Nice name

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In mathematics, a nice name is a set theoretical 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.

[edit] 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 $\langle\sigma, q\rangle \in \tau$ .