An infinite-dimensional vector space V∞ over a recursive field F is called fully effective if V∞ is a recursive set identified with ω upon which the operations of vector addition and scalar multiplication are recursive functions, identity is a recursive relation, and V∞ has a dependence algorithm, that is a uniformly effective procedure which when applied to x, a1,…,an, ∈ V∞ determines whether or not x is an element of {a1,…,an}* (the subspace generated by {a1,…,an}). The study of V∞, and of its lattice of r.e. subspaces L(V∞), was introduced in Metakides and Nerode [15]. Since then both V∞ and L(V∞) (and many other effective algebraic systems) have been studied quite intensively. The reader is directed to [5] and [17] for a good bibliography in this area, and to [15] for any unexplained notation and terminology.

In [15] Metakides and Nerode observed that a study of L(V∞) may in some ways be modelled upon a study of L(ω), the lattice of r.e. sets. For example, they showed how an e-state construction could be modified to produce an r.e. maximal subspace, where M ∈ L(V∞) is maximal if dim(V∞/M) = ∞ and, for all W ∈ L(V∞), if W ⊃ M then either dim(W/M) < ∞ or dim(V∞/W) < ∞.

However, some of the most interesting features of L(V∞) are those which do not have analogues in L(ω). Our concern here, which is probably one of the most striking characteristics of L(V∞), falls into this category. We say M ∈ L(V∞) is supermaximal if dim(V∞/M) = ∞ and for all W ∈ L(V∞), if W ⊃ M then dim(W/M) < ∞ or W = V∞. These subspaces were discovered by Kalantari and Retzlaff [13].

Any attempt to give “foundations”, for category theory or any domain in mathematics, could have two objectives, of course related.

(0.1) Noncontradiction: Namely, to provide a formal frame rich enough so that all the actual activity in the domain can be carried out within this frame, and consistent, or at least relatively consistent with a well-established and “safe” theory, e.g. Zermelo-Frankel (ZF).

(0.2) Adequacy, in the following, nontechnical sense:

(i) The basic notions must be simple enough to make transparent the syntactic structures involved.

(ii) The translation between the formal language and the usual language must be, or very quickly become, obvious. This implies in particular that the terminology and notations in the formal system should be identical, or very similar, to the current ones. Although this may seem minor, it is in fact very important.

(iii) “Foundations” can only be “foundations of a given domain at a given moment”, therefore the frame should be easily adaptable to extensions or generalizations of the domain, and, even better, in view of (i), it should suggest how to find meaningful generalizations.

(iv) Sometimes (ii) and (iii) can be incompatible because the current notations are not adapted to a more general situation. A compromise is then necessary. Usually when the tradition is very strong (ii) is predominant, but this causes some incoherence for the notations in the more general case (e.g. the notation f(x) for the value of a function f at x obliges one, in category theory, to denote the composition of arrows (f, g) → g∘f, and all attempts to change this notation have, so far, failed).

Some propositional logics (e.g. the classical system) can be characterized by a finite model, while others (e.g. Heyting's) which have the finite model property (FMP) can be characterized by an infinite set of finite models. Still others (e.g. certain extensions of Heyting's logic) which lack the FMP can only be characterized by a set of models, at least one of which is infinite. Yet all these logics admit finite models even though they may not be characterized by them. (For example, they all admit the 2-element Boolean algebra as a model in the sense that all their theorems are valid on that algebra when the propositional connectives are interpreted in the usual manner.) The object of the present paper is to give a (not too artificial) example of a propositional logic which is consistent and which admits only infinite models. It therefore lacks the FMP in a very strong sense. Such a propositional logic, I shall call hyperinfinite. The existence of hyperinfinite logics was already plausible from a result in abstract algebra which says that there are varieties of algebras of which the only finite element is the trivial algebra (see [3]).

I wish to thank Professor A. S. Troelstra, Amsterdam, for comments on an early version of this paper. The constructive criticism of two anonymous referees has also been useful.

