In this paper we introduce the notion of a set algebra satisfying a system E of equations. After defining a notion of freeness for such algebras, we show that, for any system E of equations, set algebras that are free in the class of structures satisfying E exist and are unique up to a bisimulation. Along the way, analogues of classical set-theoretic and algebraic properties are investigated.

We extend earlier work (both individual and joint with Shelah) and prove three theorems about the class of measurable limits of compact cardinals, where a compact cardinal is one which is either strongly compact or supercompact. In particular, we construct two models in which every measurable limit of compact cardinals below the least supercompact limit of supercompact cardinals possesses non-trivial degrees of supercompactness. In one of these models, every measurable limit of compact cardinals is a limit of supercompact cardinals and also a limit of strongly compact cardinals having no non-trivial degree of supercompactness. We also show that it is consistent for the least supercompact cardinal κ to be a limit of strongly compact cardinals and be so that every measurable limit of compact cardinals below κ has a non-trivial degree of supercompactness. In this model, the only compact cardinals below κ with a non-trivial degree of supercompactness are the measurable limits of compact cardinals.

When analyzing database query languages a roperty, of theories, the pseudo-finite homogeneity property, has been introduced and applied (cf. [3]). We show that a stable theory has the pseudo-finite homogeneity property just in case its expressive power for finite states is bounded. Moreover, we introduce the corresponding pseudo-finite saturation property and show that a theory fails to have the finite cover property if and only if it has the pseudo-finite saturation property.

We construct a degree which fails to be computably enumerable in any computably enumerable set strictly below .

Guarded fragments of first-order logic were recently introduced by Andréka, van Benthem and Németi; they consist of relational first-order formulae whose quantifiers are appropriately relativized by atoms. These fragments are interesting because they extend in a natural way many propositional modal logics, because they have useful model-theoretic properties and especially because they are decidable classes that avoid the usual syntactic restrictions (on the arity of relation symbols, the quantifier pattern or the number of variables) of almost all other known decidable fragments of first-order logic.

Here, we investigate the computational complexity of these fragments. We prove that the satisfiability problems for the guarded fragment (GF) and the loosely guarded fragment (LGF) of first-order logic are complete for deterministic double exponential time. For the subfragments that have only a bounded number of variables or only relation symbols of bounded arity, satisfiability is Exptime-complete. We further establish a tree model property for both the guarded fragment and the loosely guarded fragment, and give a proof of the finite model property of the guarded fragment.

It is also shown that some natural, modest extensions of the guarded fragments are undecidable.

Hrushovski's generalization of the Fraisse construction has provided a rich source of examples in model theory, model theoretic algebra and random graph theory. The construction assigns to a dimension function δ and a class K of finite (finitely generated) models a countable ‘generic’ structure. We investigate here some of the simplest possible cases of this construction. The class K will be a class of finite graphs; the dimension, δ(A), of a finite graph A will be the cardinality of A minus the number of edges of A. Finally and significantly we restrict to classes which are δ-invariant. A class of finite graphs is δ-invariant if membership of a graph in the class is determined (as specified below) by the dimension and cardinality of the graph, and dimension and cardinality of all its subgraphs. Note that a generic graph constructed as in Hrushovski's example of a new strongly minimal set does not arise from a δ-invariant class.

We show there are countably many δ-invariant (strong) amalgamation classes of finite graphs which are closed under subgraph and describe the countable generic models for these classes. This analysis provides ω-stable generic graphs with an array of saturation and model completeness properties which belies the similarity of their construction. In particular, we answer a question of Baizhanov (unpublished) and Baldwin [5] and show that this construction can yield an ω-stable generic which is not saturated. Further, we exhibit some ω-stable generic graphs that are not model complete.

Many known tools for proving expressibility bounds for first-ordér logic are based on one of several locality properties. In this paper we characterize the relationship between those notions of locality. We note that Gaifman's locality theorem gives rise to two notions: one deals with sentences and one with open formulae. We prove that the former implies Hanf's notion of locality, which in turn implies Gaifman's locality for open formulae. Each of these implies the bounded degree property, which is one of the easiest tools for proving expressibility bounds. These results apply beyond the first-order case. We use them to derive expressibility bounds for first-order logic with unary quantifiers and counting. We also characterize the notions of locality on structures of small degree.

In this paper, we show that there is no primitive recursive decision procedure for the implication-conjunction fragments of the relevant logics R, E and T, as well as for a family of related logics. The lower bound on the complexity is proved by combining the techniques of an earlier paper on the same subject [20] with a method used by Lincoln, Mitchell, Scedrov and Shankar in proving that propositional linear logic is undecidable.

The decision problem for the pure implicational fragments of E and R were solved by Saul Kripke in a tour de force of combinatorial reasoning, published only as an abstract [9]. Belnap and Wallace extended Kripke's decision procedure to the implication-negation fragment of E in [3]; an account of their decision method is to be found in [1, pp. 124–139]. The decision method extends immediately to the implication/negation fragment of R. In fact, in the case of R we can go farther: Meyer in his thesis [13] showed how to translate the logic LR, which results from R by omitting the distribution axiom, into R→⋀, so that the decision procedure can be extended to all of LR. This decision procedure has been implemented as a program Kripke by Thistlewaite, McRobbie and Meyer [17]. The program is not simply a straightforward implementation of the decision procedure; finite matrices are used extensively to prune invalid nodes from the search tree.

We show that it is consistent with ZFC + ¬CH that there is a maximal cofinitary group (or, maximal almost disjoint group) G ≤ Sym(ω) such that G is a proper subset of an almost disjoint family A ⊆ Sym(ω) and ‖G‖ < ‖A‖. We also ask several questions in this area.

We investigate the system TRC of untyped illative combinatory logic that is equiconsistent with New Foundations. We prove that various unstratified combinators do not exist in TRC.