 This example showing that the principle of the excluded middle not only does not hold but is in fact false in intuitionism, leads to the refutation of many basic properties of the continuum. Consider for example the real number \(r_\alpha\) that is the limit of the sequence consisting of the numbers \(r_n\) as given in the section on weak counterexamples, where the \(A(m)\) in the definition is taken to be the statement \(\alpha(m)=0\). Then the refutation above implies that \(\neg\forall\alpha(r_\alpha=0 \vee r_\alpha\neq 0)\), and it thereby refutes the law of trichotomy: