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

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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

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  • Kunen, Kenneth (1980). Set theory: an introduction to independence proofs. Studies in logic and the foundations of mathematics. Vol. 102. Elsevier. p. 208. ISBN 0-444-85401-0.