The Principle of Dependent Choice is shown to be equivalent to: the Baire Category Theorem for Čech-complete spaces (or for complete metric spaces); the existence theorem for generic sets of forcing conditions; and a proof-theoretic principle that abstracts the “Henkin method” of proving deductive completeness of logical systems. The Rasiowa-Sikorski Lemma is shown to be equivalent to the conjunction of the Ultrafilter Theorem and the Baire Category Theorem for compact Hausdorff spaces.

Given a prime p, we exhibit a Horn sentence Hp which expresses “the cardinality is not p” and has size O(p5 log p).

We will study syntactical and semantical properties of propositional logics weaker than the intuitionistic, in which the contraction rule (or, the exchange rule or the weakening rule, in some cases) does not hold. Here, the contraction rule means the rule of inference of the form

if we formulate our logics in a Gentzen-type formal system. Some syntactical properties of these logics have been studied firstly by the second author in [11], in connection with the study of BCK-algebras (for information on BCK-algebras, see [9]). There, it turned out that such a syntactical method is a powerful and promising tool in studying BCK-algebras. Using this method, considerable progress has been made since then (see, e.g., [8], [18], [27]).

In this paper, we will study these logics more comprehensively. We notice here that the distributive law

does not hold necessarily in these logics. By adding some axioms (or initial sequents) and rules of inference to these basic logics, we can obtain a lot of interesting nonclassical logics such as Łukasiewicz's many-valued logics, relevant logics, the intuitionistic logic and logics related to BCK-algebras, which have been studied separately until now. Thus, our approach will give a uniform way of dealing with these logics. One of our two main tools in doing so is Gentzen-type formulation of logics in syntax, and the other is semantics defined by using partially ordered monoids.

This paper is a continuation of our previous work in [12]. The results, and some applications, have been described in the announcement [13]; it may be useful to discuss here, a little more fully, the nature and purpose of this work.

We are concerned basically with three kinds of algorithmic problem: (1) isomorphism problems, (2) “orbit problems”, and (3) “effective generation”.

(1) Isomorphism problems. Here we have a class of algebraic objects of some kind, and ask: is there a uniform algorithm for deciding whether two arbitrary members of are isomorphic? In most cases, the answer is no: no such algorithm exists. Indeed this has been one of the most notable applications of methods of mathematical logic in algebra (see [26, Chapter IV, §4] for the case where is the class of all finitely presented groups). It turns out, however, that when consists of objects which are in a certain sense “finite-dimensional”, then the isomorphism problem is indeed algorithmically soluble. We gave such algorithms in [12] for the following cases:

= {finitely generated nilpotent groups};

= {(not necessarily associative) rings whose additive group is finitely generated};

= {finitely Z-generated modules over a fixed finitely generated ring}.

Combining the methods of [12] with his own earlier work, Sarkisian has obtained analogous results with the integers replaced by the rationals: in [20] and [21] he solves the isomorphism problem for radicable torsion-free nilpotent groups of finite rank and for finite-dimensional Q-algebras.

Interpretability in reflexive theories, especially in PA, has been studied in many papers; see e.g. [3], [6], [7], [10], [11], [15], [26]. It has been shown that reflexive theories exhibit many nice properties, e.g. (1) if T, S are recursively enumerable reflexive, then T is interpretable in S iff every Π1 sentence provable in T is provable in S; and (2) if S is reflexive, T is recursively enumerable and locally interpretable in S (i.e. every finite part of T is interpretable in S), then T is globally interpretable in S (Orey's theorem, cf. [3]).

In this paper we want to study such statements for nonreflexive theories, especially for finitely axiomatizable theories (which are never reflexive). These theories behave differently, although they may be quite close to reflexive theories, as e.g. GB to ZF. An important fact is that in such theories one can define proper cuts. By a cut we mean a formula with one free variable which defines a nonempty initial segment of natural numbers closed under the successor function. The importance of cuts for interpretations in GB was realized already by Vopěnka and Hájek in [30]. Pioneering work was done by Solovay in [24]. There he developed the method of “shortening of cuts”. Using this method it is possible to replace any cut by a cut which is contained in it and has some desirable additional properties; in particular it can be closed under + and ·. This introduces ambiguity in the concept of arithmetic in theories which admit proper cuts, namely, which cut (closed under + and ·) should be called the arithmetic of the theory? Cuts played the crucial role also in [20].

