In general, a forcing extension of a model \(M\) is obtained by adding to \(M\) a generic subset \(G\) of some partially ordered set \(\mathbb{P}\) that belongs to \(M\). In the above example, \(\mathbb{P}\) would be the set of all finite increasing sequences of natural numbers, seen as finite approximations to the infinite sequence \(r\), ordered by \(\subseteq\); and \(G\) would be the set of all finite initial segments of \(r\).