Given an $\omega\_1$-tree $T$ in the ground model, can Laver forcing add a cofinal branch to $T$? Assume GCH in the ground model.


Definitions:


An $\omega\_1$-tree is a well-founded tree of height $\omega\_1$ with all levels countable. A cofinal branch of $T$ is a maximal chain of type $\omega\_1$. Laver forcing $\mathbb{L}$ consists of subtrees $T$ of $\omega^{<\omega}$ where, for some $x\in T$, every $y<\_T x$ has only one child and every $y\geq\_T x$ has $\aleph\_0$ children. $q\leq\_{\mathbb{L}} p$ iff $q\subset p$.


Related results:


Silver proved in 1971 that a countably closed forcing cannot add a cofinal branch to $T$: given $p\_\varnothing$ forcing a new cofinal branch $B$, construct conditions $(p\_s:s\in 2^{<\omega+1})$ where $p\_t\leq p\_s$ for $s\subset t$ but for $s\perp t$ nodes $b\_s\perp b\_t$ of $T$ are respectively forced into $B$ by $p\_s$ and $p\_t$. This contradicts countability of any level $T\_\delta$ chosen high enough that $b\_s\in T\_{<\delta}$ for all finite $s$.


I can modify the above argument to show that Sacks forcing (perfect subtrees of $2^{<\omega}$) does not add a cofinal branch to $T$. Given $s\in 2^n$ and $p\_s$, I find, for each $y\in p\_s$ minimal with respect to having $n$ splitting nodes below it, conditions $$q\_{sy0},q\_{sy1}\leq p\_s|y=\{x\in p\_s: x\not\perp y\}$$ that respectively force nodes $c\_{sy0},c\_{sy1}$ into $B$ such that $c\_{sy0}\perp c\_{sz1}$ for all $y,z$. Then $p\_{si}=\bigcup\_y q\_{syi}$ preserves the first $n$ 'levels' of splitting nodes of $p\_s$, ensuring that $\{p\_{f|n}:n<\omega\}$ has a common extension $q\_f$ for each $f\in 2^{\omega}$. Finally, choose $\delta$ sufficiently high and then extend each $q\_f$ to $p\_f$ deciding $B$ at level $\delta$. If $f\not=g$, then $p\_f$ and $p\_g$ disagree about $B$ at level $\delta$.


The above fusion argument also applies to Miller forcing; just replace 'splitting' with 'infinitely splitting.' However, for Laver forcing, $p\_{si}=\bigcup\_y q\_{syi}$ above may fail to be a condition because the 'trunks' of the $q\_{syi}$ may be too long. 



Does Laver forcing add cofinal branches to $\omega_1$-trees?
No.


In our paper <https://arxiv.org/abs/1409.4596> with Jindra Zapletal on Y-stuff we prove that Laver forcing is Y-proper. Consequently it has the $\omega\_1$-approximation property (i.e., does not add fresh sets of size $\omega\_1$ in Joel's terminology) and hence adds no new branches to trees.


PS: Thanks to Wolfgang for pointing me to this question.


