  A prominent area of research in classical set theory is that of large cardinals (see the entry on  set theory). In constructive contexts, the ordinals are not linearly ordered. (For the notion of constructive ordinal and a brief discussion of its properties, see the supplementary document on:  Set-theoretic Principles Incompatible with Intuitionistic Logic.)   As a consequence, cardinal numbers do not play the same role as in the classical  setting.