See Jech 244 Also called **Forcing with Perfect Trees** Produces a real of minimal degree of constructibility. If forced over $L$, the generic filter yields a real $a$ such that $a \notin L$ and such that for every real $x \in L[a]$, either $x \in L$ or $a \in L[x]$. # Definition >[!info] Definition >A nonempty [[trees|tree]] $T \subseteq 2^{<\omega}$ is *perfect* if for every $t \in T$ there exists an $s \supset t$ such that both $s^\frown 0$ and $s^\frown 1$ are in $T$. >[!note] Remark > The set of all paths in a perfect tree is a perfect set in the Cantor space $\{0,1\}^\omega$. Let $\mathbb{P}$ be the set of all perfect trees $p \subset 2^{<\omega}$. $p$ is stronger than $q$ if and only if $p \subset q$. If $G$ is a generic set of perfect trees, let $ f=\bigcup\{s:(\forall p \in G) s \in p\} . $ The function $f: \omega \rightarrow\{0,1\}$ is called a *Sacks real*. Note that $V[G]=V[f]$. ## Important notions >[!info] Definition >Let $p$ be a tree. >- The *stem* of $p$ is the maximal $s\in p$ such that for every $t\in p$, $t \subseteq s \lor s \subseteq t$. >- If $s$ is a node in $p$, let $p\restriction s$ denote the tree $\{t \in p: t \subset s$ or $t \supset s\}$. ^stem >[!info] Definition >Let $p$ be a perfect tree. >- A node $s \in p$ is a *splitting node* if both $s^{\frown} 0 \in p$ and $s^{\frown} 1 \in p$ >- A splitting node $s$ is an *$n$th splitting node* if there are exactly $n$ splitting nodes $t$ such that $t \subset s$. > (A perfect tree has $2^{n-1} n$th splitting nodes.) >- For each $n \geq 1$, let $p \leq_n q$ if and only if $p \leq q$ and every $n$th splitting node of $q$ is an $n$th splitting node of $p$. >- A *fusion sequence* is a sequence of conditions $\left\{p_n\right\}_{n=0}^{\infty}$ such that $p_n \leq_n p_{n-1}$ for all $n \geq 1$. **Lemma.** If $\left\{p_n\right\}_{n=0}^{\infty}$ is a fusion sequence then $\bigcap_{n=0}^{\infty} p_n$ is a perfect tree. >[!info] Definition >- If $A$ is a set of incompatible nodes of $p$ and for each $s \in A, q_s$ is a perfect tree such that $q_s \subset p\restriction s$, then the *amalgamation of $\left\{q_s: s \in A\right\}$ into $p$* is the perfect tree > $ \left\{t \in p: \text { if } t \supset s \text { for some } s \in A \text { then } t \in q_s\right\} \text {. } $ > (Replace in $p$ each $p\restriction s$ by $q_s$.) # Properties - $|P|=2^{\aleph_0}$ - If we assume $\mathrm{CH}$ in the ground model, $\mathbb{P}$ satisfies the $\aleph_2$ [[Chain conditions|chain condition]] and all cardinals $\geq \aleph_2$ are preserved. - Satisfies [[Baumgartner's Axiom A]]. - [[Proper forcing]] - Preserves $\aleph_{1}$ >[!info] Definition > A generic filter $G$ is minimal over the ground model $M$ if for every set of ordinals $X$ in $M[G]$, either $X \in M$ or $G \in M[X]$. **Theorem(Sacks).** When forcing with perfect trees, the generic filter is minimal over the ground model.