 where \(\theta_n\) is an FO-sentence saying that there are at least n elements in the universe. If you take any finite subset \(\Phi'\) of \(\Phi\), and M is a universe whose cardinality is the largest n such that \(\theta_n\) belongs to \(\Phi'\), then all sentences in \(\Phi'\) are true in M. But no universe can make all sentences in \(\Phi\) true. And this shows that \(Q_0\) is not definable in FO, i.e., that \(\FO(Q_0) \not\leq \FO\), since otherwise we could replace \(\Phi\) by an equivalent set of FO-sentences, but FO does have the compactness property, so that it impossible.