#forcing #forcing_notion #tree #souslin #kurepa # Forcing an $\omega_{1}$-Souslin tree ![[Trees#^souslin]] $\mathbb{P}$: **Conditions**: trees $T$ such that one of the following hold 1. $T=\varnothing$ 2. There is $\gamma<\omega_{1}$ such that $T \subseteq 2^{\leq \gamma}$ and for every $\alpha<\gamma$ every $t\in T_{\alpha}$, has at least $2$ distinct extensions in $T_{\gamma}$. **Relation**: end-extension. **Lemma.** $\mathbb{P}$ is $\sigma$-[[Closed forcing|closed]]. **Lemma.** Hence we get an $\omega_{1}$-tree. **Lemma.** The generic will be Souslin. # Killing an $\omega_{1}$-Souslin tree Let $(T,\leq)$ be a Souslin tree. **Lemma.** The set $X=\{ x \in T \mid x^{\uparrow} \text{ is countable} \}$ is countable. **Theorem.** $\mathbb{P}=(T \smallsetminus X,\geq)$ is [[Chain conditions#^ccc|c.c.c]] and forces a branch through $T$. >[!note] Remark > If $\mathbb{P}$ is a forcing notion such that $\mathbb{P}\times \mathbb{P}$ is c.c.c. then $\mathbb{P}$ cannot add an $\omega_{1}$ branch to an $\omega_{1}$ tree. > Hence the square of a Souslin tree is not Souslin. # Forcing an $\omega_{1}$-Kurepa tree ![[Trees#^kurepa]] Let $\mathbb{P}$ be the forcing to add a Souslin tree. Define $\mathbb{Q}$ by using $\mathbb{P}$ as the working part. **Conditions:** $(T,I,f)$ where - $T\in \mathbb{P}$. - $I \in [\omega_{2}]^{\aleph_{0}}$ - $f:I\to T_{\gamma}$ finite-to-1 where $\gamma+1=\mathrm{ht}(T)$ **Relation** $(T',I',f')\leq (T,I,f)$ iff - $T'$ end-extends $T$ - $\forall i\in I$, $f(i) \subseteq f'(i)$ If $G$ is generic, $T^{*}$ the generic tree. $\forall\tau<\omega_{2}$ let $B(\tau)=\{ f(\tau) \mid (T,I,f) \in G\}$. Then this is a branch in $T^{*}$, and for $\tau \ne \tau'$ , $B(\tau)\ne B(\tau')$ . So $T^{*}$ is a Kurepa tree. **Lemma.** The following sets are dense: - $\forall\alpha <\omega_{1}$ $D_{\alpha}=\{ (T,I,f) \mid \mathrm{ht}(T)\geq \alpha \}$ (so we get an $\omega_{1}$ tree). - $\forall \tau<\omega_{2}$ $E_{\tau}=\{ (T,I,f) \mid \tau\in I\}$ (so we get $\aleph_{2}$ branches). **Lemma.** $\mathbb{Q}$ is $\sigma$-[[Closed forcing|closed]]. **Lemma.** CH implies $\mathbb{Q}$ is $\aleph_{2}$-c.c. # Killing all $\omega_{1}$-Kurepa trees **Lemma.** If $\mathbb{P}$ is $\aleph_{1}$-closed then it doesn't add new $\aleph_{1}$-branches to an $\aleph_{1}$-tree. Strategy: devise $\mathbb{P}$ such that for every $\aleph_{1}$-tree $T$ in $V^{\mathbb{P}}$ we can write $\mathbb{P}=\mathbb{P}_{1}\times \mathbb{P}_{2}$ such that $T\in V^{\mathbb{P}_{1}}$ is not Kurepa and $\mathbb{P}_{1},\mathbb{P}_{2}$ are $\aleph_{1}$-closed. Let $\theta$ strongly inaccessible, and use the [[Collapse forcing#Levy collapse|Levy collapse]] $\mathrm{Coll}(\omega_{1},<\theta)$. For every $T\in V^{\mathbb{P}}$ there is $\mu<\theta$ such that the number of branches is forced to be less than $\mu$. Then we can write: $\mathrm{Coll}(\omega_{1},<\theta)\equiv \mathrm{Coll}(\omega_{1},<\mu)\times\mathrm{Coll}(\omega_{1},[\mu,\theta))$ with the desired properties.