Most working mathematicians agree that the collections whose members are numbers (of some fixed sort) are sets: and further the sets whose members are such sets, and the sets of those etc. This kind of argument thinks sets as cumulative type structures in which we begin by recognising some simple type of collections as sets and then construct from these in stages more and more complex type of sets. The following collections are sets.

Level 0: Some individuals (about which we presuppose nothing).

Level 1: All collections whose members are individuals.

Level 2: All collections whose members are in level 0 or 1.

At each level we take all collections whose members are in earlier levels. Thus, if the individuals are natural numbers, the collection {1, 2} occurs at level 1, the collection {1, {2}} occurs are level 2. {1, {2}, {1, {2}}} at level 3, and so on.

Note that every set occurs at every level after the one at which it is introduced, and it is taken the same at every occurrence. Two obvious questions arise with this definition. What shall we take as individuals? And, how far do the levels go? A cumulative type theory of sets has been developed by taking no individuals at all at level 0. (This is, in fact equivalent to taking the empty set as the only set at level 1).