Monday, January 26, 2015

Spencinar: Forcing clubs in stationary subsets of $P_\kappa(\lambda)$

The first part of this talk follows closely the paper "Forcing closed unbounded sets" by Uri Abraham and Saharon Shelah ([AS]). The second part will follow the paper "Nonsplitting subset of $P_\kappa(\kappa^+)$ by Moti Gitik ([G]).

We have previously seen that given a stationary $S\subseteq \omega_1$, there is a forcing which has the $\omega_2$-c.c. and is $<\omega_1$-distributive (and hence preserves all cardinals) which adds a club $C\subseteq S$, namely the forcing of closed bounded subsets of $S$, ordered by end-extension (Baumgartner--Harrington--Kleinberg).

This generalizes to higher regular cardinals $\kappa$ assuming some cardinal arithmetic (GCH with $\kappa$ the successor of a regular cardinal suffices) and fatness of $S$, which says that for any club $E\subseteq \kappa$, $S\cap E$ contains closed subsets of arbitrarily large order type $<\kappa$ (this is a result of J. Stavi).

The focus of this talk will be to obtain analogues for these theorems for subsets of $P_\kappa(\lambda)=\{x\subseteq \lambda: |x|<\kappa\}$.

This situation is not as clear as for subsets of ordinals, and in fact there are some sets whose stationarity is quite absolute (By stationary and club I mean here in the sense of Jech, see this previous post). For example, Theorem 6 in [AS] gives a stationary subset of $P_{\aleph_1}(\omega_2)$ whose stationarity is preserved by any forcing which preserves $\aleph_2$:

Theorem: Let $W\subseteq V$ be a transitive inner model such that $\omega_2^W=\omega_2^V$, and let $S=(P_{\aleph_1}(\omega_2))^W$. Then $S$ is stationary in $V$.

Proof: The idea of the proof is to fix an arbitrary club $C\subset P_{\aleph_1}(\omega_2)$ and find submodels which are ordinals, or somehow coded by ordinals in $W$. Then these submodels will automatically be in $S$.

By Kueker's theorem, there is a function $F:[\omega_2]^{<\omega}\rightarrow \omega_2$ so that any $x\in P_{\aleph_1}(\omega_2)$ closed under $F$ is in $C$ (this uses the fact that we're working on $\aleph_1$ and $\aleph_2$, but this is a minor technical point). Let $\alpha$ be an ordinal closed under $F$. If $\alpha$ is countable, then we're done; otherwise $\alpha$ has cardinality $\aleph_1$, hence also $W$-cardinality $\aleph_1$, so fix a bijection $h:\omega_1\rightarrow \alpha$ in $W$. Then there is $\xi<\omega_1$ with $h[\xi]$ closed under $F$, and $h[\xi]$ is in $W$ and hence in $S$, so we're done again. $\Box$

In [G], it's shown that any forcing which adds reals to $W$ destroys the club-ness of $S$.

This paper gives some examples of sets whose stationary can be destroyed. We won't go through those results here. Instead, we now turn towards a result in [G] which also gives examples of this kind.

Let $\kappa$ be supercompact in $W$, and let $V$ be obtained from $W$ by Radin forcing. We won't need to know much about Radin forcing, just:
  • the continuum function and all cardinalities are preserved,
  • there is a club $C\subseteq \kappa$ of $W$-inaccessible cardinals,
  • $\kappa$ remains (sufficiently) supercompact in $V$. 
From now on, work in $V$. Let $A$ be any subset of the set of all $t\in P_\kappa(\kappa^+)\cap W$ such that $V\vDash ``|t| \textrm{ is a successor cardinal}"$.

We will find a forcing that adds a club in $S:=P_\kappa(\kappa^+)\setminus A$ which is $<\kappa$-distributive (this is crucial, so that $P_\kappa(\kappa^+)$ itself does not change), and has the $\kappa^+$-c.c.

Define the poset $\mathbb{P}$ to be the collection of subsets of $S$ of size $<\kappa$ which have a maximum element and are closed under increasing unions, ordered by end-extension. 

Claim: $\mathbb{P}$ is $<\kappa$-distributive.