I characterise various model-theoretic properties of types, in complete theories of modules, in terms of the algebraic structure of pure-injective modules. More specifically, I consider the generalised RK-order, and the relation of domination between types, orthogonality of types, and regular types. It will be seen that, essentially, it is the stationary types over pure-injective models which bear the algebraic structural information.

For background in the model theory of modules, see [4], [5], [13], [17], [19], and for the model-theoretic background [10], [11], [12], [18]. More specific references will be given in the text. I summarise some principal definitions and results below. First, though, let me describe the main results.

Throughout this paper, R will be a ring with 1; the language will be that for (right R-) modules. All types will be complete types in a complete theory, T, of R-modules. The notation is reserved for an extremely saturated model of T, inside which all “small” situations may be found. Thus, sets of parameters are just subsets of , and all models will be elementary substructures of . A positive primitive, or pp, formula is one which is equivalent to a formula of the form

with the rij ∈ R.

Suppose that p and q are 1-types over a pure-injective model M.

We initiate the study of model theory in the absence of the Axiom of Choice, using the Axiom of Determinateness as a powerful substitute. We first show that, in this context, is no more powerful than first-order logic. The emphasis then turns to upward Löwenhein-Skolem theorems; ℵ1 is the Hanf number of first-order logic, of , and of a strong fragment of , The main technical innovation is the development of iterated ultrapowers using infinite supports; this requires an application of infinite-exponent partition relations. All our theorems can be proven from hypotheses weaker than AD.

Probably the two most famous examples of elementary embeddings between inner models of set theory are the embeddings of the universe into an inner model given by a measurable cardinal and the embeddings of the constructible universe L into itself given by 0#. In both of these examples, the “target model” is a subclass of the “ground model” (and in the latter case they are equal). It is not hard to find examples of embeddings in which the target model is not a subclass of the ground model: if is a generic ultrafilter arising from forcing with a precipitous ideal on a successor cardinal κ, then the ultraproduct of the ground model via collapses κ. Such considerations suggest a classification of how close the target model comes to “fitting inside” the ground model.

Definition 1.1. Let M and N be inner models (transitive, proper class models) of ZFC, and let j: M → N be an elementary embedding. The co-critical point of j is the least ordinal λ, if any exist, such that there is X ⊆ λ, X ∈ N but X ∉ M. Such an X is called a new subset of λ.

It is easy to see that the co-critical point of j: M → N is a cardinal in N.

We define, in the spirit of Fenstad [2], a higher type computation theory, and show that countable recursion over the continuous functionals forms such a theory. We also discuss Hyland's proposal from [4] for a scheme with which to supplement S1–S9, and show that this augmented set of schemes fails to generate countable recursion. We make another proposal to which the methods of this section do not apply.

In this paper we investigate the first order metatheory of the classical propositional logic. In the first section we prove that the first order metatheory of the classical propositional logic is undecidable. Thus as a mathematical object even the simplest of logics is, from a logical standpoint, quite complex. In fact it is of the same complexity as true first order number theory.

This result answers negatively a question of J. F. A. K. van Benthem (see [van Benthem and Doets 1983]) as to whether the interpolation theorem in some sense completes the metatheory of the calculus. Let us begin by motivating the question that we answer. In [van Benthem and Doets 1983] it is claimed that a folklore prejudice has it that interpolation was the final elementary property of first order logic to be discovered. Even though other properties of the propositional calculus have been discovered since Craig's orginal paper [Craig 1957] (see for example [Reznikoff 1965]) there is a lot of evidence for the fundamental nature of the property. In abstract model theory for example one finds that very few logics have the interpolation property. There are two well-known open problems in this area. These are

1. Is there a logic satisfying the full compactness theorem as well as the interpolation theorem that is not equivalent to first order logic even for finite models?

2. Is there a logic stronger than L(Q), the logic with the quantifierthere exist uncountably many, that is countably compact and has the interpolation property?

Introduction. This paper contains a new proof of the Church-Rosser theorem for the typed λ-calculus, which also applies to systems with infinitely long terms.

