Nice name

In set theory, a nice name is used in forcing to impose an upper bound on the number of subsets in the generic model. It is used in the context of forcing to prove independence results in set theory such as Easton's theorem.

This article is about mathematics. For the name "Nice", see Nice (disambiguation).

Formal definition

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 in ${\displaystyle M}$, ${\displaystyle \tau }$,

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

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

(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 ${\displaystyle \langle \sigma ,q\rangle \in \tau }$.

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