The hyperinfinite propositional logic to be described is obtained from Positive Logic—the negative-free part of Heyting's logic—by adjoining certain axioms which govern the use of a unary modal connective.

We study the effects of Vopěnka's principle on properties of model theoretic logics. We show that Vopěnka's principle is equivalent to the assumption that every finitely generated logic has a compact cardinal. We show also that it is equivalent to the assumption that every such logic has a global Hanf number.

Before stating the results we would like to thank the referee for reorganizing the whole paper and changing its original logical terminology to an algebraical terminology.

Let Z* be a proper elementary extension of the integral domain Z. For any b ∈ Z*∖Z we define Z〈b〉 as the following subring of Z*:

It is easy to prove that the natural divisibility relation on Z〈b〉 is the restriction to Z〈b〉 of the divisibility on Z* (see Lemma 1.1 below). The much stronger statement that Z〈b〉 is algebraically closed in Z* does not hold for all b. Similarly, there are many b's such that Z〈b〉 is not a recursive ring, e.g. if the set of standard primes dividing b is not recursive then Z〈b〉 cannot be a recursive ring under any bijection with N.

Our main result is the following.

Theorem. There exists an element b ∈ Z*∖Z such that

(1) Z〈b〉 is algebraically closed in Z*,

(2) Z〈b〉 is a recursive ring, and

(3) each infinite element of Z〈b〉 is divisible by a standard integer > 1 (so the only “primes” of Z〈b〉 are the standard ones).

An easy consequence of (1) is that certain induction axioms are satisfied by Z〈b〉. This explains the title of the paper. In the last section of the paper, §3, we shall say more about this.

In the December 1982 issue of this Journal Weitkamp [W] posed some questions concerning the incomparability of certain “r.e.” sets for the notion of Kleene reducibility. He asked whether the incomparability of, for example, the Friedman set F (defined below) and the set WI0 (the set of reals coding wellfounded trees of admissible height) was equivalent to the existence of 0#, since forcing over L with a set of conditions could not achieve this. We answer this by showing that in a certain class generic extension of L they are comparable, but 0# does not exist. This is an application of Jensen's coding theorem (cf. [BJW]), using a modified construction due to René David [D]. Indeed the result here is a simple application of his result. Define F as follows:

Harrington showed, in effect, that one could not add a cone of Turing degrees to this set by forcing with sets of conditions over L. The method used here does add a cone of Turing degrees to a much simpler set RI1 (defined below)—and indeed the whole process could be viewed as forcing over L to obtain the determinacy of certain rather simple sets. It is the determinacy of the game with payoff set RI1 that ensures the comparability of F and WI0 (amongst many others).

We shall refrain from repeating all the basic definitions and lemmas since the reader can readily refer to [W]; we shall give the basic necessities.

S. Ulam asked about the number of nonisomorphic projective algebras with κ generators. This paper answers his question for projective algebras of finite dimension at least three and shows that there are the maximum possible number, continuum many, of nonisomorphic one-generated structures of finite dimension n, where n is at least three, of the following kinds: projective set algebras, projective algebras, diagonal-free cylindric set algebras, diagonal-free cylindric algebras, cylindric set algebras, and cylindric algebras. The results of this paper extend earlier results to the collection of cylindric set algebras and provide a uniform proof for all the results. Extensions of these results for dimension two are discussed where some modifications on the hypotheses are needed. Furthermore for α ≥ 2, the number of isomorphism classes of regular locally finite cylindric set algebras of dimension α of the following two kinds are computed: ones of power κ for infinite κ ≥ ∣α∣, and ones with a single generator.

