sets is the union of the entire tower of Grothendieck universes. (This is necessarily a proper class, but each Grothendieck universe is a set because it is an element of some larger Grothendieck universe.) However, one does not work directly with the "category of all sets". Instead, theorems are expressed in terms of the category SetU whose objects are the elements of a sufficiently large Grothendieck universe U, and are then shown not to depend on the particular choice of U. As a foundation for category theory, this approach is well matched to a system like Tarski–Grothendieck set theory in which one cannot reason directly about proper classes; its principal disadvantage is that a theorem can be true of all SetU but not of Set. Various other solutions, and variations on the above, have been proposed. The same issues arise with other concrete categories, such as the category of groups or the category of topological spaces. See also Category of topological spaces Set theory Small set (category theory) Notes References Lawvere, F.W. An elementary theory of the category of sets (long version) with commentary External links A231344 Number of morphisms in full subcategories of Set spanned by {{}, {1}, {1, 2}, ..., {1, 2, ..., n}} at OEIS. Foundations of mathematics Sets Basic concepts in set theory