We prove (in ZFC Set Theory) that all infinite games whose winning sets are of the following forms are determined:

(1) (A − S) ∪ B, where A is , , and the game whose winning set is B is “strongly determined” (meaning that all of its subgames are determined).

(2) A Boolean combination of sets and sets smaller than the continuum.

This also enables us to show that strong determinateness is not preserved under complementation, improving a result of Morton Davis which required the continuum hypothesis to prove this fact.

Various open questions related to the above results are discussed. Our main conjecture is that (2) above remains true when is replaced by “Borel”.

Let 〈, ≤ 〉 be the usual structure of the degrees of unsolvability and 〈, ≤ 〉 the structure of the T-degrees of partial functions defined in [7]. We prove that every countable distributive lattice with a least element can be isomorphically embedded as an initial segment of 〈, ≤ 〉: as a corollary, the first order theory of 〈, ≤ 〉 is recursively isomorphic to that of 〈, ≤ 〉. We also show that 〈, ≤ 〉 and 〈, ≤ 〉 are not elementarily equivalent.

Contrary to what is stated in Lemma 7.1 of [8], it is shown that some Boolean algebras of finitary logic admit finitely additive probabilities that are not σ-additive. Consequences of Lemma 7.1 are reconsidered. The concept of a C-σ-additive probability ℬ (where ℬ and C are Boolean algebras, and ℬ ⊆ C) is introduced, and a generalization of Hahn's extension theorem is proved. This and other results are employed to show that every S̄(L)-σ-additive probability on s̄(L) can be extended (uniquely, under some conditions) to a σ-additive probability on S̄(L), where L belongs to a quite extensive family of first order languages, and S̄(L) and s̄(L) are, respectively, the Boolean algebras of sentences and quantifier free sentences of L.

It is proven that the following statement:

“there exists a club C ⊆ κ such that every α ∈ C is an inaccessible cardinal in L and, for every δ a limit point of C, C ∩ δ is almost contained in every club of δ of L”

is equiconsistent with a weakly compact cardinal if δ = ℵ1, and with a weakly compact cardinal of order 1 if δ = ℵ2.

The starting point of this work was Saracino and Wood's description of the finitely generic abelian ordered groups [S-W].

We generalize the result of Saracino and Wood to a class ∑UH of subdirect products of substructures of elements of a class ∑, which has some relationships with the discriminator variety V(∑t) generated by ∑. More precisely, let ∑ be an elementary class of L-algebras with theory T. Burris and Werner have shown that if ∑ has a model companion then the existentially closed models in the discriminator variety V(∑t) form an elementary class which they have axiomatized. In general it is not the case that the existentially closed elements of ∑UH form an elementary class. For instance, take for ∑ the class ∑0 of linearly ordered abelian groups (see [G-P]).

We determine the finitely generic elements of ∑UH via the three following conditions on T:

(1) There is an open L-formula which says in any element of ∑UH that the complement of equalizers do not overlap.

(2) There is an existentially closed element of ∑UH which is an L-reduct of an element of V(∑t) and whose L-extensions respect the relationships between the complements of the equalizers.

(3) For any models A, B of T, there exists a model C of TUH such that A and B embed in C.

(Condition (3) is weaker then “T has the joint embedding property”. It is satisfied for example if every model of T has a one-element substructure. Condition (3) implies that ∑UH has the joint embedding property and therefore that the class of finitely generic elements of ∑UH is complete.)

Let T be a (countable, complete, quantifier eliminable) ω-stable theory; an analysis of T, and consequently a classification of ω-stable theories, can be done by looking at the Boolean algebras B(M) of definable subsets of its countable models M (as usual, we often confuse a definable subset of M with the class of formulas defining it). If M ⊨ T, ∣M∣ = ℵ0, then, for every LM-formula ϕ(v) and for every ordinal α, we define a relation

(CB = Cantor-Bendixson, of course) by induction on α:

CB-rank ϕ(v) ≥ 0 if ϕ(M) ≠ ∅

CB-rank ϕ(v) ≥ λ for λ a limit ordinal, if CB-rank ϕ(v) ≥ for all v < λ;

CB-rank ϕ(v)≥ α + 1 if, for all n ∈ ω,

(*) there are LM-formulas ϕ0(v), …, ϕn − 1(v) such that

It is well known that the ω-stability of T implies that, for every consistent LM-formula ϕ(v), there is exactly one ordinal α < ω1 such that CB-rank ϕ(v) ≥ α and CB-rank ϕ(v)≱α + 1. Therefore we define:

CB-rank ϕ(v) = α

CB-degree ϕ(v) = d if d is the maximal n ∈ ω satisfying (*); and

CB-type ϕ(v) = (α, d).

