Since 1933, when Kolmogorov laid the foundations for probability and statistics as we know them today [1], it has been recognized that propositions asserting that such and such an event occurred as a consequence of the execution of a particular random experiment tend to band together and form a Boolean algebra. In 1936, Birkhoff and von Neumann [2] suggested that the so-called logic of quantum mechanics should not be a Boolean algebra, but rather should form what is now called a modular ortholattice [3]. Presumably, the departure from Boolean algebras encountered in quantum mechanics can be attributed to the fact that in quantum mechanics, one must consider more than one physical experiment, e.g., an experiment measuring position, an experiment measuring charge, an experiment measuring momentum, etc., and, because of the uncertainty principle, these experiments need not admit a common refinement in terms of which the Kolmogorov theory is directly applicable.

Mackey's Axioms I–VI for quantum mechanics [4] imply that the logic of quantum mechanics should be a σ-orthocomplete orthomodular poset [5]. Most contemporary practitioners of quantum logic seem to agree that a quantum logic is (at least) an orthomodular poset [6], [7], [8], [9], [10] or some variation thereof [11]. P. D. Finch [12] has shown that every completely orthomodular poset is the logic arising from sets of Boolean logics, where these sets have a structure similar to the structures generally given to quantum logic. In all of these versions of quantum logic, a fundamental relation, the relation of compatibility or commutativity, plays a decisive role.

It is well known that the laws of logic governing the sentence connectives—“and”, “or”, “not”, etc.—can be expressed by means of equations in the theory of Boolean algebras. The task of providing a similar algebraic setting for the full first-order predicate logic is the primary concern of algebraic logicians. The best-known efforts in this direction are the polyadic algebras of Halmos (cf. [2]) and the cylindric algebras of Tarski (cf. [3]), both of which may be described as Boolean algebras with infinitely many additional operations. In particular, there is a primitive operator, cκ, corresponding to each quantification, ∃υκ. In this paper we explore a version of algebraic logic conceived by A. H. Copeland, Sr., and described in [1], which has this advantage: All operators are generated from a finite set of primitive operations.

Following the theory of cylindric algebras, we introduce, in the natural way, the classes of Copeland set algebras (SCpA), representable Copeland algebras (RCpA), and Copeland algebras of formulas. Playing a central role in the discussion is the set, Γ, of all equations holding in every set algebra. The reason for this is that the operations in a set algebra reflect the notion of satisfaction of a formula in a model, and hence an equation expresses the fact that two formulas are satisfied by the same sequences of objects in the model. Thus to say that an equation holds in every set algebra is to assert that a certain pair of formulas are logically equivalent.

The notion of an almost strongly minimal theory was introduced in [1]. Such a theory is a particularly simple sort of ℵ1-categorical theory. In [1] we characterized this simplicity in terms of the Stone space of models of T. Here, we characterize almost strongly minimal theories which are not ℵ0-categorical in terms of D. M. R. Park's notion [4] of a theory with the strong elementary intersection property. In addition we prove a useful sufficient condition for an elementary theory to be an almost strongly minimal theory. Our notation is from [1] but this paper is independent of the results proved there. We do assume familiarity with §1 and §2 of [2].

In [4], Park defines a theory T to have the strong elementary intersection property (s.e.i.p.) if for each model C of T and each pair of elementary submodels of C either is an elementary submodel of C. T has the nontrivial strong elementary intersection property (n.s.e.i.p.) if for each triple C, as above Park proves the following two statements equivalent:

In this paper we show that in Zermelo-Fraenkel set theory (ZF) sets of reals are determinate.

Before proceeding to the proof it will be helpful to consider some previous work in this area. The first major result was obtained by Gale and Stewart [3] who showed that in ZF open games are determinate. This was then successively improved by Wolfe [4] to (and so of course ) and then by Morton Davis [1] to . The results of Morton Davis further showed that countable unions of sufficiently ‘simple’ determinate sets are also determinate. At this time, however, sets did not appear sufficiently simple for this method to be applied in order to get determinacy.

The next major advance was made by D. A. Martin who showed, using indiscernibles, that with large cardinal assumptions games are equivalent to certain open games (i.e., player I (II) has a winning strategy for the game iff I (II) has a winning strategy for the open game). Thus, by the Gale–Stewart result, games are determinate. Martin's result also showed that under these large cardinal assumptions sets are sufficiently simple for the Morton Davis method to be applied to them.

This paper is devoted to a description of the way in which ultraproducts can be used in proofs of various well-known Σ1-compactness theorems for infinitary languages ℒA associated with admissible sets A; the method generalises the ultra-product proof of compactness for finitary languages.

