In the case of the consistency proof of the negation of the CH, one starts from a model \(M\) and adds \(\aleph_2\) new subsets of \(\omega\), so that in the generic extension the CH fails. In this case one needs to use an appropriate partial ordering \(\mathbb{P}\) so that the \(\aleph_2\) of \(M\) is not collapsed, i.e., it is the same as the \(\aleph_2\) of the generic extension, and thus the generic extension \(M[G]\) will satisfy the sentence that says that there are \(\aleph_2\) real numbers.