Forcing (mathematics)

Forcing is a technique in set theory, invented by Paul Cohen. It was first used to prove independence of Continuum hypothesis from the axioms of Set theory, and is nowadays one of the basic techniques in the field.

In it, a forcing relation between "conditions" and statements of set theory is considered. Each "condition" is finite piece of information - the idea is that only finite pieces are relevant for consistency, since by Compactness theorem a theory is consistent if and only if any finite subset of its axioms is consistent. Then, we can pick an infinite set of consistent conditions to extend our model. Thus, assuming consistency of set theory, we prove consistency of the theory extended with this infinite set.

Perhaps more clearly, the method can be explained in terms of Boolean valued models. In it, any statement is assigned a truth value from some infinite Boolean algebra, rather than just a true/false value. Then an ultrafilter is picked in this Boolean algebra, which assigns values true/false to statements of our theory. The point is that the resulting theory has a model which contains this ultrafilter, which can be understood as a model obtained by extending the old one with this ultrafilter. By picking Boolean valued model in appropriate way, we can get a model which has desired property. In it, only statements which must be true (are "forced" to be true) will be true, in a sense (since it has this extension/minimality property).