# Definition A [[forcing#^e01f21|forcing notion]] $\mathbb{P}$ is *proper* iff for all $\kappa$ and all [[Club sets and stationary sets#Club and stationary subsets of $P_{ omega_{1}}(X)$|stationary]] $\mathcal{S} \subseteq \mathcal{P}_{\omega_{1}}(\kappa)$, $\mathbb{P}\Vdash$ "$\mathcal{S}$ is stationary". ## A combinatorial characterisation (see Kunen V.7.2 p. 382, Jech pg. 602) >[!info] Definition > Let $\mathbb{P}$ be a forcing notion, $\theta$ a regular cardinal gt;2^{\mathbb{P}}$ and $M$ s.t. $(\mathbb{P}, \leq, \mathbb{1}) \in$ $M \preccurlyeq H(\theta)$ >We say that $p \in \mathbb{P}$ is *$(M, \mathbb{P})$-generic* iff for all dense $D \subseteq \mathbb{P}$ such that $D \in M$, the set $D\cap M$ is predense below $p$ - there is no $q\leq p$ such that $q\bot D\cap M$). >[!note] Remark >In the Definition we get an equivalent notion if the "all dense $D \subseteq \mathbb{P}$ " is replaced by any one of "all dense open $D \subseteq \mathbb{P}$ " or "all predense $D \subseteq \mathbb{P}$ " or "all maximal antichains $D \subseteq \mathbb{P}$ ". **Example.** If $\mathbb{P} \in M \preccurlyeq H(\theta)$ and $\mathbb{P}$ is ccc, then every $p \in \mathbb{P}$ is $(M, \mathbb{P})$ generic. **Example** If $\mathbb{P} \in M \preccurlyeq H(\theta)$ and $\mathbb{P}$ is countably closed and $p \in M$ and $M$ is countable, then there is a $q \leq p$ that is $(M, \mathbb{P})$-generic. Also, if $\mathbb{P}$ is atomless, then this $q$ cannot be in $M$. **Lemma.** Let $\lambda$ be sufficiently large, let $M \prec H_\lambda$ be such that $P \in M$, and let $q \in P$. The following are equivalent: 1. $q$ is $(M, P)$-generic. 2. If $\dot{\alpha} \in M$ is an ordinal name then $q \Vdash \dot{\alpha} \in M$, i.e. $\forall r \leq q \exists s \leq r \exists \beta \in M s \Vdash \dot{\alpha}=\beta .$ 3. $q \Vdash \dot{G} \cap M$ is a filter on $P$ generic over $M$. **Theorem (Combinatorial characterisation).** TFAE 1. $\mathbb{P}$ is proper 2. for all regular $\theta$ with $\mathbb{P} \in H(\theta)$, there is a club $\mathcal{C} \subseteq \mathcal{P}_{\omega_{1}}(H(\theta))$ such that for all $M \in \mathcal{C}$: $\mathbb{P}\in M$, $M \prec H(\theta)$ and for all $p \in \mathbb{P} \cap M$, there is a $q \leq p$ that is $(M, \mathbb{P})$-generic. 3. for every $p \in \mathbb{P}$, every sufficiently large $\lambda$ ($\lambda>\left( 2^{|\mathbb{P}|} \right)^{+}$) and every countable $M \prec\left(H_\lambda, \in,<\right)$ containing $\mathbb{P}$ and $p$, there exists a $q \leq p$ that is $(M, \mathbb{P})$-generic. ## A characterisation via games (See Jech pg. 603) >[!info] Definition >Let $\mathbb{P}$ be a forcing notion and let $p \in \mathbb{P}$. The *proper game* (for $\mathbb{P}$, below $p$) is played as follows: > I plays $\mathbb{P}$-names $\dot{\alpha}_n$ for ordinal numbers, and II plays ordinal numbers $\beta_n$. Player II wins if there exists a $q \leq p$ such that $ q \Vdash \forall n \exists k \dot{\alpha}_n=\beta_k . $ **Theorem.** A forcing notion $\mathbb{P}$ is proper if and only if for every $p \in \mathbb{P}$, II has a winning strategy for the proper game. # Properties **Lemma.** If $\mathbb{P}$ is proper then every countable set of ordinals in $V[G]$ is included in a set in $V$ that is countable in $V$. (Jech pg. 602) %% **Lemma.** If $\mathbb{P}$ is proper, $\kappa$ arbitrary and $p\Vdash \dot{A}\in \mathcal{P}_{\omega_{1}}(\kappa)$ then there is $B\in \mathcal{P}_{\omega_{1}}(\kappa)$ and $q\leq p$ such that $q \Vdash \dot{A} \subseteq B$.% (Kunen pg. 380)%% In particular, $\mathbb{P}$ preserves $\omega_{1}$. ## Preservation - Preserved under countable support iterations. # Relations ## Implied by - [[Chain conditions#^ccc|Countable chain condition]] - [[Closed forcing#^c-closed|Countably closed]] - [[Baumgartner's Axiom A]] ## Implies - $\omega_{1}$-preserving - [[Semiproper forcing|Semiproper forcing]]