Let $\langle D_\beta: \beta<\alpha\rangle$ be a sequence of dense open subsets of $\mathbb{P}$ for some $\alpha<\kappa$.  Then we will prove that $\bigcap_{\beta<\alpha} D_\beta$ is also dense. So let $x\in \mathbb{P}$.

Let $M\prec (H(\theta),\in,\triangleleft,A,\langle D_\beta\rangle,x,\ldots)$ be an elementary substructure of size $\kappa$ so that $M\cap \kappa^+\in \kappa^+$ and $M$ is closed under $<\kappa$-sequences. Let $g\in W$ be a bijection from $\kappa$ onto $M\cap \kappa^+$. Using $<\kappa$-closure of $M$, construct $\langle M_i:i<\kappa\rangle$ a continuous IA chain of elementary submodels of $M$ of size $<\kappa$ so that $\alpha+1\subseteq M_0$ and $g``i\subseteq M_i$. There is a club $E\subseteq \kappa$ such that for every $i\in E$, $i\in C$ and $M_i\cap \kappa^+=g``i$. Let $\langle i_\beta:\beta<\kappa\rangle$ be the increasing enumeration of $E$, and let $N_\beta=M_{i_\beta}$.

We will inductively construct a decreasing sequence of conditions $\langle x_\beta:\beta\le \alpha\rangle$ such that:

  •  $x_0\le x$,
  •  $x_{\beta+1}\in D_\beta$,
  •  $N_\beta\cap \kappa^+=\max(x_\beta)$,
  •  $x_\beta\in N_{\beta+1}$.

Take $x'_0\in N_0$, $x'_0\le x$, and let $x_0=x'_0\cup\{N_0\cap \kappa^+\}$.

Now we construct $x_\beta$, assuming that $x_\gamma$ has been constructed for every $\gamma<\beta$. If $\beta=\gamma+1$ for some $\gamma$, then pick $x'_\beta$ to be the $\vartriangleleft$-least in $N_\beta\cap D_\beta$ extending $x_\gamma$, and define $x_\beta=x'_\beta\cup\{N_\beta\cap \kappa^+\}$. This is a valid condition since $N_\beta\cap \kappa^+$ has $W$-cardinality in $C$, so it can't be a member of $A$.

If $\beta$ is limit, then let $x'_\beta$ be the closure of $\bigcup \{x_{\gamma}:\gamma<\beta\}$ under increasing unions, and $x_\beta=x'_\beta\cup\{N_\beta\cap \kappa^+\}$. By internal approachability, $x_\beta\in N_{\beta+1}$.

It is easy to see that:
Claim: For every $\gamma<\beta$, $x_\beta$ is an end-extension of $x_{\gamma}$.

We now check that $x_\beta\in \mathbb{P}$. If not, then there is $t\in x_\beta\cap A$. Since $t\subseteq N_\beta\cap \kappa^+$, we must have $|t|^W\le i_\beta$. Furthermore, $|t|^W$ is a successor cardinal of $W$ and $i_\beta$ is W-inaccessible, so $|t|^W<i_\beta$.

Since $i_\beta$ is $W$-regular, there is some $\gamma<\beta$ with $t\subseteq g``i_\gamma=N_\gamma\cap \kappa^+$. But this is impossible by the claim.  

The $\kappa^+$-c.c. follows by a standard $\Delta$-system argument.

Gitik used this forcing as the building block of an iteration to produce a stationary subset $Z$ of $P_\kappa(\kappa^+)$ so that the non-stationary ideal restricted to that stationary set is $\kappa^+$-saturated, which is very interesting in light of results that show that the whole nonstationary ideal is not saturated. Another way to look at this is that it shows the consistency of the failure of a natural analogue of Solovay's splitting theorem for stationary subsets of $P_\kappa(\lambda)$.

Assuming that $\kappa$ is supercompact, the set $Z$ of all $t\in P_\kappa(\kappa^+)\cap W$ such that $W\vDash ``|t| \textrm{ is a successor cardinal}"$ is stationary. The idea of the iteration is that we will destroy the stationary of the "bad sets" to while maintaining the stationary of $Z$. Maintaining the stationarity of $Z$ is achieved through extending the supercompactness embedding to the final model, but this is not an easy task in this case.

An earlier version of this post had several occurrences of $V$ which should have been $W$.