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

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

(1) $\text{<mrow data-mjx-texclass="ORD">dom</mrow>}(\eta )\subseteq \text{<mrow data-mjx-texclass="ORD">dom</mrow>}(\tau )$

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

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

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