In this paper we will define a Lagrangian for scalar and gauge fields on causal sets, based the selection of an Alexandrov set in which variations appropriate expressions terms either field or holonomies around suitable loops take least value. For these fields, find that values Lagrangians covariant form.

In this paper the machinery and results developed in [Awodey et al, 2004] are extended to the study of constructive set theories. Specifically, we introduce two constructive set theories BCST and CST and prove that they are sound and complete with respect to models in categories with certain structure. Specifically, basic categories of classes and categories of classes are axiomatized and shown...

Elementary set theory can be studied informally and intuitively, and so can be taught in primary schools using, say, Venn diagrams. The intuitive approach silently assumes that all objects in the universe of discourse satisfying any defining condition form a set. This assumption gives rise to antinomies, the simplest and best known of which being Russell's paradox. Axiomatic set theory was orig...

Mathematicians manipulate sets with confidence almost every day of their working lives. We do so whenever we work with sets of real or complex numbers, or with vector spaces, topological spaces, groups, or any of the many other set-based structures. These underlying set-theoretic manipulations are so automatic that we seldom give them a thought, and it is rare that we make mistakes in what we d...

Since its inception in 1965, the theory of fuzzy sets has advanced in a variety of ways and in many disciplines. Applications of this theory can be found, for example, in artificial intelligence, computer science, medicine, control engineering, decision theory, expert systems, logic, management science, operations research, pattern recognition, and robotics. Mathematical developments have advan...