One important example is Suslin’s Hypothesis (SH). Cantor had shown that every linearly ordered set \(S\) without endpoints that is dense (i.e., between any two different elements of \(S\) there is another one), complete (i.e., every subset of \(S\) that is bounded above has a supremum), and with a countable dense subset is isomorphic to the real line. Suslin conjectured that this is still true if one relaxes the requirement of containing a countable dense subset to being ccc, i.e., every collection of pairwise-disjoint intervals is countable. In the early 1970s, Thomas Jech produced a consistent counterexample using forcing, and Ronald Jensen showed that a counterexample exists in \(L\). About the same time, Robert Solovay and Stanley Tennenbaum (1971) developed and used for the first time the iterated forcing technique to produce a model where the SH holds, thus showing its independence from ZFC. In order to make sure that the SH holds in the generic extension, one needs to destroy all counterexamples, but by destroying one particular counterexample one may inadvertently create new ones, and so one needs to force again and again; in fact one needs to go on for at least \(\omega_2\)-many steps. This is why a forcing iteration is needed.