Paradoxical set

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 $M\models$ ZFC be transitive, $(\mathbb {P} ,<)$ a forcing notion in $M$, and suppose $G\subseteq \mathbb {P}$ is generic over $M$. Then for any $\mathbb {P}$-name in $M$, $\tau$,

$\eta$ is a nice name for a subset of $\tau$ if $\eta$ is a $\mathbb {P}$-name satisfying the following properties:

(1) ${\textrm {dom}}(\eta )\subseteq {\textrm {dom}}(\tau )$

(2) For all $\mathbb {P}$-names $\sigma \in M$, $\{p\in \mathbb {P} |\langle \sigma ,p\rangle \in \eta \}$ forms an antichain.

(3) (Natural addition): If $\langle \sigma ,p\rangle \in \eta$, then there exists $q\geq p$ in $\mathbb {P}$ such that $\langle \sigma ,q\rangle \in \tau$.

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