The ordinary proof of the Church-Rosser theorem for the general untyped calculus goes as follows (see [1]). If is the binary reduction relation between the terms we define the one-step reduction 1 in such a way that the following lemma is valid.

Lemma. For all terms a and b we have: ab if and only if there is a sequence a = a0, …, an = b, n ≥ 0, such that aiiai + 1for 0 ≤ i < n.

We then prove the Church-Rosser property for the relation 1 by induction on the length of the reductions. And by combining this result with the above lemma we obtain the Church-Rosser theorem for the relation .

Unfortunately when we come to infinite terms the above lemma is not valid anymore. The difficulty is that, assuming the hypothesis for the infinitely many premises of the infinite rule, there may not exist an upper bound for the lengths n of the sequences ai = a0, …, an = bi (i < α); cf. the infinite rule (iv) in §6.

A completely new idea in the case of the typed λ-calculus would be to exploit the type structure in the way Tait did in order to prove the normalization theorem. In this we succeed by defining a suitable predicate, the monovaluedness predicate, defined over the type structure and having some nice properties. The key notion permitting to define this predicate is the notion of I-form term (see below). This Tait-type proof has a merit, namely that it can be extended immediately to the case of infinite terms.

Nielsen [7] has proved that every subgroup of a free group of finite rank is free. The theorem was later strengthened by Schreier [8] by eliminating the finiteness restriction on the rank. Several proofs of this theorem (known as the Nielsen-Schreier theorem, henceforth denoted by NS) have appeared since Schreier's 1927 paper (see [1] and [2]). All proofs of NS use the axiom of choice (AC) and it is natural to ask whether NS is equivalent to AC. Läuchli has given a partial answer to this question by proving [6] that the negation of NS is consistent with ZFA (Zermelo-Fraenkel set theory weakened to permit the existence of atoms). By the Jech-Sochor embedding theorem (see [3] and [4]) ZFA can be replaced by ZF. Some form of AC, therefore, is needed to prove NS. The main purpose of this paper is to give a further answer to this question.

In §2 we prove that NS implies ACffin (the axiom of choice for sets of finite sets). In §3 we show that a strengthened version of NS implies AC and in §4 we give a partial list of open problems.

Let y be a set; ∣y∣ denotes the cardinal number of y and (y) is the power set of y. If p is a permutation of y and t ∈ y, the p-orbit of t is the set {pn(t): n is an integer}. If

we call p a cyclic permutation of y. If f is a function with domain y and x ⊆ y, f″x denotes the set {f(t):t ∈ x}. If A is a subset of a group (G, °) (sometimes (G, °) will be denoted by G) then A−1= {x−1:x ∈ A} and [A] denotes the subgroup of G generated by A.

Since the invention of β-logic by J. Y. Girard, a lot of work has been done concerning different aspects of this logic, β-completeness (classical and intuitionistic), interpolation theorems, and completeness of Lβω are some examples of this (see [2], [4] and [5]).

In this paper we examine the question, posed by J. Y. Girard, of the Gödel-functional interpretation for this new logic.

The central notion in β-logic is the notion of the β-rule. The β-rule is a functor which appropriately groups x-proofs, for every ordinal x. An x-proof is like a proof in ω-logic but instead of the ω-rule, with premises indexed by ω, we use the x-rule, with premises indexed by x.

In order to obtain the Gödel-functional interpretation of the β-rule, we need, first, a functional interpretation of the x-proofs, which require functionals using the x-rule for their construction (the x-functionals) and, second, an appropriate grouping of these x-functionals by means of a functor (the β-functionals).

We use the letters x,y,z, … and sometimes the Greek letters α and γ to denote ordinals. ON is the category of ordinals. The objects are the ordinals, and the morphisms from x to y are the members of I(x, y), which is the set of all strictly increasing functions from x to y. ON < ω denotes the restriction of ON to ω, the set of finite ordinals. We denote direct systems, in ON or in more general categories, by (xi, fij) where fij is the morphism from xi, to xj. If the direct limit exists we denote it by (x, fi), where fi is the morphism from xi to x. We write (x, fi) =lim(xifij). In ON the direct limits are unique.