 Given the definition of a partial model, a partially interpreted language is a pair \((L,M)\) where \(L\) is a first-order language and \(M\) is a partial model of \(L\). Kripke recursively defines a sequence of partially interpreted languages \(L_0, L_1, L_2,\ldots\), only differing in their interpretation of the truth predicate \(T\). The first language, \(L_0\), is taken to be an arbitrary language in which both the extension and anti-extension of \(T\) are the empty set. Thus in \(L_0\), the truth predicate is completely undefined. For any \(\alpha\), the language \(L_{\alpha +1}\) is like \(L_{\alpha}\) except that \(T\) is interpreted by the extension/anti-extension pair \((U,V)\) given by: