 A more serious case of non-absoluteness is the sentence \(\theta_{\textrm{CH}}\) of  §5.3.  The sentence \(\theta_{\textrm{CH}}\) of the empty vocabulary has a model if and only if the Continuum Hypothesis is true. If \(T\subseteq \ZFC\) is finite, then there are countable transitive models \(M\subseteq M'\) such that one, say M, satisfies CH and the other, in this case \(M'\), does not (by Cohen 1966). In M the sentence \(\theta_{\textrm{CH}}\) has a model \(\ma\), that is, \(M\models \textrm{“}\ma\models\theta_{\textrm{CH}}\textrm{”}\). In \(M'\) the same sentence has no models, in particular \(M'\not\models \textrm{“}\ma\models\theta_{\textrm{CH}}\textrm{”}\). Thus the property of a model satisfying a second-order sentence is not absolute relative to ZFC. Even the property of a second-order sentence having a model is not absolute relative to ZFC, because in M the sentence \(\theta_{\textrm{CH}}\) has a model but in \(M'\) not. For first order logic the property of a sentence of having a model is co-re and therefore arithmetical. Arithmetical properties are always absolute relative to ZFC. The situation is similar if the Continuum Hypothesis is replaced by the Axiom of Choice (see  §5.4).