The present work may perhaps be seen as a point of convergence of two historically distinct sequences of results. One sequence of results started with the work of Tennenbaum [59] who showed that there could be no nonstandard recursive model of the system PA of first order Peano arithmetic. Shepherdson [65] on the other hand showed that the system of arithmetic with open induction was sufficiently weak to allow the construction of nonstandard recursive models. Between these two results there remained for many years a large gap occasioned by a general lack of interest in weak systems of arithmetic. However Dana Scott observed that the addition alone of a nonstandard model of PA could not be recursive, while more recently McAloon [82] improved these results by showing that even for the weaker system of arithmetic with only bounded induction, neither the addition nor the multiplication of a nonstandard model could be recursive.

Another sequence of results starts with the work of Lessan [78], and independently Jensen and Ehrenfeucht [76], who showed that the structures which may be obtained as the reducts to addition of countable nonstandard models of PA are exactly the countable recursively saturated models of Presburger arithmetic. More recently, Cegielski, McAloon and the author [81] showed that the above result holds true if PA is replaced by the much weaker system of bounded induction.

However in both the case of the Tennenbaum phenomenon and in that of the recursive saturation of addition the problem remained open as to how strong a system was really necessary to generate the required phenomenon. All that was clear a priori was that open induction was too weak to produce either result.

Hofstadter has introduced the notion of an explicit Henkin sentence. Roughly speaking, an explicit Henkin sentence not only asserts its own provability, as ordinary Henkin sentences do, but explicitly provides a detailed description of a proof. We provide, in this paper, a precise formalization of Hofstadter's notion and then show that true explicit Henkin sentences exist.

If dominating functions in ωω are adjoined repeatedly over a model of GCH via a finite-support c.c.c. iteration, then in the resulting generic extension there are no long towers, every well-ordered unbounded family of increasing functions is a scale, and the splitting number (and hence the distributivity number ) remains at ω1.

Our purpose is to delineate the extensions (normal and otherwise) of the propositional modal logic K5. We associate with each logic extending K5 a finitary index, in such a way that properties of the logics (for example, inclusion, normality, and tabularity) become effectively decidable properties of the indices. In addition we obtain explicit finite axiomatizations of all the extensions of K5 and an abstract characterization of the lattice of such extensions.

This paper refines and extends the Ph.D. thesis [2] of the first-named author, who wishes to acknowledge his debt to Brian F. Chellas for his considerable efforts in directing the research culminating in [2] and [3]. We also thank W. J. Blok and Gregory Cherlin for observations which greatly simplified the proofs of Theorem 3 and Corollary 10.

By a logic we mean a set of formulas in the countably infinite set Var of propositional variables and the connectives ⊥, →, and □ (other connectives being used abbreviatively) which contains all the classical tautologies and is closed under detachment and substitution. A logic is classical if it is also closed under RE (from A↔B infer □A ↔□B) and normal if it is classical and contains □ ⊤ and □ (P → q) → (□p → □q). A logic is quasi-classical if it contains a classical logic and quasi-normal if it contains a normal logic. Thus a quasi-normal logic is normal if and only if it is classical, and if and only if it is closed under RN (from A infer □A).

For any A ⊂ R, the Banach game B(A) is the following infinite game on reals: Players I and II alternately play positive real numbers a1; a2, a3, a4,… such that for n > 1, an < an−1. Player I wins iff ai exists and is in A.

This type of game was introduced by Banach in 1935 in the Scottish Book [15], Problem 43. The (rather vague) problem which Banach posed was to characterize those sets A for which I (II) has a winning strategy in B(A). (There are three parts to Problem 43. In the first, Mazur defined a game G**(A) for every set A ⊂ R and conjectured that II has a winning strategy in G**(A) iff A is meager and I has a winning strategy in G**(A) iff A is comeager in some neighborhood; this conjecture was proved by Banach. Presumably Banach had this result in mind when he asked the question about B(A), and hoped for a similar type of characterization.) Incidentally, Problem 43 of the Scottish Book appears to be the first time infinite games of any sort were studied by mathematicians.

