 Then \((N,A,g,a)\models\theta_{\textrm{PA}}(U,G,z)\) if and only if \((N,A,g,a)\) is isomorphic to a structure \((M,\oN,s,0)\) where \(\oN\subseteq M\) and \(s(n)=n+1\). Thus we have used second-order logic to say that the U-part of any model of \(\theta_{\textrm{PA}}(U,G,z)\), together with the function G and the individual z are isomorphic to the natural numbers \(\oN\) with their successor function and zero. There are many ways to see that no first order sentence can have this property. For example, by the Compactness Theorem such a sentence would have also a non-standard model where some element is different from each in the sequence z, \(G(z)\), \(G(G(z))\), \(G(G(G(z)))\), \(\ldots\). Such a model cannot be isomorphic to the natural numbers \(\oN\) with their successor function and zero. Hereupon we may conclude that the Compactness Theorem does not hold for second-order logic in the form that it holds for first order logic.