 The sequence \(L_0, L_1, L_2,\ldots\) has an important property: For each \(\alpha\), the interpretation of \(T\) in \(L_{\alpha +1}\) extends the interpretation of \(T\) in \(L_{\alpha}\), that is, both the extension and anti-extension of \(T\) increase (or stay the same) when moving from \(L_{\alpha}\) to \(L_{\alpha +1}\). This means that one can define a new partially interpreted language \(L_{\omega}\) by letting the extension of \(T\) be the union of all the extensions of \(T\) in \(L_0, L_1, L_2,\ldots\); and similarly for the anti-extension. Thus in \(L_{\omega}\), the interpretation of \(T\) extends the interpretation that \(T\) receives in all previous languages. This furnishes a strategy for continuing the iterative construction of a truth predicate into the transfinite: For each successor ordinal \(\alpha +1\), define \(L_{\alpha +1}\) from \(L_{\alpha}\) exactly as in the finite case above; and for each limit ordinal \(\sigma\), define \(L_{\sigma}\) from the preceding languages \((L_i)_{i\lt \sigma}\) in the same way as \(L_{\omega}\) was defined (for a detailed explanation of the ordinal numbers and their use in this context, see the entry on  the revision theory of truth).  A simple cardinality consideration now shows that this transfinite sequence of languages will eventually stabilise: There is an ordinal \(\gamma\) such that \(L_{\gamma} = L_{\gamma +1}\). Hence the following instance of (4) is obtained: