Editing Forcing (mathematics) (section)

=== The role of the model ===
In order for the above approach to work smoothly, <math>M</math> must in fact be a [[Standard model (set theory)|standard transitive model]] in <math>V</math>, so that membership and other elementary notions can be handled intuitively in both <math>M</math> and <math>V</math>. A standard transitive model can be obtained from any standard model through the [[Mostowski collapse lemma]], but the existence of any standard model of <math>\mathsf{ZFC}</math> (or any variant thereof) is in itself a stronger assumption than the consistency of <math>\mathsf{ZFC}</math>.

To get around this issue, a standard technique is to let <math>M</math> be a standard transitive model of an arbitrary finite subset of <math>\mathsf{ZFC}</math> (any axiomatization of <math>\mathsf{ZFC}</math> has at least one [[axiom schema]], and thus an infinite number of axioms), the existence of which is guaranteed by the [[Reflection principle#In ZFC|reflection principle]]. As the goal of a forcing argument is to prove [[consistency]] results, this is enough since any inconsistency in a theory must manifest with a derivation of a finite length, and thus involve only a finite number of axioms.