Since the axiom \(V=L\) implies the AC and the CH, any model of the negation of the AC or the CH must violate \(V=L\). So, let’s illustrate the idea of forcing in the case of building a model for the negation of \(V=L\). We start with a transitive model \(M\) of ZFC, which we may assume, without loss of generality, to be a model of \(V=L\). To violate \(V=L\) we need to expand \(M\) by adding a new set \(r\) so that, in the expanded model, \(r\) will be non-constructible. Since all hereditarily-finite sets are constructible, we aim to add an infinite set of natural numbers. The first problem we face is that \(M\) may contain already all subsets of \(\omega\). Fortunately, by the Löwenheim-Skolem theorem for first-order logic, \(M\) has a countable elementary submodel \(N\). So, since we are only interested in the statements that hold in \(M\), and not in \(M\) itself, we may as well work with \(N\) instead of \(M\), and so we may assume that \(M\) itself is countable. Then, since \(\mathcal{P}(\omega)\) is uncountable, there are plenty of subsets of \(\omega\) that do not belong to \(M\). But, unfortunately, we cannot just pick any infinite subset \(r\) of \(\omega\) that does not belong to \(M\) and add it to \(M\). The reason is that \(r\) may encode a lot of information, so that when added to \(M\), \(M\) is no longer a model of ZFC, or it is still a model of \(V=L\). To avoid this, one needs to pick \(r\) with great care. The idea is to pick \(r\) generic over \(M\), meaning that \(r\) is built from its finite approximations in such a way that it does not have any property that is definable in \(M\) and can be avoided. For example, by viewing \(r\) as an infinite sequence of natural numbers in the increasing order, the property of \(r\) containing only finitely-many even numbers can be avoided, because given any finite approximation to \(r\)—i.e., any finite increasing sequence of natural numbers—one can always extend it by adding more even numbers, so that at the end of the construction \(r\) will contain infinitely-many even numbers; while the property of containing the number 7 cannot be avoided, because when a finite approximation to \(r\) contains the number 7, then it stays there no matter how the construction of \(r\) proceeds. Since \(M\) is countable, there are such generic \(r\). Then the expanded model \(M[r]\), which includes \(M\) and contains the new set \(r\), is called a generic extension of \(M\). Since we assumed \(M\) is a transitive model of \(V=L\), the model \(M[r]\) is just \(L_\alpha (r)\), where \(\alpha\) is the supremum of the ordinals of \(M\). Then one can show, using the forcing relation between finite approximations to \(r\) and formulas in the language of set theory expanded with so-called names for sets in the generic extension, that \(M[r]\) is a model of ZFC and \(r\) is not constructible in \(M[r]\), hence the axiom of constructibility \(V=L\) fails.