Contemporary methods of Boolean-valued analysis, due to their nature, involve rather bulky logical technique. We can say that, from a pragmatic viewpoint, this technique might distract the user-analyst from a concrete aim: to apply the results of Boolean-valued analysis for solving analytical problems.

Various function spaces are common in functional analysis, and so the intention is natural of replacing an abstract Boolean-valued system by some function analog, a model whose elements are functions and in which the basic logical operations are calculated “pointwise.”

In the present chapter, a solution is proposed to the above problem. To this end, we introduce and study the new notion of continuous polyverse, the latter being a continuous bundle of models of set theory. It is shown that the class of continuous sections of a continuous polyverse is a Boolean-valued system satisfying all basic principles of Boolean-valued analysis and, conversely, every Boolean-valued algebraic system can be represented as the class of sections of a suitable continuous polyverse.