Boolean-Valued Set-Theoretic Systems: General Formalism and Basic Technique

This article is devoted to the study of Boolean-valued universe as an algebraic system. We start with logical backgrounds notion and present formalism extending syntax Boolean truth values by use definable symbols, internal classes, outer terms external classes. Next, we enrich collection research tools technique partial elements corresponding joins, mixings ascents. Passing on set-theoretic signature, prove that bounded formulas are absolute for transitive subsystems. also introduce intensional, predicative, cyclic regular systems, examine maximum principle, analyze its relationship ascent mixing principles. The main applications relate over arbitrary extensional A close interrelation established between such a intensional hierarchy. existence uniqueness up unique isomorphism show conditions in axiomatic characterization logically independent. describe structure means several cumulative hierarchies. Another application, based quantifier hierarchy formulas, improves transfer principle canonical embedding universe.