 The Completeness Theorem is one of the cornerstones of our understanding of first order logic. Also many extensions of first order logic have a Completeness Theorem, most notably the extension of first order logic by the  generalized quantifier  “There are uncountably many” and the  infinitary language  \(L_{\omega_1\omega}\). We shall now see that there is no hope of a Completeness Theorem for second-order logic. Later in  §9.1  we shall modify the semantics so that a Completeness Theorem can be proved.