This paper establishes another very general completeness result for the logics within the field of K4. With each finite transitive frame ℭ we may associate a formula — Bℭ which validates just those frames ℑ in which ℭ is not in a certain sense embeddable (to be exact, ℭ is not the p-morphic image of any subframe of ℑ. By a subframe logic we mean the result of adding such formulas as axioms to K4. The general result is that each subframe logic has the finite model property.

There are a continuum of subframe logics and they include many of the standard ones, such as T, S4, S4.3, S5 and G. It turns out that the subframe logics are exactly those complete for a condition that is closed under subframes (any subframe of a frame satisfying the condition also satisfies the condition). As a consequence, every logic complete for a condition closed under subframes has the finite model property.

It is ascertained which of the subframe logics are compact. It turns out that the compact logics are just those whose axioms express an elementary condition. Tests are given for determining whether a given axiom expresses an elementary condition and for determining what it is in case it does.

In one respect the present general completeness result differs from most of the others in the literature. The others have usually either been what one might call logic based or formula based. They have usually either been to the effect that all of the logics containing a given logic are complete or to the effect that all logics whose axioms come from a given syntactically characterized class of formulas are complete. The present result is, by contrast, what one might call frame based. The axioms of the logics to be proved complete are characterized most directly in terms of their connection with certain frames.

In this paper a model for barrecursion is presented. It has as a novelty that it contains discontinuous functionals. The model is based on a concept called strong majorizability. This concept is a modification of Howard's majorizability notion; see [T, p. 456].

We show that at each type κ ≥ 2, there exist c-irreducible functionals of c-r.e. degree, as defined in [Nor 1]. Our proofs are based on arguments due to Hinman, [Hin 1], and Dvornikov, [Dvo 1].

The theory of trees with additional unary predicates and quantification over nodes and branches embraces a rich branching time logic. This theory was reduced in the companion paper to the first-order theory of binary, bounded, well-founded trees with additional unary predicates. Here we prove the decidability of the latter theory.

Syntactic translations of classical logic C into intuitionistic logic I are well known (see [Kol25], [Gli29], [Göd32], [Kre58b], [M063], [Cel69] and [Lei71]). Harvey Friedman [Fri78] used a translation of a similar nature, from I into itself, to reprove a theorem of Kreisel [Kre58a] that various theories based on I are closed under Markov's rule: if ¬¬∃x.α is a theorem, where x is a numeric variable and α is a primitive recursive relation, then ∃x.α is a theorem. Composing this with Gödel's translation from classical to intuitionistic theories, it follows that the functions provably recursive in the classical version of the theories considered are provably recursive already in their intuitionistic version. This conservation result is important in that it guarantees that no information about the convergence of recursive functions is lost when proofs are restricted to constructive logic, thus removing a potential objection to the use of constructive logic in reasoning about programs (see [C078] for example). Conversely, no objection can be raised by intuitionists to proofs of formulas that use classical reasoning, because such proofs can be converted to constructive proofs (this has been exploited extensively; see [Smo82]).

Proofs of closure under Markov's rule had required, until Friedman's proof, a relatively sophisticated mathematical apparatus. The chief method used Godel's “Dialectica” interpretation (see [Tro73, §3]). Other proofs used cut-elimination, provable reflection for subsystems [Gir73], and Kripke models [Smo73]. Moreover, adapting these proofs to new theories had required that the underlying meta-mathematical techniques be adapted first, not always a trivial step.

In algebra, the properties of having the (strong) amalgamation property and epis being surjective are well investigated; see the survey [10]. In algebraic logic it is shown that to these algebraic properties there correspond interesting logical properties, see e.g. [15], [12], [4], and [8, p. 311, Problem 10 and the remark below it]. In the present paper we show that the varieties Crsα (of cylindric-relativised set algebras) and Boα (of Boolean algebras with operators) have the strong amalgamation property. These contrast to the following result proved in Pigozzi [15]: No class K with Gsα ⊆ K ⊆ CAα has amalgamation property. Note that Gsα ⊆ Crsα ⊆ Boα and CAα ⊆ Boα. For related results see [3], [1], [16], [11]. For more connections with logic and abstract model theory see [14] and §4.3 of [9].

