Boolean-valued Models: why should I care?

Question

I was exposed to Boolean-valued models during an undergraduate course on Boolean algebras and set theory, and I was told that they were very useful e.g. in forcing and in building models of set theory. While I obviously recognize the importance of these applications, I couldn't seem to find any use of BVMs outside of them, which seemed a little strange given how flexible they are. Are there other applications of the theory of Boolean-valued models, for example, in classical model theory?

Answer 1

Bell mentioned Boolean-valued analysis in his book Boolean-Valued Models and Independence Proofs in Set Theory. It introduced real analysis over the Boolean-valued model and its application to other fields like real analysis or quantum physics.

For example, consider the Boolean algebra $\mathbb{B}$ given by the set of all measurable subsets of $\mathbb{R}$ up to measure 0 (that is, two measurable subsets are identified when their symmetric difference is of measure 0.)

We know that $V^\mathbb{B}$ is a model of ZFC, and surprisingly, real numbers of $V^\mathbb{B}$ corresponds with measurable functions over $\mathbb{R}$. Therefore, we can translate facts on real numbers (which is a theorem of ZFC, so valid on $V^\mathbb{B}$) into facts on measurable functions. For example, the least upper bound principle can be translated into the following theorem (see Theorem 7.2. of Bell):

Let $U$ be a non-empty set of measurable functions, such that there is a measurable function $f$ such that $g\le f$ almost everywhere, for any $g\in U$.

Then there is a measurable function $h$ such that

1. $g\le h$ almost everywhere for all $g\in U$, and

2. If $k$ is a measurable function such that $g\le k$ almost everywhere for all $g\in U$, then $h\le k$ almost everywhere.

(That is, $h$ is a 'least upper bound' of $U$.)

On the other hand, Boolean-valued models are essentially equivalent to forcing. Hence I think finding any other application of the Boolean-valued model outside set theory might be difficult.

(It is fair to mention that there are lots of applications of forcing to other fields of logic like model theory, topology, analysis, or algebra. If forcing coincides with the Boolean-valued models, then we may say there are lots of applications of Boolean-valued models. However, I believe these applications lie in the natural extension of usage of forcing in set theory, like providing independence results. Moreover, I doubt most of these results based on Boolean-valued models, not poset-styled forcing.)

I have heard that forcing is a special case of sheaf models. Hence we may find any other application of Boolean-valued models by searching applications of sheaf models or topoi into other fields, which is out of my scope.