User blog comment:Vel!/1+w/@comment-5982810-20141011203204/@comment-2033667-20141011203700

Because otherwise we get multiple, distinct values of the right-hand side of \((\omega^{\alpha_1} + \omega^{\alpha_2} + \cdots + \omega^{\alpha_{k - 1}} + \omega^{\alpha_k})[n] = \omega^{\alpha_1} + \omega^{\alpha_2} + \cdots + \omega^{\alpha_{k - 1}} + \omega^{\alpha_k}[n]\), which leads to inconsistencies. For example, if we remove this restriction we get \((1 + \omega)[n] = 1 + \omega[n]\) and \((1 + \omega)[n] = \omega[n]\), so \(1 + \omega[n] = \omega[n]\) and \(1 = 0\). Cantor normal form is designed to be unique representation.