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  "body": "\n## Summary\n\nA Boolean-valued model is like a first-order structure, but each statement gets a truth value in a complete [[Boolean algebra|0XADJKM699]].\nIn an ordinary structure the only values are \"true\" and \"false\".\nWhen the Boolean algebra has two elements, it is essentially an ordinary structure.\nBoolean-valued models of set theory, written $V^B$, give a way to present [[forcing|F2DTWW9HP7]] that needs no countable transitive model.\nThey were developed by Scott, Solovay and Vopěnka.\n\n## Explanations\n\n### L1 human\n\nIn an ordinary structure every statement is either true or false.\nIn a Boolean-valued model a statement can be \"true to some degree\", and the degrees form a Boolean algebra.\nFor 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.\nOne can think of the model as a family of ordinary structures indexed by $S$, glued together.\n",
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