The compactness theorems we consider are (§2) the Barwise Compactness Theorem for ℒA when A is countable admissible [1], and (§3) the Cofinality (ω) Compactness Theorem of Barwise and Karp [2] and [4]. Our proof of the Barwise theorem unfortunately has the defect that it relies heavily on the Completeness Theorem for ℒA. This defect has, however, been avoided in the case of the Cf(ω) Compactness Theorem, so we have a purely model-theoretic proof.

Although there are many characterizations of weakly compact cardinals (e.g. in terms of indescnbability and tree properties as well as compactness) the most interesting set-theoretic (combinatorial) one is in terms of partition relations. To be more precise we define for κ and α cardinals and n an integer the partition relation of Erdös, Hajnal and Rado [2] as follows:

For every function F: [κ]n→ α (called a partition of [κ]n, the n-element subsets of κ, into α pieces), there exists a set C⊆ κ (called homogeneous for F) such that card C = κ and F″[C]n≠ α, i.e. some element of the range is omitted when F is restricted to the n-element subsets of C. It is the simplest (nontrivial) of these relations, i.e. , that is the well-known equivalent of weak compactness.1

Two directions of inquiry immediately suggest themselves when weak compactness is described in terms of these partition relations: (a) Trying to strengthen the relation by increasing the superscript—e.g., —and (b) trying to weaken the relation by increasing the subscript—e.g., . As it turns out, the strengthening to is only illusory for using the equivalence of to the tree property one quickly sees that implies (and so is equivalent to) for every n. Thus is the strongest of these partition relations. The second question seems much more difficult.

Let α be a limit ordinal with the property that any “recursive” function whose domain is a proper initial segment of α has its range bounded by α. α is then called admissible (in a sense to be made precise later) and a recursion theory can be developed on it (α-recursion theory) by providing the generalized notions of α-recursively enumerable, α-recursive and α-finite. Takeuti [12] was the first to study recursive functions of ordinals, the subject owing its further development to Kripke [7], Platek [8], Kreisel [6], and Sacks [9].

Infinitary logic on the other hand (i.e., the study of languages which allow expressions of infinite length) was quite extensively studied by Scott [11], Tarski, Kreisel, Karp [5] and others. Kreisel suggested in the late '50's that these languages (even which allows countable expressions but only finite quantification) were too large and that one should only allow expressions which are, in some generalized sense, finite. This made the application of generalized recursion theory to the logic of infinitary languages appear natural. In 1967 Barwise [1] was the first to present a complete formalization of the restriction of to an admissible fragment (A a countable admissible set) and to prove that completeness and compactness hold for it. [2] is an excellent reference for a detailed exposition of admissible languages.

In [5] Scott asked if there was a proof theoretic proof of his interpolation theorem. The purpose of this paper is to provide such a proof, working with the first order system LK of Gentzen [2]. Our method is an extension of the one in Maehara [3] for Craig's interpolation theorem. We will also sketch the original model theoretic proof and show how Scott used his result to obtain a definability theorem of Svenonius [7].

A language for LK contains the usual logical symbols: , ∧, ∨, ⊃, ∀, ∃; countably many free variables a0, a1, … and bound variables x0, x1, …; and some or all of the following nonlogical symbols: n-ary predicates ; n-ary functions ; and individual constants c0, c1, …. Semiterms are defined as follows: (1) Free variables, bound variables and individual constants are semiterms. (2) If f is an n-ary function and s1 …, sn are semiterms, then f(s1 …, sn), is a semiterm. A term is a semiterm that does not contain a bound variable. Formulas are defined as follows: (1) If R is an n-ary predicate and t1 …, tn are terms, then R(t1 …, tn) is a formula. (2) If A and B are formulas, then A, A ∧ B, A ∨ B and A ⊃ B are formulas. (3) If A(t) is a formula which has zero or more occurrences of the term t, and if x is a bound variable not contained in A(t), then ∀xA(x) and ∃xA(x) are formulas where A(x) is obtained from A(t) by substituting x for t at all indicated places.

In previous articles ([4], [5]) it has been shown that the deductive system developed by Aristotle in his “second logic” (cf. Bochenski [2, p. 43]) is a natural deduction system and not an axiomatic system as previously had been thought [6]. It was also pointed out that Aristotle's logic is self-sufficient in two senses: First, that it presupposed no other logical concepts, not even those of propositional logic; second, that it is (strongly) complete in the sense that every valid argument formable in the language of the system is demonstrable by means of a formal deduction in the system. Review of the system makes the first point obvious. The purpose of the present article is to prove the second. Strong completeness is demonstrated for the Aristotélian system.

Ackermann's set theory [1], called here A, involves a schema

where φ is an ∈-formula with free variables among y1, …, yn and w does not appear in φ. Variables are thought of as ranging over classes and V is intended as the class of all sets.

S is a kind of comprehension principle, perhaps most simply motivated by the following idea: The familiar paradoxes seem to arise when the class CP of all P-sets is claimed to be a set, while there exists some P-object x not in CP such that x would have to be a set if CP were. Clearly this cannot happen if all P-objects are sets.

