Editing Forcing (mathematics) (section)

== Meta-mathematical explanation ==

In forcing, we usually seek to show that some [[sentence (mathematical logic)|sentence]] is [[Consistency proof|consistent]] with <math> \mathsf{ZFC} </math> (or optionally some extension of <math> \mathsf{ZFC} </math>). One way to interpret the argument is to assume that <math> \mathsf{ZFC} </math> is consistent and then prove that <math> \mathsf{ZFC} </math> combined with the new [[sentence (mathematical logic)|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. Therefore, assuming the consistency of <math> \mathsf{ZFC} </math>, we prove the consistency of <math> \mathsf{ZFC} </math> extended by this infinite set.