Nice name

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, τ,

η is a nice name for a subset of τ if η 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 $\langle\sigma, q\rangle \in \tau$.

Reference

• Kenneth Kunen (1980) Set theory: an introduction to independence proofs, Volume 102 of Studies in logic and the foundations of mathematics (Elsevier) ISBN 0444854010, p.208