Nice name

Let ${\displaystyle M\models }$ ZFC be transitive, ${\displaystyle (\mathbb {P} ,<)}$ a forcing notion in ${\displaystyle M}$, and suppose ${\displaystyle G\subseteq \mathbb {P} }$ is generic over ${\displaystyle M}$.

Then for any ${\displaystyle \mathbb {P} }$-name ${\displaystyle \tau }$ in ${\displaystyle M}$, we say that ${\displaystyle \eta }$ is a nice name for a subset of ${\displaystyle \tau }$ if ${\displaystyle \eta }$ is a ${\displaystyle \mathbb {P} }$-name satisfying the following properties:

(1) ${\displaystyle \operatorname {dom} (\eta )\subseteq \operatorname {dom} (\tau )}$

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

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

• 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.