Now, Levy [2] and Reinhardt [3] showed that A* (A with regularity) is in some sense equivalent to ZF. But the strong replacement axiom of Gödel-Bernays set theory intuitively ought to be a theorem of A* although in fact it is not (Levy's work shows this). Strong replacement can be formulated as

This lack of A* can be remedied by replacing S above by

where ψ and φ are ∈-formulas and x is not in ψ and w is not in φ. ψv is ψ with quantifiers relativized to V, and y and z stand for y1, …, yn and z1, …, zm.

In this paper we deal with infinitary languages which allow arbitrary infinite disjunctions and conjunctions, but finite strings of quantifiers only. Furthermore, we shall assume that the primitive predicate symbols are finitary. (Cf. [K] and §2 for further information and unexplained notations.)

Our main result shows that every “reasonable” language of this type is, in a certain sense, reducible to one of the same type which allows countable disjunctions and conjunctions only. More precisely if we let θ be the first measurable cardinal and μ ≤ θ, we have the following result partially announced in [R1]:

Theorem. Every theory in a Lμω language is equivalent to some theory in a Lω1ω language in the sense that atomic formulas of one language can be mapped into formulas of the other in such a way that, for every set, these maps establish a bijective correspondence between their models having that set as a universe.

The paper is divided into three sections. In the first, we derive the main result from a theorem of Sikorski on ω-homomorphisms of μ-complete boolean algebras. In the second, we give a refined version of Sikorski's theorem from which cardinality bounds can be given for the Lω1ω theories obtained by our reduction. In the last, we use a Skolem ultrapower construction to give examples for which these bounds are actually attained. As a corollary, we obtain the converse of our main result.

Let ⊥, →, and □ be primitive, and let us have a countable supply of propositional letters. By a (modal) logic we understand a proper subset of the set of all formulas containing every tautology and being closed under modus ponens and substitution. A logic is regular if it contains every instance of □A ∧ □B ↔ □(A ∧ B) and is closed under the rule

A regular logic is normal if it contains □⊤. The smallest regular logic we denote by C (the same as Lemmon's C2), the smallest normal one by K. If L and L' are logics and L ⊆ L′, then L is a sublogic of L', and L' is an extension of L; properly so if L ≠ L'. A logic is quasi-regular (respectively, quasi-normal) if it is an extension of C (respectively, K).

A logic is Post complete if it has no proper extension. The Post number, denoted by p(L), is the number of Post complete extensions of L. Thanks to Lindenbaum, we know that

There is an obvious upper bound, too:

Furthermore,

.

We have come to believe that propositional modal logic (with the usual relational semantics) must be understood as a rather strong fragment of classical second-order predicate logic. (The interpretation of propositional modal logic in second-order predicate logic is well known; see e.g. [2, §1].) “Strong” refers of course to the expressive power of the languages, not to the deductive power of formal systems. By “rather strong” we mean sufficiently strong that theorems about first-order logic which fail for second-order logic usually fail even for propositional modal logic. Some evidence for this belief is contained in [2] and [3]. In the former is exhibited a finitely axiomatized consistent tense logic having no relational models, and the latter presents a finitely axiomatized modal logic between T and S4, such that □p → □2p is valid in all relational models of the logic but is not a thesis of the logic. The result of [2] is strong evidence that bimodal logic is essentially second-order, but that of [3] does not eliminate the possibility that unimodal logic only appears to be incomplete because we have not adopted sufficiently powerful rules of inference. In the present paper we present stronger evidence of the essentially second-order nature of unimodal logic.

Fraenkel-Mostowski models are a particularly simple and conceptual tool for proving consistency results involving the axiom of choice, AC. These models satisfy the theory, FM, of a well founded universe of sets built from a ground set of individuals. Zermelo-Fraenkel set theory, ZF, is the extension of FM in which the set of individuals is assumed to be empty. In this paper we show that there is a large class of statements whose consistency with ZF can be proven directly by means of a Fraenkel-Mostowski model.

A statement, Φ, of set theory is said to be transferable if there is a metatheorem: If Φ is true in a Fraenkel-Mostowski model then Φ is consistent with ZF. Jech and Sochor introduced, in [12], the class of boundable statements and proved them to be transferable. Most existential contradictions of AC are boundable. It remains to find criteria under which Ψ ∧ Φ is transferable where Ψ is a universal consequence of AC and Φ is an existential contradiction of AC. To this end we give two classes of statements. Each class is closed under conjunction, contains the boundable statements, and contains a number of universal consequences of AC. Nearly every Fraenkel-Mostowski consistency in the literature falls into one of these two classes.

In §2 we give two generalizations of the boundable statements. In §§3 and 4 the classes of transferable statements are discussed. In §5 we discuss the transfer problem and prove a metatheorem concerning nontransferable statements.