Though researchers in the field of abstract computational complexity theory have utilized many of the tools of recursive function theory in the development of their field, the early results obtained (e.g., see [8]) seemed to be rather independent of results in recursion theory (at least to the extent that the results were not uniformly interesting to both varieties of theorists). It seems to have been generally accepted, however, that strong parallels of one form or another must exist between the two fields. Indeed, recent results of Blum and Marques [7], Morris [13], Soare [15] and Bennison [1], [3] have revealed a striking correspondence between complexity theoretic properties and recursion theoretic properties. These results are not contrived, but rather link together interesting properties which had arisen naturally and independently in their respective fields. This paper presents the results of research aimed at finding a recursion theoretic characterization for a complexity theoretic property which had arisen from the study of the speed-up phenomenon.

In abstract computational complexity theory we are concerned with categorizing computable functions or sets according to their relative difficulty of computation. The phrase “difficult to compute” may take on different meanings depending on which criteria (complexity theoretic properties) we use to define what it means for a function or set to be hard to compute. In the abstract setting, however, such criteria should yield the same classes of functions or sets regardless of the underlying abstract complexity measure (in the sense of Blum [4], e.g., tape, time, etc.). In other words, such criteria should be measure-independent. In this paper we will be considering one way of defining “difficult to compute”. Namely, we shall say that a function or set is difficult to compute if it does not have a recursively enumerable complexity sequence as defined by Meyer and Fischer [12]. For a property to have a recursion theoretic characterization it must be measure-independent, for a recursion theoretic property is, by its very nature, measure-independent. It will not be immediately obvious whether or not the property of having an r.e. complexity sequence is measure-independent. We attack this question by first considering an alternative definition of an r.e. complexity sequence, one which is easily seen to be measure-independent.

In this series of two papers we wish to discuss two new methods of constructing models for an arbitrary theory T of stationary logic, L(aa). The original work in constructing models for this logic appears in Barwise-Kaufmann-Makkai [BKM] (which we shall refer to as BKM for brevity). In that paper the authors prove completeness, compactness, and omitting types theorems for L(aa) by a technique similar to that used by Keisler [1970] in constructing models of L(Q1), logic with the generalized quantifier “there exist uncountably many”. In this paper we give an alternative method of constructing models using a method of model-theoretic forcing similar to that of Bruce [1978]. In a succeeding paper we will give yet another method of constructing models using definable ultrapowers.

As applications of this new forcing construction we give new proofs of the completeness and omitting types theorems for LB(aa), where LB(aa) is any countable fragment of Lω1ω(aa). We also give new proofs of versions of the Barwise completeness and compactness theorems for LA(aa) where A is a countable admissible set. In proofs of these theorems the model constructions via forcing are used in a way very similar to that of consistency properties in Lω1ω.

Aside from the ease in handling the model theory of Lω1ω(aa) we see two other advantages in this sort of model construction. The first is in the ability to prove versions of theorems relativized to formulas of low complexity (e.g. relativized omitting types theorems). The second is that we have a high degree of control over the construction of standard models and hence have a better knowledge of the structure of the model (see the remarks following the proof of Lemma 3.3), something that should prove useful in constructing more complicated models.

Two questions on rings of strictly upper triangular matrices arising from B. Rose's work [5] are answered in this paper. An n × n matrix (αi, j) is strictly upper triangular if αi, j = 0 whenever i ≥ j. The ring of strictly upper triangular n × n matrices with entries from a field F will be denoted by Sn(F). Throughout this paper n will be an integer greater than 2. B. Rose [5] has shown that the complete theory of Sn(F) for an algebraically closed field F is ℵ1categorical. The first main result of this paper is that the rings Sn(F) and Sn(K) for fields F and K are isomorphic or elementarily equivalent if and only if F and K are isomorphic or elementarily equivalent, respectively (Corollary 1.6 and Theorem 2.2). This result shortens the proof of B. Rose's categoricity theorem [5, Theorem 7] by avoiding the co-stability considerations; furthermore, this result yields a proof of the converse of this categoricity theorem. The second main result is that the theory of rings of strictly upper triangular n × n matrices over algebraically closed fields is the model-completion of the theory of rings of strictly upper triangular n × n matrices over arbitrary fields (Theorem 2.5). This answers affirmatively the two conjectures at the end of [5].

