 The mathematical continuum as conceived by Brouwer displays a number of features that seem bizarre to the classical eye. For example, in the Brouwerian continuum the usual law of comparability, namely that for any real numbers \(a, b\) either \(a \lt b\) or \(a = b\) or \(a \gt b\), fails. Even more fundamental is the failure of the law of excluded middle in the form that for any real numbers \(a, b\), either \(a = b\) or \(a \ne b\). The failure of these seemingly unquestionable principles in turn vitiates the proofs of a number of basic results of classical analysis, for example the Bolzano-Weierstrass theorem, as well as the theorems of monotone convergence, intermediate value, least upper bound, and maximum value for continuous  functions.[40]