 Behind the failure of the Completeness Theorem is the fact that second-order logic is—almost by definition—able to say that a set is the power set of another set. What this means is the following: Let L be a vocabulary. Let E be a binary and U a unary relation symbol, both not in L. Let \(\theta_{\textrm{Pow}}(U,E)\) be the second-order formula (\ref{ex6}). Let us consider the conjunction 