 For first order logic it can be proved that “\(\mm\models_s\phi\)” is an absolute property of \(\mm\), s and \(\phi\) relative to ZFC. In fact, this is true even if ZFC is replaced by the weaker Kripke-Platek set theory KP (see Barwise 1972a). The usual Tarski Truth Definition of first order logic can be written in \(\Delta_1^{\text{ZFC}}\) form. For second-order logic the propositions “\(\phi\) is a second-order formula”, “\(\mm\) is an L-structure” and “s is an assignment” are all absolute relative to ZFC as well, but “\(\mm\models_s\phi\)” is not, as we shall now see. This is a crucial property of second-order logic. One may consider it a weakness, if one thinks of absoluteness as a desirable property, or a strength, if one takes non-absoluteness as a sign of expressive power. Whether a weakness or not, it is an important feature of second-order logic and a dominating topic in discussions about it. Second-order logic is, however, not the only non-absolute logic. For example, \(L(Q_1)\) (see the entry on  generalized quantifiers)  and \(L_{\omega_1\omega_1}\) (see then entry on  infinitary languages)  are non-absolute, while \(L(Q_0)\) and \(L_{\omega_1\omega}\) are absolute. The reason why \(L(Q_0)\) is absolute is essentially that the property of a set being infinite is absolute in transitive models of ZFC. Although the set of sentences of \(L_{\omega_1\omega}\) may vary from one transitive model of ZFC to another, the truth of a sentence of \(L_{\omega_1\omega}\) in a model is absolute in transitive models of ZFC essentially because the truth has a simple inductive definition which only refers to subformulas of the sentence and elements of the model.