 Second-order logic has some remarkable qualities which we will now uncover in detail. Above we characterized the structure of the natural numbers by means of (\ref{ind}). This application of second-order logic is perhaps the most famous and most important. We now reformulate this in a slightly more general form in order to get more power out of it. Suppose U is a unary relation variable (“the set of natural numbers”), G a unary function variable (“the successor function”), and z an individual variable (“the zero”). Let \(\theta_{\textrm{PA}}(U,G,z)\) be the sentence 