In recent years there has been a proliferation of logics which extend first-order logic, e.g., logics with infinite sentences, logics with cardinal quantifiers such as “there exist infinitely many…” and “there exist uncountably many…”, and a weak second-order logic with variables and quantifiers for finite sets of individuals. It is well known that first-order logic has a limited ability to express many of the concepts studied by mathematicians, e.g., the concept of a wellordering. However, first-order logic, being among the simplest logics with applications to mathematics, does have an extensively developed and well understood model theory. On the other hand, full second-order logic has all the expressive power needed to do mathematics, but has an unworkable model theory. Indeed, the search for a logic with a semantics complex enough to say something, yet at the same time simple enough to say something about, accounts for the proliferation of logics mentioned above. In this paper, a number of proposed strengthenings of first-order logic are examined with respect to their relative expressive power, i.e., given two logics, what concepts can be expressed in one but not the other?

For the most part, the notation is standard. Most of the notation is either explained in the text or can be found in the book [2] of Chang and Keisler. Some notational conventions used throughout the text are listed below: the empty set is denoted by ∅.

Consistency properties and their model existence theorems have provided an important method of constructing models for fragments of L∞ω. In [E] Ellentuck extended this construction to Suslin logic. One of his extensions, the Borel consistency property, has its extra rule based not on the semantic interpretation of the extra symbols but rather on a theorem of Sierpinski about the classical operation . In this paper we extend that consistency property to the game logic LG and use it to show how one can extend results about and its countable fragments to LG and certain of its countable fragments. The particular formation of LG which we use will allow in the game quantifier infinite alternation of countable conjunctions and disjunctions as well as infinite alternation of quantifiers. In this way LG can be viewed as an extension of Suslin logic.

In recent years model theorists have been studying various sheaf-theoretic notions as they apply to model theory. For quite a while however, a sheaf of structures was considered to be just a local homeomorphism between topological spaces such that each stalk Sx = p−1(x) is a model-theoretic structure and such that certain maps are continuous. Some of the model-theoretic work done with this notion of a sheaf of structures are the papers by Carson [2] and Macintyre [7]. Soon came the idea of considering a sheaf of structures not just as a collection of structures glued together in some continuous way, but rather as some sort of generalized structure. A significant model-theoretic study of sheaves in this new sense became possible only after the development of the theory of topoi. As F.W. Lawvere pointed out in [6], this represents the advance of mathematics (in our case the advance of model theory) from metaphysics to dialectics.

A topos is the rather ingenious evolution of the notion of a Grothendieck topos [13]. It provides us with the idea that an object of a topos (e.g. the topos of sheaves over a topological space) may be thought of as a generalized set. Furthermore, all first-order logical operations have an interpretation in a topos, hence we may talk about generalized structures. Angus Macintyre suggested that some of his model-theoretic results about sheaves of structures may be understood better and perhaps simplified by doing model theory inside a topos of sheaves.

In [5] Henkin defined a quantifier, which we shall denote by QH: linking four variables in one formula. This quantifier is related to the notion of formulas in which the usual universal and existential quantifiers occur but are not linearly ordered. The original definition of QH was

Here (QHx1x2y1y2)φ is true if for every x1 there exists y1 such that for every x2 there exists y2, whose choice depends only on x2 not on x1 and y1 such that φ(x14, x2, y1, y2). Another way of writing this is

In [5] it was observed that the logic L(QH) obtained by adjoining QH defined as in (1) is more powerful than first-order logic. In particular, it turned out that the quantifier “there exist infinitely many” denoted Q0 was definable from QH because

Beth's Definability Theorem, and consequently the Interpolation Lemma, fail for the version of quantified S5 that is presented in Kripke's [6]. These failures persist when the constant domain axiom-scheme ∀x□φ ≡ □∀xφ is added to S5 or, indeed, to any weaker extension of quantificational K.

§1 reviews some standard material on quantificational modal logic. This is in contrast to quantified intermediate logics for, as Gabbay [6] has shown, the Interpolation Lemma holds for the logic CD with constant domains and for several of its extensions. §§2—4 establish the negative results for the systems based upon S5. §5 establishes a more general negative result and, finally, §6 considers some positive results and open problems. A basic knowledge of classical and modal quantificational logic is presupposed.

Let me briefly review the relevant model theory for quantified modal logic. Further details can be found in [3] or [7].

The language is obtained from the language for classical first-order logic with identity by adding a unary operator □ for necessity. The atomic formula ‘Ex’ is used as an abbreviation for ‘∃y(y = x)’ and may be read as ‘x exists’.

It is shown that the modally first-degree formulas of quantificational S5 constitute a reduction class.

We prove there are theories (stable or countable) for which over every A there is a prime model but it is not necessarily unique. We also give a simplified proof of the uniqueness theorem for countable stable theories.

This paper proposes a generalization of several reducibilities in the sense of recursive function theory. For this purpose two structures are introduced as models of combinatory logic and reducibilities are defined in a rather natural way by means of the application operation in each model. The first model we consider is called the graph model and was introduced by Dana Scott in [11]. Reducibilities in this model are generalizations of enumeration and Turing reducibilities. The second model is called the hypergraph model and induces reducibilities which are generalizations of hyperenumeration and hyperarithmetical reducibilities.

The possibility of formulating reducibilities in this way is not new. For the graph model it is implicit in [7] and for the hypergraph model in [9]. On the other hand it is possible to look at the present method as a variant of the well-known technique of relativization in recursive function theory. We think that this does not exhaust the power of the method, which is conceptually elegant and provides a natural frame for the results of this paper.

In the first part of the paper we discuss the definition and general properties of the models. Then we introduce the reducibilities in the graph model and prove several theorems which are generalizations of properties already Known for enmeration and Turing reducibilities. Next we define reducibilities in the hypergraph model and try to extend the preceding results. For this purpose we prove two theorems showing significant relations between the operators in both models. In fact we prove that each operator in the hypergraph model can be simulated on a comeager set by an operator of the graph model.

Spector [21] proved that a relation R on ω is Π11 if and only if it is (positive elementary) inductive on the structure 〈ω, +, ·〉; Kleene [8] showed that R is Π11 if and only if it is semirecursive in the type 2 object E. These two “constructive” characterizations of the Π11 relations have led to the independent study of (positive elementary) induction and recursion in E on an arbitrary structure, as natural generalizations of the theory of Π11 relations on (ω, +, · 〉.

The theory of (positive elementary) induction on an arbitrary structure was developed by Moschovakis in his book Elementary induction on abstract structures (EIAS); one of the most important theorems there is a generalization of the classical theorem of Spector [20] and Gandy [3] about the Π11 relations on ω: a relation R is inductive on an acceptable structure . if and only if there is a formula φ(Y, x) of the language of such that:

where HYP is the collection of hyperelementary relations on .

Moschovakis [17] and Kechris and Moschovakis [6] showed how to develop the theory of recursion in higher types as a chapter in the general theory of inductive definability. This approach to recursion in higher types makes it possible to use methods from the theory of inductive definability in studying recursion in E.

In this paper we establish some results about recursion in E on a structure which bolster the naturalness of this theory and contribute to its comparison with (positive elementary) induction.

Large cardinal properties divide rather strikingly into two groups. “Small” large cardinal properties such as weak compactness always relativize to L, while in contrast “large” large cardinal properties such as Ramsey are incompatible with L. These properties seem to be similar otherwise and this sense of similarity is reinforced by the fact that many of the large cardinals do exist in the L-like model L(μ). This paper will show that the division is caused by an artificially restrictive class of “constructible” sets rather than an essential difference in the properties themselves. Specifically, we consider the class K of sets constructible from mice as defined by Dodd and Jensen [3] and prove

Theorem 1. If ρ is Ramsey then ρ is Ramsey in K.

A modification of the proof will show

Theorem 2. If ρ is Jonson, then ρ is Ramsey in K.

Since Ramsey implies Jonson this shows that the notions are equiconsistent. Theorem 2 was proved by Kunen [5] under the assumption that V = L(μ).

The proof of Theorems 1 and 2 depends heavily on results of Dodd and Jensen [3] about K and mice. These results are stated without proof in §2. §2 also contains elementary (to a reader familiar with the theory of iterated ultrapowers) proofs of special cases of some of these lemmas sufficient to give a self contained proof of the following corollary of Theorem 1:

Corollary. If ρ ≤ κ,ρ is Ramsey and L(μ) ⊨ μ is a measure on κ then L(μ) ⊨ ρ is Ramsey.

This short article presents an algorithm for bracket abstraction [1] which avoids a combinatorial explosion in the size of the resulting expression when applied repeatedly for abstraction in a series of variables. It differs from a previous solution [2] in introducing only a finite number of additional combinators and in not requiring that all the variables to be abstracted be treated together in a single operation.