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.
Read more about this topic: Forcing (mathematics)
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