Paradoxical set

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In set theory, a nice name is a concept used in forcing to impose an upper bound on the number of subsets in the generic model. It is a technical concept 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 , ,

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

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

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