A computer handles λ-terms more easily if these are translated into combinatory terms. This translation process is called bracket abstraction. The simplest abstraction algorithm—the (fab) algorithm of Curry (see Curry and Feys [6])—is lengthy to implement and produces combinatory terms that increase rapidly in length as the number of variables to be abstracted increases.

There are several ways in which these problems can be alleviated:

(1) A change in order of the clauses in the algorithm so that (f) is performed as a last resort.

(2) The use of an extra clause (c), appropriate to βη reduction.

(3) The introduction of a finite number of extra combinators.

The original 1924 form of bracket abstraction of Schönfinkel [17], which in fact predates λ-calculus, uses all three of these techniques; all are also mentioned in Curry and Feys [6].

A technique employed by many computing scientists (Turner [20], Peyton Jones [16], Oberhauser [15]) is to use the (fab) algorithm followed by certain “optimizations” or simplifications involving extra combinators and sometimes special cases of (c).

Another is either to allow a fixed infinite set of (super-) combinators (Abdali [1], Kennaway and Sleep [10], Krishnamurthy [12], Tonino [19]) or to allow new combinators to be defined one by one during the abstraction process (Hughes [7] and [8]).

A final method encodes the variables to be abstracted as an n-tuple—this requires only a finite number of combinators (Curien [5], Statman [18]).

Eine auf neue Art und mit neuen symbolischen Mitteln durchgeführte Analyse des Prozesses der Prädikation ergibt, daβ das für die logisch-mengentheoretischen Paradoxa nach Art des Russellschen essentielle Moment weder das der Selbstprädikation als solches ist noch irgendetwas, was man anderweitig bisher für sie verantwortlich gemacht hat, sondern vielmehr der Umstand, daβ der Algorithmus der Prädikation ebensowohl wie zu einem natürlichen Abschluβ auch auf einen unendlichen Regreβ führen kann. Damit werden die genannten Paradoxa als den bekannten erledigten der mathematischen unendlichen Prozesse wesentlich gleichartig und der gleichen Behandlung zugänglich erkannt. Das neue Kalkülmittel der limitierten Variablen bewirkt – ohne irgendwelche metalogische Regulative ad hoc – die automatische Sicherung des logischen Kalküls gegen die früheren Verführungen.

§1. The set Cn(R, A + X) is the set of all expressions deducible, on the ground of formulas A, from the class X of formulas of the implicational propositional calculus, by means of rules belonging to the set R. In other words, Cn(R, A + X) is the set of expressions obtained from the set X by the help of a two-parameter function Cn[R, A]. The formulais called the classical deduction theorem.

The classical deduction theorem is true for the system 〈R, A〉 (where R is the set of primitive rules, A is the set of axioms of the propositional calculus) if it holds for the function Cn[R, A].

Assuming CH. Hindman [2] showed that the existence of certain ultrafilters on the power set of the natural numbers is equivalent to Hindman's Theorem. Adapting this work to a countable setting formalized in RCA0, this article proves the equivalence of the existence of certain ultrafilters on countable Boolean algebras and an iterated form of Hindman's Theorem, which is closely related to Milliken's Theorem. A computable restriction of Hindman's Theorem follows as a corollary.

(Xα: α < ω2) ⊂ ℘(ω1) is a strong chain in ℘(ω1)/Fin if and only if Xβ – Xα is finite and Xα – Xβ is uncountable for each β < α < ω1. We show that it is consistent that a strong chain in ℘(ω1) exists. On the other hand we show that it is consistent that there is a strongly almost-disjoint family in ℘(ω1) but no strong chain exists: is used to construct a c.c.c forcing that adds a strong chain and Chang's Conjecture to prove that there is no strong chain.

