Nice name
In set theory, a nice name is used in forcing to impose an upper bound on the number of subsets in the generic model. It is used in the context of forcing to prove independence results in set theory such as Easton's theorem. Formal definition
Let  ZFC be transitive,  a forcing notion in , and suppose  is generic over . Then for any -name  in , we say that  is a nice name for a subset of  if  is a -name satisfying the following properties:

(1) 

(2) For all -names ,  forms an antichain. (3) (Natural addition): If , then there exists  in  such that . References

Forcing (mathematics)