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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.

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