The current paper studies the formal properties of the Global Reflection Principle, to wit the assertion “All theorems of $\mathrm {Th}$ are true,” where $\mathrm {Th}$ is a theory in the language of arithmetic and the truth predicate satisfies the usual Tarskian inductive conditions for formulae in the language of arithmetic. We fix the gap in Kotlarski’s proof from [15], showing that the Global Reflection Principle for Peano Arithmetic is provable in the theory of compositional truth with bounded induction only ( $\mathrm {CT}_0$ ). Furthermore, we extend the above result showing that $\Sigma _1$ -uniform reflection over a theory of uniform Tarski biconditionals ( $\mathrm {UTB}^-$ ) is provable in $\mathrm {CT}_0$ , thus answering the question of Beklemishev and Pakhomov [2]. Finally, we introduce the notion of a prolongable satisfaction class and use it to study the structure of models of $\mathrm {CT}_0$ . In particular, we provide a new model-theoretical characterization of theories of finite iterations of uniform reflection and present a new proof characterizing the arithmetical consequences of $\mathrm {CT}_0$ .

We study the generic theory of algebraically closed fields of fixed positive characteristic with a predicate for an additive subgroup, called $\mathrm {ACFG}$ . This theory was introduced in [16] as a new example of $\mathrm {NSOP}_{1}$ nonsimple theory. In this paper we describe more features of $\mathrm {ACFG}$ , such as imaginaries. We also study various independence relations in $\mathrm {ACFG}$ , such as Kim-independence or forking independence, and describe interactions between them.

We present a framework for studying the concept of independence in a general context covering database theory, algebra and model theory as special cases. We show that well-known axioms and rules of independence for making inferences concerning basic atomic independence statements are complete with respect to a variety of semantics. Our results show that the uses of independence concepts in as different areas as database theory, algebra, and model theory, can be completely characterized by the same axioms. We also consider concepts related to independence, such as dependence.

It is well known that if a postulate LLCpp or a rule of procedure ├α→├Lα (from a thesis α of a considered system to infer a thesis Lα of that system) is added to Lewis's modal system S3, we get his system S4 (see [7], p. 148). It is also known that the addition of this rule to S2 and SI has the effect of converting these systems into the Gödel-Feys-von Wright system T (M). My purpose in this paper is first to draw attention to some other ways of converting S3 and S2 into S4 and T respectively, as well as to extend a theorem of Halldén on S3, S4 and S7 to S2, T and S6. The results reached will also apply to the systems S3.5, S5 and S7.5, where S3.5 and S7.5 are obtained, respectively, from S3 and S7 by the addition of the postulate CNLpLNLp; an answer will be given to the question of irreducible modalities in S3.5. Moreover, two results will be proved that bear on the problem whether the systems S2 and T can be axiomatized by means of a finite number of axiom schemata and material detachment as the only primitive rule of inference.

In this paper we investigate the relationship between the number of countable and decidable models of a complete theory. The number of decidable models will be determined in two ways, in §1 with respect to abstract isomorphism type, and in §2 with respect to recursive isomorphism type.

Definition 1. A complete theory is (α, β) if the number of countable models of T, up to abstract isomorphism, is β, and similarly the number of decidable models of T is α.

Definition 2. A model is ω-decidable if ∣∣= ω and for an effective listing {θn∣n < ω} of all sentences in the language of Th() augmented by new constant symbols i*, i < ω, {n ∣〈, i〉i<ω ⊨ θn} is recursive, where i interprets i* (in these terms, is decidable if is abstractly isomorphic to an ω-decidable model).

Definition 3. A complete theory is (α, β)r if it is (γ, β) for some γ and it has exactly αω-decidable models up to recursive isomorphism.

Specifically we will show in §1 that there is a (2, ω) theory, and in §2 that although there is a (2, 2ω) theory, there is no (2, β)r theory for any β, β < 2ω.

The first-order theory of a string automatic structure is known to be decidable, but there are examples of string automatic structures with nonelementary first-order theories. We prove that the first-order theory of a string automatic structure of bounded degree is decidable in doubly exponential space (for injective automatic presentations, this holds even uniformly). This result is shown to be optimal since we also present a string automatic structure of bounded degree whose first-order theory is hard for 2EXPSPACE. We prove similar results also for tree automatic structures. These findings close the gaps left open in [28] by improving both the lower and the upper bounds.

