Abstract:
The category of individual sets and the category of strong individual sets are proved to be much similar to the category of sets by using method of category theory. For example, for each cardinality α, there exists an individual set (resp., a strong individual set) Xα such that |Xα|=α,where |Xα| is the cardinality of Xα; individual sets and strong individual sets are closed under the operations of subsets and powers; both the category of nonempty individual sets and the category of nonempty strong individual sets are complete monoidal topoi. Superstructure functor V and ultrapower functor HF are constructed and the following conclusions are obtained: (1) For any nonempty strong individual sets X and Y,a map g:X→Y is an injection (resp., a surjection) if and only if V(g) is an injection (resp., a surjection); (2) For any sets X and Y,a map g:X→Y is an injection (resp., a surjection) if and only if HF(g) is an injection (resp., a surjection).
