 In constructive mathematics a problem is counted as solved only if an explicit solution can, in principle at least, be produced. Thus, for example, “There is an \(x\) such that \(P(x)\)” means that, in principle at least, we can explicitly produce an \(x\) such that \(P(x)\). This fact led to the questioning of certain principles of classical logic, in particular, the law of excluded middle, and the creation of a new logic, intuitionistic logic (see entry on  intuitionistic logic).  It also led to the introduction of a sharpened definition of real numbers—the constructive real numbers. A constructive real number is a sequence of rationals \((r_n) = r_1, r_2,\ldots\) such that, for any \(k\), a number \(n\) can be computed in such a way that \(\lvert r_{n+p} - r_{n}\rvert \le 1/k\). Each rational number a may be regarded as a real number by identifying it with the real number \((\alpha , \alpha ,\ldots)\). The set \(R\) of all constructive real numbers is the constructive real line.