In his paper [3] Hindley shows that strong reduction in combinatory logic (see [1] for the basic discussion of this notion) cannot be axiomatized with a finite number of axiom schemes, and then he presents an infinite system of axiom schemes for strong reduction. Hindley lists one axiom and six axiom schemes, together with a method (clause (viii) below) for generating further axiom schemes from these. The results of this note are that different applications of clause (viii) yield different axiom schemes, and that the property of being an axiom is decidable.

We deal with a restricted form WC-N′ of the weak continuity principle, a version BT′ of Baire's theorem, and a boundedness principle BD-N. We show, in the spirit of constructive reverse mathematics, that WC-N′, BT′ + ¬LPO and BD-N + ¬LPO are equivalent in a constructive system, where LPO is the limited principle of omniscience.

Kleene [7] and Kreisel [8] defined independently the countable (continuous) functionals. Kleene [7] defined the countable functionals of type k to be total functionals of type k acting in a continuous way when restricted to countable arguments of type k − 1. He also defined the associates for countable functionals. They are functions α: N → N containing information about how the functional acts on countable arguments. Kleene [7] showed that the countable functionals are closed under the computations derived from S1–S9 of his paper [6], and that every computable functional has a recursive associate.

Kreisel defined the continuous functionals to be equivalence-classes of associates. By his definition it is meaningless to let a continuous functional act upon anything but continuous arguments.

One disadvantage of Kleene's approach is that two different functionals may have the same associates We will later see that there may be two functionals φ1 and φ2 with the same associates but such that the relations

are not the same.

In more recent papers on the countable functionals it is normal to regard the hierarchy 〈Ct(k)〉kϵN of countable functionals as a type-structure such that the functionals in Ct(k + 1) are maps from Ct(k) to N(Ct(0) = N), see e.g. Bergstra [1] and Gandy and Hyland [3].

We will then enjoy the streamlined formalism of a type-structure in which S1–S9 have meaning, but avoid the ambiguities of Kleene's original approach.

We will presuppose a brief familiarity with the theory of the countable functionals.

We define elementary extension and elementary equivalence in Positive Logic.

We prove that RCA0 + RRT ⊬ ACA0 where RRT is the Rainbow Ramsey Theorem for 2-bounded colorings of triples. This reverse mathematical result is based on a cone avoidance theorem, that every 2-bounded coloring of pairs admits a cone-avoiding infinite rainbow, regardless of the complexity of the given coloring. We also apply the proof of the cone avoidance theorem to the question whether RCA0 + RRT ⊦ ACA0 and obtain some partial answer.

We introduce a very weak language on p-adic fields K, which is just rich enough to have exactly the same definable subsets of the line K that one has using the ring language. (In our context, definable always means definable with parameters.) We prove that the only definable functions in the language are trivial functions. We also give a definitional expansion of in which K has quantifier elimination, and we obtain a cell decomposition result for -definable sets.

Our language can serve as a p-adic analogue of the very weak language (<) on the real numbers, to define a notion of minimality on the field of p-adic numbers and on related valued fields. These fields are not necessarily Henselian and may have positive characteristic.

I have argued in [1] that a concept bearing some resemblance to ‘p is the answer to d’ (p a proposition and d a question) can be defined wherever d has the form,

‘For which a's is it the case that A (a)?’ (Qa)A(a)

where a is a variable and A a wff containing a. To say that p is the true and complete answer to (Qa)A(a) is expressed as saying that p is logically equivalent to the true conjunction of A(a) or ~A(a) for each a. It is defined as;

Such a concept of answer is like Belnap's [2] direct true answer to a complete list question, or like Harrah's use [3] (p. 43) of the notion of a state description. The main difference between my approach and that of Belnap and Harrah is that while they are concerned to develop a formal metalanguage for discussion of questions and answers I am concerned to express, as far as possible in existing systems, certain interrogative statements; in particular statements of the form ‘— is the (an) answer to —’.

While the account in [1] does give a formal analysis of one ‘answer’ concept there are respects in which it is inadequate.

1. Since it uses entailment (or strict implication) to define the relation between p the answer and d the question we can shew that if p is the answer to d and q is logically equivalent to p then q is the answer to d.

We answer in the affirmative the following question of Jörg Brendle: If there is a Σ21 mad family, is there then a Π11 mad family?