Some theory of Boolean valued models

Boolean valued structures are defined and some of their properties are studied. Completeness and compactness theorems are proved and Lowenheim-Skolem theorems are looked at. It is seen that for any consistent theory T and cardinal number KT there is a model N of T a "universal" model) such that for any model M of T with M <, K, M can be written as a quotient of N. A theory T is shown to be open if and only if given structures M c N, if N is a model of T, then M is a model of T, T is shown to be existential if and only if the union of every chain of models of T is a model of T. The prefix problem and obstructions to elementary extensions are examined. Various forms of completeness are compared and, finally, an example is given where Boolean valued models are used to prove a theorem of Mathematics (Hilbert's 17-th Problem) without using the Axiom of Choice. Throughout, it is seen that good Boolean valued structures (for all Φ [(Ev[sub j])Φ][sub M] = I(Φ,a][sub M] for some a Є U[sub M]) behave very much like relational structures and much of the theory depends upon their existence.Science, Faculty ofMathematics, Department ofGraduat