
> 
> Suppose $S \subset \omega\_{1}$ is stationary co-stationary. Then there is a forcing notion $P\_{S}$ which shoots a closed unbounded $C \subset S$ without collapsing cardinals (or
>  changing cofinalities).
> 
> 
> 
> > 
> > Let $P$ = $Levy$($\aleph\_{0}$， $< \aleph\_{1}$). So P is essentially adding $\aleph\_{1}$ Cohen reals. If $G\_{P} \subset P$ is (directed and) generic over V, then $G\_{P}$ is also generic over $L$ (the constructible universe) as $P \in L$ and also over $L[S]$. By Theorem I 6.7 the forcing P satisfies the countable chain condition and $\aleph\_{1}^{L[G\_{P}]} = \aleph\_{1}^{V} = \aleph\_{1}^{V[G\_{P}]}$ and $V$, $V[G\_{P}]$ have the same cardinals and cofinalities. Let
> > 
> > 
> > $Q$ = {$C$ : $C$ is a closed bounded subset of $S$ which belongs to $L[S, G\_{P}]$} and
> >  $C\_{1} < C\_{2}$ iff $C\_{1} = C\_{2} \cap (Max(C\_{1}) + 1)$.
> > 
> > 
> > Clearly $Q$ is a forcing notion of power $\aleph\_{1}$, so it cannot collapse cardinals or regularity of cardinals except possibly $\aleph\_{1}$ (all finite subsets of $S$ belong to $Q$). So we shall prove that $P \* \dot{Q}$ does not collapse $\aleph\_{1}$. So let (in $V$) $N \prec (H(\lambda), \epsilon)$, $N$ countable, $(p, \dot{q}) \in P \* \dot{Q} \in N$, $(p, \dot{q}) \in N$ , $\delta := N \cap \omega\_{1} \in S$ and suppose $G\_{P} \subset P$ is generic over $V$ and $p \in G\_{P}$. Note that as $S$ is a stationary subset of $\omega\_{1}$, there is such $N$ (in $V$). So it is enough to find $(p',\dot{q'}) \geq (p, \dot{q})$ which is $(N, P \* \dot{Q})$-generic. As $P$ satisfies the countable chain condition, p is $(N, P)$-generic (by 2.9), hence $N[G\_{P}] \cap V = N$. Clearly $\dot{Q}[G\_{P}] \in L[S, G\_{P}] \subset V[G\_{P}]$, now $N[G\_{P}]$ does not necessarily belong to $L[S, G\_{P}]$, but $N[G\_{P}] \cap L[S, G\_{P}]$ is $N[G\_{\gamma}] \cap L\_{\delta}[S, G\_{P}] = L\_{\delta}[S, G\_{P}] \in L[S, G\_{P}]$ and is a countable set in $L[S, G\_{P}]$.
> > 
> > 
> > In $L[S, G\_{P}]$ we have an enumeration of $\dot{Q}[G\_{P}] \cap N[G\_{P}]$ (of length $\omega$), say $\langle q\_{n} : n < \omega \rangle$ (but not of the set of dense subsets); in fact we have it even in $L[S, G\_{P} \upharpoonright(\delta + 1)]$ (and as we use $Levy(\aleph\_{0}, < \aleph\_{1})$ not $Levy(\aleph\_{0}, < \kappa)$ even in $L[S, G\_{P} \upharpoonright \delta]$). Now in $L[S, G\_{P}]$ there is a Cohen generic real over $V[G\_{P} \upharpoonright (\delta + 1)]$ say $r^{\*} \in$ $^{\omega}{\omega}$ and we use it to construct a sequence $\bar{C} = \langle C\_{n} : n < \omega \rangle$ such that $C\_{n} \in \dot{Q}[G\_{P}]$ in $L[S, G\_{P}]$ i.e. $\bar{C} \in L[S, G\_{P}]$, e.g. we choose $C\_{n}$ by induction on $n$; we let $C\_{0} = \dot{q}[G\_{P}]$ and we let $C\_{n+1}$ be $q\_{m(n)}$ where $m(n)$ is the first natural number $m$ such that: $m \geq r^{\*}(n)$ and $\dot{Q}[G\_{P}] \models$ "$C\_{n} \leq q\_{m}$". Let 
> > 
> > 
> > $G$ := {$q$ : $q \in \dot{Q}[G\_{P}]$, and $q \in N[G\_{P}]$ and for some $n$, $\dot{Q}[G\_{P}] \models$ "$q \leq C\_{n}$"}. 
> > 
> > 
> > Clearly $G \subset N[G\_{P}] \cap \dot{Q}[G\_{P}]$ is generic over $V[G\_{P} \upharpoonright (\delta + l)]$. So $q^{\dagger} = (\bigcup\_{n} C\_{n}) \cup \{\delta\} \in \dot{Q}[G\_{P}]$ is $(N[G\_{P}], \dot{Q}[G\_{P}])$-generic. Going to names we finish.
> > 
> > 
> > 
> 
> 
> 


A few questions regarding the proof above (Theorem 4.4 of Shelah's PIF). Here '$p \geq q$' means '$p$ extends $q$'.


First let $A \in N[G\_{P}]$ be a maximal antichain of $\dot{Q}[G\_{P}]$.


1. How do we ensure that the $G$ constructed at the end of the proof intersect $N[G\_{P}] \cap A$?
2. If we cannot ensure the above, how do we show that $q^{\dagger}$ is ($N[G\_{P}]$, $\dot{Q}[G\_{P}]$)-generic?


I think there is some crucial properties of Cohen reals that I am missing here, such as how Cohen-genericity translate to $\dot{Q}[G\_{P}]$-genericity, if at all.


Much thanks in advance.



Question about Shelah's version of "Shooting a club" found in PIF
Sorry this is sketchy, but I hope it helps!


A Cohen condition $p$ of length $n$ is going to determine $\langle C\_m:m<n\rangle$. We know there's going to be an $i<\omega$ such that $q\_i$ extends both $C\_{n-1}$ and a member of $N[G\_P]\cap A$. Extend $p$ to a Cohen condition $p'$ with $p'(n)=i$. If $r^\*$ is any Cohen generic extending $p'$, then running the construction with $r^\*$ guarantees that $C\_n$ extends a member of $N[G\_P]\cap A$. So a density argument tells us the construction is forced to work.


The point is that in Shelah's construction, $m(n)$ is going to be equal to $r^\*(n)$ very often because $r^\*$ is Cohen.


