Forcing

Summary

Forcing is a method to build new models of set theory by adding a new set to a given model. One starts from a model $M$ of ZFC and a partial order in $M$, whose elements are approximations of the new set. A generic filter $G$ over $M$ collects compatible approximations, and the extension $M[G]$ is the smallest transitive model of ZFC that contains $M$ and $G$. Paul Cohen introduced forcing to prove that the continuum hypothesis cannot be proved in ZFC, and it is now the main tool for independence proofs.