In 1922 Skolem delivered an address before the Fifth Congress of Scandinavian Mathematicians in which he pointed out what he called a “relativity of set-theoretic notions”. This “relativity” has frequently been regarded as paradoxical; but today, although one hears the expression “the Löwenheim-Skolem Paradox”, it seems to be thought of as only an apparent paradox, something the cognoscenti enjoy but are not seriously troubled by. Thus van Heijenoort writes, “The existence of such a ‘relativity’ is sometimes referred to as the Löwenheim-Skolem Paradox. But, of course, it is not a paradox in the sense of an antinomy; it is a novel and unexpected feature of formal systems.” In this address I want to take up Skolem's arguments, not with the aim of refuting them but with the aim of extending them in somewhat the direction he seemed to be indicating. It is not my claim that the “Löwenheim-Skolem Paradox” is an antinomy in formal logic; but I shall argue that it is an antinomy, or something close to it, in philosophy of language. Moreover, I shall argue that the resolution of the antinomy—the only resolution that I myself can see as making sense—has profound implications for the great metaphysical dispute about realism which has always been the central dispute in the philosophy of language.

The structure of my argument will be as follows: I shall point out that in many different areas there are three main positions on reference and truth: there is the extreme Platonist position, which posits nonnatural mental powers of directly “grasping” forms (it is characteristic of this position that “understanding” or “grasping” is itself an irreducible and unexplicated notion); there is the verificationist position which replaces the classical notion of truth with the notion of verification or proof, at least when it comes to describing how the language is understood; and there is the moderate realist position which seeks to preserve the centrality of the classical notions of truth and reference without postulating nonnatural mental powers.

In [K] Keisler introduces a set theoretical relation WO(δ, x), where δ is an ordinal. This relation is characterized in ZF set theory by the following properties:

(1) WO(0, x) if and only if there is a wellordering on x.

(2) For δ > 0, WO(δ, x) if and only if there is a function ƒ with domain an ordinal λ such that:

We denote WO∞ the collection denned by: WO∞(x) = ∃δ WO(δ, x).

In [K] Keisler shows that countable transitive models of ZF + ∃x ¬ WO∞(x) have transitive uncountable elementary extensions with the same ordinals. Moreover for transitive models, satisfying ∃x ¬ WO∞(x) is also a necessary condition for the existence of an elementary extension with the same ordinals. (See [K bis] and also [K-M] where the connection with forcing is analysed.)

Sanchis [Sa] and Diller [Di] have introduced an interesting relation between terms for primitive recursive functionals of finite type in order to obtain a computability proof. The method is as follows. First the relation is shown to be well-founded. Then computability of each term is obtained by transfinite induction over the relation.

Their relation is given by various clauses which define the (immediate) successors of a term. Hence, starting with a term H and repeatedly applying the successor relation one generates the tree of H (say). The problem is to prove that the trees are well-founded. In their proofs Sanchis and Diller use the axiom of bar induction. Diller also gives a proof using transfinite induction: each tree is shown to have a length b < φω(0) (the first ω-critical number) and the induction is over ordinals less than or equal to b.

For proof-theoretic purposes it would be desirable to employ a method of proof which is more elementary than the axiom of bar induction and also to obtain a smaller ordinal bound. The purpose of the following is to provide such a method. We show that for a primitive recursive term H of finite type the tree generated by the relation of Sanchis and Diller has length less than ε0. Our method is similar to that used by Tait [Ta] in his theory of infinite terms. The immediately natural metamathematics is elementary intuitionistic analysis with the axiom of choice and transfinite induction over the relevant ordinal notations. We give several versions of the basic list of successor clauses, and for one version we show how to carry out the treatment in Skolem arithmetic (with primitive recursion of lowest type).

We prove the independence of a strong partition relation on ℵω, answering a question of Erdös and Hajnal. We then give an almost complete answer to the free subset problem.

Modular machines were introduced in [1] and [2], where they were used to give simple proofs of various unsolvability results in group theory. Here we discuss the degrees of the halting, word, and confluence problems for modular machines, both for their own sake and in the hope that the results may be useful in group theory (see [4] for an application of a related result to group theory).

