p κλ: f(x) ∈ x}) = 1, there is an α < λ such that ü({x ∈ p κλ: f(x) = α}) = 1. Finally, a fine measure on p κλ has the partition property if every F: [p κλ]2 → 2 has a homogeneous set of measure one.

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.

There is a mistake in [6], discovered by Robert V. Kohn: contrary to the claim on p. 714, the set {◊X i : i ≥ 2} is not independent in S3. As a result three problems thought to be settled are open once more, viz., whether the Post numbers of S2, S3, and K4 are denumerable. In [6] it was claimed they are not. Now we shall prove that that claim is indeed correct for S2 and K4 but, surprisingly enough, not for S3.

Several people have independently been studying Boolean ultrapowers recently; see for example [2], [3], [4], [6]. Boolean ultrapowers are a quite natural generalization of the well-known usual ultrapowers, but it seemed to be unknown whether every Boolean ultrapower is isomorphic to an ultrapower. We give a negative answer to that question. We further show that a Boolean ultrapower by an ℵ1-regular ultrafilter need not be ℵ2-universal, i.e. that Theorem 4.3.12 of [1] does not hold for Boolean ultrapowers.

Let B be a complete Boolean algebra (we identify the algebra with its underlying set), whose operations are denoted by +, ·, −, 0, 1, Σ, Π Let be a structure for some language ℒ.

Let a be an admissible ordinal and let ∧ ≤ α be a limit ordinal. A sequence of a-r.e. degrees is said to be ascending, simultaneous and of length ∧ if (i) there is an α-recursive function t: α × ∧ → α such that, for all ϒ < ∧, A ϒ = {t(σ, ϒ)∣ σ < α} is of degree a ϒ; (ii) if ϒ < ⊤ < ∧, then a ϒ ≤α a τ and (iii) for all ϒ < ∧, there is a ⊤ > ϒ with a ϒ, >α a ϒ. Lerman [4] showed that such an exists for every ∧ ≤ α. An upper bound a of is an α-r.e. degree in which every element of is α-recursive. a is minimal if there is no α-r.e. degree b <α a which is also an upper bound of . Sacks [6] proved that every ascending sequence of simultaneously ω-r.e. degrees of length ω cannot have 0ω′, the complete ω-r.e. degree, as a minimal upper bound. In contrast, Cooper [2] showed that there exists an ascending sequence of simultaneously ω-r.e. degrees of length to having a minimal upper bound which is an ω-r.e. degree. In this paper we investigate the behavior of ascending sequences of simultaneously α-r.e. degrees for admissible ordinals α > ω. Call α Σ∞-admissibIe if it is Σ n -nadmissible for all n. Let Φ(∧) say: No ascending sequence of simultaneously α-r.e. degrees of length ∧ can have 0α′, the complete α-r.e. degree, as a minimal upper bound. Our main result in this paper is:

Let α be either a constructible cardinal with σ2ci(α) < α or Σ∞-admissible. Then σ2cf(α) is the least ordinal ν for which every ∧ ≤ α of cofinality ν (over L α) can satisfy Φ(∧).