 A special property of second-order characterizable structures is that their reducts are also second-order characterizable, because we can use existential second-order quantifiers to “guess” the missing relations and functions. Therefore it is interesting to find characterizable structures that have as many (but only finitely many) relations and functions as possible. We can endow \(\oN\) with any finite number of recursive functions \(f_1,\ldots, f_n \) and relations \(R_1,\ldots, R_m\) obtaining the structure \((\oN,f_1,\ldots, f_n,R_1,\ldots, R_m)\) and this is second-order characterizable. We can endow \(\oR\) with any familiar analytic functions such as trigonometric functions or any other functions given by a convergent power series the coefficients of which are given by a recursive function, and the result is second-order characterizable.