In the course of the analysis, I found it convenient to compare degrees of these problems for a Turing machine T and for a Turing machine T1 obtained from T by enlarging the alphabet but retaining the same quintuples (or quadruples). The results were surprising. The degree for a problem of T1 depends not just on the corresponding degree for T, but also on the degrees of the corresponding problems when T is restricted to a semi-infinite tape (both semi-infinite to the right and semi-infinite to the left). For the halting and confluence problems, the Turing degrees of the problems for these three machines can be any r.e. degrees. In particular the halting problem of T can be solvable, while that of T1 has any r.e. degree.

A machine M (in the general sense) consists of a countable set of configurations (together with a numbering, which we usually take for granted), a recursive subset of configurations called the terminal configurations, and a recursive function, written C ⇒ C′, on the set of configurations. If, for some n ≥ 0, we have C = C0 ⇒ C1 ⇒ … ⇒ Cn = C′, we write C → C′. We say M halts from C if C → C′ for some terminal C′.

In a previous paper [9] a demonstrably consistent type-free system of combinatory logic CΔ was presented. This system contained a moderately strong exten-sionality principle, the usual equations of combinatory logic, Boolean propositional connectives, unrestricted quantifiers, an unrestricted abstraction principle (“comprehension axiom”), and a limited principle of excluded middle. It also contained elementary arithmetic in its entirety, avoiding arithmetical Gödel incompleteness [14] by being ω-complete. CΔ probably can be shown to contain certain important parts of mathematical analysis, such as the theory of continuous functions, which are contained by the somewhat similar (but nonextensional) system K′ [6], [7]. A stronger system CΓ, having these same properties and more, will be formulated in the present paper. Elsewhere [13] it will be shown that the system Q of my book, Elements of combinatory logic (Yale University Press, New Haven and London, 1974), referred to below as ECL, is essentially a subsystem of CΓ, so that the consistency of CΓ guarantees that of Q.

CΓ will be stronger than both CΔ and Q in having an operator ‘ɿ’ (inverted Greek iota) which denotes the inverse of equality in the sense that ‘ɿ(= a) = a’ is a theorem of the system. Thus ‘ɿ’ denotes a function or operator which operates on a unit class to give the only member of that class. (Here ‘=a’ denotes the unit class whose only member is denoted by ‘a’.) If uninverted Greek iota, ‘ι’, is used as an abbreviation for ‘=’, the above equation could be written as ‘ɿ(ιa) = a’. Here ‘ιa’ is analogous to Bertrand Russell's well-known notation ‘ιa’, denoting a unit class. This same operator ‘ɿ’ makes it possible to transform one-many or many-one relations into the corresponding functions or operators, a kind of transformation not usually available in systems of combinatory logic [1], [2]. It also provides a method for restricting functions by restricting the corresponding relations.

Let A be an abelian group and B a pure injective pure extension of A. Then there is a homomorphic image C of B over A which is a pure injective hull of A; C can be constructed by using Zorn's lemma to find a suitable congruence on B. In a paper [4] which greatly generalises this and related facts about pure injectives, Walter Taylor asks (Problem 1.5) whether one can find a “construction” of C which is more concrete than the one mentioned above; he asks also whether the points of C can be explicitly described. In this note I return the answer No.

The continuous or countable functionals were independently denned by Kleene [9] and Kreisel [10]. They were intended as a suitable basis for constructive mathematics, and thus it is interesting to investigate various notions of recursion on the countable functionals.

There have been two main streams in this investigation, the study of countable recursion and the study of computability or Kleene-recursion.

Countable recursion is the theory of recursion on the associates. Gandy and Hyland [3] and Hyland [7] are good sources for the recent development of countable recursion.

This paper will mostly be concerned with Kleene-recursion on the countable functionals as denned in Kleene [8] and [9]. We assume some familiarity with the countable functionals and associates, as presented in Kleene [9], Bergstra [1] or any other paper on the subject.

Pioneering work with recursion in nonnormal objects was done by Grilliot [4], who proved that a functional F of type 2 is normal if and only if its 1-section (that is the set of functions recursive in F) is closed under ordinary jump, and if and only if F is continuous on 1-section (F).

Hinman [6] constructed a countable functional that is not recursively equivalent to a function, and thereby showed that recursion in nonnormal functionals is an extension of ordinary recursion in functions.

In [6], Hinman asked if there are functionals with topless 1-sections, i.e. with no maximal elements in the semi-lattice of degrees. This was answered in the affirmative by Bergstra [1], using a spoiling construction. Thus the class of 1-sections of functionals extends the class of 1-sections of functions.

We prove independence results concerning the number of nonisomorphic models (using the S-chain condition and S-properness) and the consistency of “ there is a universal linear order of power ℵ1”. Most of these results were announced in [Sh 4], [Sh 5].

In subsequent papers we shall prove an analog f MA for forcing which does not destroy stationary subsets of ω1 investigate -properness for various filters and prove the consistency with G.C.H. of an axiom implying SH (for ℵ1), and connected results.

Keisler, in [1], defined a set F(D) of infinite cardinals for every ultrafilter D on a set I, and, assuming GCH, gave a sufficient condition for a set C of infinite cardinals to have the form F(D) for suitable D and I. In this paper we prove a similar theorem (Theorem 1) under considerably weaker assumptions. Our main tool is a construction of elementary end extensions of Boolean ultrapowers of ω outlined by Shelah in [6, Exercise VI.3.35]. Hence this paper will mostly be concerned with Boolean ultrapowers of ω.

This paper is primarily concerned with ℵ0-categoricity of theories of partially ordered sets. It contains some general conjectures, a collection of known results and some new theorems on ℵ0-categoricity. Among the latter are the following.

Corollary 3.3. For every countable ℵ0-categoricalthere is a linear order of A such that (, <) is ℵ0-categorical.

Corollary 6.7. Every ℵ0-categorical theory of a partially ordered set of finite width has a decidable theory.

Theorem 7.7. Every ℵ0-categorical theory of reticles has a decidable theory.

There is a section dealing just with decidability of partially ordered sets, the main result of this section being

Theorem 8.2. If (P, <) is a finite partially ordered set and KP is the class of partially ordered sets which do not embed (P, <), then Th(KP) is decidable iff KP contains only reticles.

Let T extend the theory P of Peano arithmetic, and suppose . Form from a model , in analogy to the way in which the ordered ring of integers is formed from the standard model of arithmetic. Let P′ and T′ be the corresponding analogues of P and T respectively. Now consider the group . In [5] MacDowell and Specker set out to determine the structure of such groups. (The precise statement in [5] refers to the ring of integers rather than the ordered ring. However, as pointed out to us by J. Knight, since Lagrange's Theorem that a positive integer is the sum of four squares is provable in the analogue of P′ for rings (see, for example, the proof in [7, p. 102]), the set of positive elements is definable in the ring, and consequently, so is the ordering. Thus, for the present purpose it makes no difference which of the two structures is used. Of course, one needs the ordering to discuss end extensions, as considered in [5]. On the other hand, one should be aware that in Pr′ one cannot define an ordering, where the theory Pr′ is the theory of the group of integers with distinguished element 1, 〈Z, +, 1〉. The constant 1 is needed so that divisibility mod n can be expressed. We will return to this point later.) In §1 we shall outline the results in this direction obtained in [5].

Lipshitz and Nadel, unaware that a similar question had been posed and investigated in [5] (though, of course aware that [5] contained the celebrated results on end extensions) set out to characterize those models 〈A, +〉 of Pr = Presburger Arithmetic (the complete theory of 〈ω, +〉) which can be expanded to models 〈A, +, ·, 0, 1, ≤〉 of P. They were able to give a complete characterization for countable models 〈A, +〉 in [4], which we describe in §2.

The usefulness of measurable cardinals in set theory arises in good part from the fact that an ultraproduct of wellfounded structures by a countably complete ultrafilter is wellfounded. In the standard proof of the wellfoundedness of such an ultraproduct, one first shows, without any use of the axiom of choice, that the ultraproduct contains no infinite descending chains. One then completes the proof by noting that, assuming the axiom of choice, any partial ordering with no infinite descending chain is wellfounded. In fact, the axiom of dependent choices (a weakened form of the axiom of choice) suffices. It is therefore of interest to ask whether some use of the axiom of choice is needed in order to prove the wellfoundedness of such ultraproducts or whether, on the other hand, their wellfoundedness can be proved in ZF alone. In Theorem 1, we show that the axiom of choice is needed for the proof (assuming the consistency of a strong partition relation). Theorem 1 also contains some related consistency results concerning infinite exponent partition relations. We then use Theorem 1 to show how to change the cofinality of a cardinal κ satisfying certain partition relations to any regular cardinal less than κ, while introducing no new bounded subsets of κ. This generalizes a theorem of Prikry [5].