The lambda calculus was conceived by Church, and to some extent by Curry (see [1] and [2]), as a general theory of functions, more precisely as a formalization of two basic functional concepts: application and abstraction. This claim to universality was impossible to maintain, for it soon became apparent that functions in the calculus are required to have properties which are by no means universal. Self-application is certainly a very special feature, but more relevant from our point of view is the existence of fixed points, which is definitely a nonuniversal property of functions. This situation was not ignored, but was considered rather as a form of inconsistency very much related to the paradoxes of set theory (for example, see the discussion of the paradoxical combinator in [2]).

More recently, the lambda calculus has been studied by people in computer science, and as a result the algorithmic nature of the system has been more explicitly recognized. Eventually, the right approach was taken by D. Scott (see [4] and [5]) who restricted the objects of the calculus to continuous functionals over complete lattices, which can be combined in structures where application and abstraction are properly interpreted. It soon became clear that complete lattices were not necessary, and more general structures were introduced. On the other hand the requirement of continuity was essential for the construction, and has remained a basic feature. It is one of the purposes of this paper to show that in the general situation continuity can be replaced by monotonicity.

The work of Scott was extended by Wadsworth [6], who gave a precise formulation of the operational semantics, and proved the equivalence to Scott's interpretation.

We give resource bounded versions of the Completeness Theorem for propositional and predicate logic. For example, it is well known that every computable consistent propositional theory has a computable complete consistent extension. We show that, when length is measured relative to the binary representation of natural numbers and formulas, every polynomial time decidable propositional theory has an exponential time (EXPTIME) complete consistent extension whereas there is a nondeterministic polynomial time (NP) decidable theory which has no polynomial time complete consistent extension when length is measured relative to the binary representation of natural numbers and formulas. It is well known that a propositional theory is axiomatizable (respectively decidable) if and only if it may be represented as the set of infinite paths through a computable tree (respectively a computable tree with no dead ends). We show that any polynomial time decidable theory may be represented as the set of paths through a polynomial time decidable tree. On the other hand, the statement that every polynomial time decidable tree represents the set of complete consistent extensions of some theory which is polynomial time decidable, relative to the tally representation of natural numbers and formulas, is equivalent to P = NP. For predicate logic, we develop a complexity theoretic version of the Henkin construction to prove a complexity theoretic version of the Completeness Theorem. Our results imply that that any polynomial space decidable theory Δ possesses a polynomial space computable model which is exponential space decidable and thus Δ has an exponential space complete consistent extension. Similar results are obtained for other notions of complexity.

We prove that the strong Martin conjecture is false. The counterexample is the first-order theory of infinite atomic Boolean algebras. We show that for this class of Boolean algebras, the classification of their (ω + ω)-elementary theories can be reduced to the classification of the elementary theories of their quotient algrbras modulo the Frechét ideals.

We point out that the theory of difference fields with algebraically closed fixed field has no model companion.

Let T be a countable, complete, ω-stable, nonmultidimensional theory. By Lascar [7], in Teq there is in every dimension of T a type with Lascar rank ωα for some α. We give sufficient conditions for α to coincide with the level of that dimension in Pillay's [10] RK-hierarchy of dimensions computed in Teq. In particular, this is fulfilled for modules.

We conclude here the treatment of forcing in recursion theory begun in Part I and continued in Part II of [31]. The numbering of sections is the continuation of the numbering of the first two parts. The bibliography is independent.

In Part I our language was a first-order language: the only set we considered was the (set constant for the) generic set. In Part II a second-order language was introduced, and we had to interpret the second-order variables in some way. What we did was to consider the ramified analytic hierarchy, defined by induction as:

A0 = {X ⊆ ω: X is arithmetic},

Aα+1 = {X ⊆ ω: X is definable (in 2nd order arithmetic) over Aα},

Aλ = ⋃α<λAα (λ limit),

RA = ⋃αAα.

We then used (a relativized version of) the fact that (Kleene [27]). The definition of RA is obviously modeled on the definition of the constructible hierarchy introduced by Gödel [14]. For this we no longer work in a language for second-order arithmetic, but in a language for (first-order) set theory with membership as the only nonlogical relation:

L0 = ⊘,

Lα+1 = {X: X is (first-order) definable over Lα},

Lλ = ⋃α<λLα (λ limit),

L = ⋃αLα.

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]).

In [3] a definition of negation was presented for the system K′ of extended basic logic [1], but it has since been shown by Peter Päppinghaus (personal communication) that this definition fails to give rise to the law of double negation as I claimed it did. The purpose of this note is to revise this defective definition in such a way that it clearly does give rise to the law of double negation, as well as to the other negation rules of K′.

Although Päppinghaus's original letter to me was dated September 19, 1972, the matter has remained unresolved all this time. Only recently have I seen that there is a simple way to correct the definition. I am of course very grateful to Päppinghaus for pointing out my error in claiming to be able to derive the rule of double negation from the original form of the definition.

The corrected definition will, as before, use fixed-point operators to give the effect of the required kind of transfinite induction, but this time a double transfinite induction will be used, somewhat like the double transfinite induction used in [5] to define simultaneously the theorems and antitheorems of system CΓ.

We prove an Ax-Kochen-Ershov like transfer principle for groups acting on groups. The simplest case is the following: let B be a soluble group acting on an abelian group G so that G is a torsion-free divisible module over the group ring ℤ[B], then the theory of B determines the one of the two-sorted structure 〈G,B,*〉, where * is the action of B on C. More generally, we show a similar principle for structures 〈G,B,*〉, where G is a torsion-free divisible module over the quotient of ℤ[B] by the annulator of G.

Two applications come immediately from this result:

First, for not necessarily commutative domains, where we consider the action of a subgroup of the invertible elements on the additive group. We obtain then the decidability of a weakened structure of ring, with partial multiplication.

The second application is to pure groups. The semi-direct product of G by B is bi-interpretable with our structure 〈G,B,*〉. Thus, we obtain stable decidable groups that are not linear over a field.

The basic notions of the theory of term rewriting are defined for terms that may involve function letters of infinite arity. A sufficient condition for completeness is derived, and its use demonstrated by the example of abstract clones over infinitary signatures.