An example of using $\Delta_1$ terms in Boolean valued analysis

Syntactic tools related to $\Delta_1$ terms are demonstrated by application to Boolean valued analysis. As an example, the question is considered of what approaches to defining the field $\mathbb R$ of reals and what complete Boolean algebras $B$ provide the explicit inclusion $\mathbb R^\land\subset\mathbb R$ inside the Boolean valued universe $\mathbb V^{(B)}$.