 Gilbert argues that, given \(\mathcal{S}'_{N'}(A)\), the smooth-reasoner counterparts of the agents of \(N\) actually satisfy a much stronger condition, namely mutual knowledge \(\mathbf{K}^{\alpha}_{N'}(A)\) to the level of any ordinal number \(\alpha\), finite or infinite. When this stronger condition is satisfied, the proposition \(A\) is said to be open* to the agents of \(N\). With the concept of open*-ness, Gilbert gives her definition of common knowledge.