BA denotes the class of all Boolean algebras. Let α be any ordinal. From now on, throughout in the paper, α is an arbitrary but fixed ordinal. Recall from [7, p. 430, Definition 2.7.1] that an α-dimensional BA with operators, a Boα, is an algebra = 〈A, + −, ci, dij〉i, j ∈ α of the same similarity type as CAα's such that , is a BA and the operations ci (i ∈ α) are additive, i.e., ⊨ ci(x + y) = cix + ciy for all i ⊨ α. If ⊨ Boα then I is called the Boolean reduct of . Note that BA = Bo0. A Boα is said to be normal if {ci 0 = 0: i ∈ α} is valid in it, and a Boα is said to be extensive if {x ≤ cix: i ∈ α} is valid in it. Boα's were introduced in [17].

The class Crsα of all cylindric-relativised set algebras is defined in Definition 1.1.1 (iii) of [8, p. 4]. We give a definition in the present paper, too—see Definition 5 below. It is shown in [13] that ICrsα is a variety.

Our main result is (i) of Theorem 1 below, but we obtain (ii)–(vi), too, as a byproduct from the proof.

The purpose of this paper is not to produce a survey of systems with a universal set: we do not yet understand them well enough. Rather it will concentrate on one particular aspect of them: the curious circumstance that there are half-a-dozen or so distinct proofs of ~ AC available in set theories with a universal set. This began to emerge in 1953 when Specker published in [2] a proof of ~ AC in Quine's system NF. Until recently this was an isolated phenomenon and poorly understood. The proof answered few questions and seemed rather ad hoc, thus inviting an investigation to determine whether this was an artefact caused by the particular axioms for NF, or part of a general conflict between the demands of big sets and AC.

We will start with an informal discussion of the genesis of set theories with a universal set, collecting, en route, a number of desiderata for such theories. A number of new refutations of AC in systems meeting some or all of these conditions will then be presented. The conclusion the reader is invited to draw is that any sensible set theory with a universal set will probably have trouble with AC.

Why do set theory with V e V at all? It contradicts conventional wisdom which teaches us that e is wellfounded. It must be said for this doctrine (let us abbreviate it to WOOF) that it has enabled the rapid execution of the logicist programme, so if that were the sole purpose of set theory we could consider ourselves well served. However most pure mathematicians are platonists and study things not because they are useful but simply because they are there. And the belief that V is not there can, after all, only be the result of early conditioning.

The probability logic is a logic with a natural interpretation on probability spaces (thus, a logic whose model theory is part of probability theory rather than a system for putting probabilities on formulas of first order logic). Its exact definition and basic development are contained in the paper [3] of H. J. Keisler and the papers [1] and [2] of the author. Building on work in [2], we prove in this paper the following probabilistic interpolation theorem for .

Let L be a countable relational language, and let A be a countable admissible set with ω ∈ A (in this paper some probabilistic notation will be used, but ω will always mean the least infinite ordinal). is the admissible fragment of corresponding to A. We will assume that L is a countable set in A, as is usual in practice, though all that is in fact needed for our proof is that L be a set in A which is wellordered in A.

Theorem. Let ϕ(x) and ψ(x) be formulas of LAP such that

where ε ∈ [0, 1) is a real in A (reals may be defined in the usual way as Dedekind cuts in the rationals). Then for any real d > ε¼, there is a formula θ(x) of (L(ϕ) ∩ L(ψ))AP such that

and

There are well-known theorems in mathematical logic that indicate rather profound differences between the logic of first-order languages and the logic of second-order languages. In the first-order case, for example, there is Gödel's completeness theorem: every consistent set of sentences (vis-à-vis a standard axiomatization) has a model. As a corollary, first-order logic is compact: if a set of formulas is not satisfiable, then it has a finite subset which also is not satisfiable. The downward Löwenheim-Skolem theorem is that every set of satisfiable first-order sentences has a model whose cardinality is at most countable (or the cardinality of the set of sentences, whichever is greater), and the upward Löwenheim-Skolem theorem is that if a set of first-order sentences has, for each natural number n, a model whose cardinality is at least n, then it has, for each infinite cardinal κ (greater than or equal to the cardinality of the set of sentences), a model of cardinality κ. It follows, of course, that no set of first-order sentences that has an infinite model can be categorical. Second-order logic, on the other hand, is inherently incomplete in the sense that no recursive, sound axiomatization of it is complete. It is not compact, and there are many well-known categorical sets of second-order sentences (with infinite models). Thus, there are no straightforward analogues to the Löwenheim-Skolem theorems for second-order languages and logic.

There has been some controversy in recent years as to whether “second-order logic” should be considered a part of logic, but this boundary issue does not concern me directly, at least not here.

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.

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.

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.

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.