Home > Forcing (mathematics)
Set theoryIn mathematical logic, forcing is a technique in set theory, invented by Paul Cohen. It was first used to prove independence of the continuum hypothesis from the axioms of set theory, and is nowadays one of the basic techniques in the field. There are two different but provably equivalent interpretations of forcing.
1 Partial order interpretation
In it, a forcing relation between "conditions" and statements of set theory is considered. The forcing relation is usually denoted and is read, "condition p forces sentence psi," or just, "p forces psi." Conditions are elements in a partially ordered set and this partial order is refered to as a notion of forcing . The forcing relation must satisfy several properties which guarantee that every condition forces every axiom of ZFC. Furthermore the collection of sentences forced by a generic set of conditions is consistent. Thus one can prove a statement is consistent with ZFC by selecting a notion of forcing where every generic set of conditions force the statement in question.
2 Boolean-valued model interpretation
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 a Boolean-valued model in appropriate way, we can get a model that has the 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).
3 Meta-mathematical explanation
In forcing we usually seek to show some sentence is consistent with ZFC (or optionally some extension of ZFC). One way to interpret the argument is that we assume ZFC is consistent and use it to prove ZFC combined with our new sentence is also consistent.
Each "condition" is a finite piece of information - the idea is that only finite pieces are relevant for consistency, since by the compactness theorem a theory is satisfiable if and only if every finite subset of its axioms is satisfiable. 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.
4 External links
- Tim Chow's newsgroup article Forcing for dummies is a good introduction to the concepts of forcing; it covers the main ideas, omitting technical details
