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Full Idea
Compactness is intuitively surprising, ..because one might have thought there could be some contradiction latent within some infinite set, preventing it from being satisfiable, only discovered when you consider the whole set. But this can't happen.
Gist of Idea
Compactness surprisingly says that no contradictions can emerge when the set goes infinite
Source
Theodore Sider (Logic for Philosophy [2010], 4.5)
Book Ref
Sider,Theodore: 'Logic for Philosophy' [OUP 2010], p.105