 Proposition 2.18  Let \(C^*_N\) be the greatest fixed point of \(f_E.\) Then \(C^{*}_N (E) = K^*_N (E).\) (In Barwise (1988, 1989), \(E\) is defined to be common knowledge at \(\omega\) iff \(\omega \in C^*_N(E).)\)  Proof.