Empty set

Abstract: A term is an individual constant or variable or an n-adic function letter followed by n terms. An atomic formula is an n-adic predicate letter followed by n terms. A literal is an atomic formula or the negation thereof. A clause is a set of literals and is thought of as representing the universally-quantified disjunction of its members. It will sometimes be notationally convenient1 to distinguish between the empty clause □, viewed as a clause, and ‘other’ empty sets such as the empty set of clauses, even though all these empty sets are the same set-theoretic object o. A ground clause (term, literal) is one with no variables. A clause C’ (literal, term) is an instance of another clause C (literal, term) if there is a uniform replacement of the variables in C by terms that transform C into C’.

Abstract: Dijsktra's calculus of guarded commands can be generalized and simplified by dropping the law of the excluded miracle. This paper gives a self-contained account of the generalized calculus from first principles through the semantics of recursion. The treatment of recursion uses the fixpoint method from denotational semantics. The paper relies only on the algebraic properties of predicates; individual states are not mentioned (except for motivation). To achieve this, we apply the correspondence between programs and predicates that underlies predicative programming. The paper is written from the axiomatic semantic point of view, but its contents can be described from the denotational semantic point of view roughly as follows: The Plotkin-Apt correspondence between wp semantics and the Smyth powerdomain is extended to a correspondence between the full wp/wlp semantics and the Plotkin powerdomain extended with the empty set.

Abstract: We develop a theory of stability in many-to-many matching markets. We give conditions under which the setwise-stable set, a core-like concept, is nonempty and can be approached through an algorithm. The usual core may be empty. The setwise-stable set coincides with the pairwise-stable set, and with the predictions of a non-cooperative bargaining model. The setwise-stable set possesses the canonical conflict/coincidence of interest properties from many-to-one, and one-to-one models. The theory parallels the standard theory of stability for many-to-one, and one-to-one, models. We provide results for a number of core-like solutions, besides the setwise-stable set.

Abstract: This text presents methods of modern set theory as tools that can be usefully applied to other areas of mathematics. The author describes numerous applications in abstract geometry and real analysis and, in some cases, in topology and algebra. The book begins with a tour of the basics of set theory, culminating in a proof of Zorn's Lemma and a discussion of some of its applications. The author then develops the notions of transfinite induction and descriptive set theory, with applications to the theory of real functions. The final part of the book presents the tools of 'modern' set theory: Martin's Axiom, the Diamond Principle, and elements of forcing. Written primarily as a text for beginning graduate or advanced level undergraduate students, this book should also interest researchers wanting to learn more about set theoretical techniques applicable to their fields.