This paper will not provide the reader with any answer to Banach's question. I know of no nontrivial way to characterize when player I (or II) wins, and I suspect there is none. This paper is concerned with a different (also rather vague) question: For which sets A is the Banach game B(A) determined? To say that B(A) is determined means, of course, that one of the players has a winning strategy for B(A).

A series of investigations started by H. Friedman ([5], [6]) and pursued by him, by S. Simpson, and by others (see [7], [16] and the references there) had the objective of determining what are precisely the set existence axioms needed for proving theorems of “ordinary” mathematics. The study centered around -CA0 (for “-Comprehension Axiom”), a fragment of second order arithmetic in which the main body of “ordinary” mathematics can be comfortably developed (of course, after a suitable encoding of the various concepts into numbers and sets of numbers). The main question is always this: given a theorem τ provable in -CA0, do we need all of -CA0, or, maybe, is τ provable in a weaker subsystem? A surprisingly simple pattern emerged: there seem to be just five important systems, denoted (in order of increasing strength) RCA0, WKL0, ACA0, ATR0 and -CA0, such that in most cases, whenever an ordinary mathematical theorem τ is provable in -CA0, it is either provable in RCA0 or it is equivalent to one, call it S, of the other four systems, the proof of the equivalence being done in one of the systems which is weaker than S (most typically, RCA0 or ACA0). This very interesting phenomenon—or “theme” as it was called by Friedman—has been verified for many instances in the realm of analysis and algebra (cf. the references above and the forthcoming book [17]). It is the purpose of this paper to do the same for a particular branch of model theory, namely stability theory.

In this paper we answer the question of whether all low sets with the splitting property are promptly simple. Further we try to make the role of lowness properties and prompt simplicity in the construction of automorphisms of the lattice of r.e. (recursively enumerable) sets more perspicuous. It turns out that two new properties of r.e. sets, which are dual to each other, are essential in this context: the prompt and the low shrinking property.

In an earlier paper [4] we had shown (using Soare's automorphism construction [10] and [12]) that all r.e. generic sets are automorphic in the lattice ℰ of r.e. sets under inclusion. We called a set A promptly simple if Ā is infinite and there is a recursive enumeration of A and the r.e. sets (We)e∈N such that if We is infinite then there is some element (or equivalently: infinitely many elements) x of We such that x gets into A “promptly” after its appearance in We (i.e. for some fixed total recursive function f we have x ∈ Af(s), where s is the stage at which x entered We). Prompt simplicity in combination with lowness turned out to capture those properties of r.e. generic sets that were used in the mentioned automorphism result. In a following paper with Shore and Stob [7] we studied an ℰ-definable consequence of prompt simplicity: the splitting property.

The purpose of this work is to present Gentzen-style formulations of S5 and S4 based on sequents of higher levels. Sequents of level 1 are like ordinary sequents, sequents of level 2 have collections of sequents of level 1 on the left and right of the turnstile, etc. Rules for modal constants involve sequents of level 2, whereas rules for customary logical constants of first-order logic with identity involve only sequents of level 1. A restriction on Thinning on the right of level 2, which when applied to Thinning on the right of level 1 produces intuitionistic out of classical logic (without changing anything else), produces S4 out of S5 (without changing anything else).

This characterization of modal constants with sequents of level 2 is unique in the following sense. If constants which differ only graphically are given a formally identical characterization, they can be shown inter-replaceable (not only uniformly) with the original constants salva provability. Customary characterizations of modal constants with sequents of level 1, as well as characterizations in Hilbert-style axiomatizations, are not unique in this sense. This parallels the case with implication, which is not uniquely characterized in Hilbert-style axiomatizations, but can be uniquely characterized with sequents of level 1.

These results bear upon theories of philosophical logic which attempt to characterize logical constants syntactically. They also provide an illustration of how alternative logics differ only in their structural rules, whereas their rules for logical constants are identical.

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.

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.

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.