---
tag: FVMNKJ3DT8
kind: concept
name: Boolean-valued model
msc: ["03E40", "03C90"]
sources:
  - url: https://en.wikipedia.org/wiki/Boolean-valued_model
same_as:
  wikidata: Q4943349
  wikipedia: Boolean-valued model
---

## Summary

A Boolean-valued model is like a first-order structure, but each statement gets a truth value in a complete [[Boolean algebra|0XADJKM699]].
In an ordinary structure the only values are "true" and "false".
When the Boolean algebra has two elements, it is essentially an ordinary structure.
Boolean-valued models of set theory, written $V^B$, give a way to present [[forcing|F2DTWW9HP7]] that needs no countable transitive model.
They were developed by Scott, Solovay and Vopěnka.

## Explanations

### L1 human

In an ordinary structure every statement is either true or false.
In a Boolean-valued model a statement can be "true to some degree", and the degrees form a Boolean algebra.
For example, if the algebra is the algebra of subsets of a set $S$, the truth value of a statement is the set of points of $S$ where it holds.
One can think of the model as a family of ordinary structures indexed by $S$, glued together.