Shelah’s approachability ideal (part 1)

Given an infinite cardinal $\lambda$, Shelah defines an ideal $I[\lambda]$ as follows.

Definition (Shelah, implicit in here). A set $S$ is in $I[\lambda]$ iff $S\subseteq\lambda$ and there exists a collection $\{ \mathcal D_\alpha\mid\alpha<\lambda\}\subseteq\mathcal [\mathcal P(\lambda)]^{<\lambda}$, and some club $E\subseteq\lambda$, so that for every $\delta\in S\cap E$, there exists a cofinal subset $A_\delta\subseteq\delta$ such that:

  • $\text{otp}(A_\delta)<\delta$ (in particular, $\delta$ is singular);
  • for every $\gamma<\delta$, there exists some $\alpha<\delta$ such that $A_\delta\cap\gamma\in\mathcal D_\alpha$.

In other words, $\bigcup_{\alpha<\delta}\mathcal D_\alpha$ contains all the initial segments of some small cofinal subset, $A_\delta$, of $\delta$.

Easy observations. 1) $E^\lambda_\omega\in I[\lambda]$;
2) if $\mu^{<\kappa}<\lambda$ for all $\mu<\lambda$, then $E^\lambda_\kappa\in I[\lambda]$;
3) if $S\subseteq\lambda$ is nonstationary, then $S\in I[\lambda]$;
4) if $\square_\lambda$ holds, then $I[\lambda^+]=\mathcal P(\lambda^+)$. (the definition of $\square_\lambda$ may be found in here.)
Proof hints. 1) Let $\mathcal D_\alpha:=[\alpha]^{<\omega}$ for all $\alpha<\lambda$.
2) Let $\mathcal D_\alpha:=[\alpha]^{<\kappa}$ for all $\alpha<\lambda$.
3) Take a club $E$ which is disjoint from $S$.
4) If $\langle C_\alpha\mid\alpha<\lambda^+\rangle$ is a $\square_\lambda$-sequence, then simply let $\mathcal D_\alpha:=\{\text{acc}(C_\alpha)\}\cup[C_\alpha]^{<\omega}$ for all $\alpha<\lambda^+$, and restrict your attention to $E:=\text{acc}(\lambda^+)$. $\square$

So, $I[\lambda^+]$ is rather trivial in the presence of $\square_\lambda$. On the other hand, it is a very sophisticated result of Mitchell, that $I[\lambda^+]$ may behave as the other extreme:

Theorem (Mitchell, 2009). Starting with a cardinal $\kappa$ which is $\kappa^+$-Mahlo, in some forcing extension, $I[\aleph_2]$ has the property that $S\cap E^{\aleph_2}_{\aleph_1}$ is nonstationary for every $S\in I[\aleph_2]$. $\square$

In other words, $I[\kappa^{++}]$ restricted to cofinality $\kappa^+$ may coincide with nonstationary ideal $NS^{\kappa^{++}}_{\kappa^+}$. Next, let us ask what about $I[\kappa^{++}]$ restricted to cofinality $\kappa$? it turns out that we are back on track here, and this follows from the trivial fact that $\kappa^{++}$ is a successor of a regular cardinal, together with the following less-trivial proposition:

Proposition (Shelah, 1991). $E^{\lambda^+}_{<\lambda}\in I[\lambda^+]$ for every regular cardinal $\lambda$.
Proof. Of course, we may assume that $\lambda>\aleph_0$. For every ordinal $\alpha<\lambda^+$, fix an injection $d_\alpha:\alpha\rightarrow\lambda$. Notice that for every $\alpha<\delta<\lambda^+$:

  • $\forall i<\lambda\exists j\in[i,\lambda)$ such that $d_\delta^{-1}[i]\cap\alpha\subseteq d_\alpha^{-1}[j]$;
  • $\forall i<\lambda\exists j\in[i,\lambda)$ such that $d_\delta^{-1}[j]\cap\alpha\supseteq d_\alpha^{-1}[i]$.

It follows that for all $\alpha<\delta<\lambda^+$, the following set is a club in $\lambda$: $$C_\alpha^\delta:=\{ i<\lambda\mid d_\delta^{-1}[i]\cap\alpha=d_\alpha^{-1}[i]\}.$$

Next, for all $\alpha<\lambda^+$, let $$\mathcal D_\alpha:=\{ d_\alpha^{-1}[i]\cap\gamma\mid \gamma\le\alpha,i<\lambda\}.$$ Clearly, $\{ \mathcal D_\alpha\mid \alpha<\lambda^+\}\subseteq[\mathcal P(\lambda^+)]^{\le\lambda}$. We now fix an arbitrary limit ordinal $\delta\in E^{\lambda^+}_{<\lambda}$, and show that $\bigcup_{\alpha<\delta}\mathcal D_\alpha$ contains all the initial segments of some small cofinal subset of $\delta$.
Let $u$ be a cofinal subset of $\delta$ of minimal order-type. In particular, $|u|<\lambda$, and hence $C:=\bigcap_{\alpha\in u}C^\delta_\alpha$ is a club in $\lambda$. Put $i:=\min(C\setminus\sup(d_\delta[u])+1)$, and $A_\delta:=d^{-1}_\delta[i]$. Then:

  • $A_\delta\subseteq\delta$;
  • $u\subseteq A_\delta$, and hence $\sup(A_\delta)=\delta$;
  • $|A_\delta|=|i|$, and hence $|A_\delta|<\lambda$;
  • for every $\alpha\in u$, as $i\in C^\delta_\alpha$, we have $ A_\delta\cap\alpha=d_\delta^{-1}[i]\cap\alpha=d_\alpha^{-1}[i]$;
  • for every $\gamma<\delta$, letting $\alpha:=\min(u\setminus\gamma)$, we get that $$A_\delta\cap\gamma=A_\delta\cap\alpha\cap\gamma=d_\alpha^{-1}[i]\cap\gamma\in \mathcal D_\alpha.$$

This completes the proof. $\square$

We now arrive to the main result of today’s post:

Theorem (Shelah, 1993). If $\kappa$ is a regular cardinal, and $\kappa^+<\lambda$, then $I[\lambda]$ contains a stationary subset of $E^\lambda_\kappa$.
Proof. We already know that $E^\lambda_\omega\in I[\lambda]$. We also know that $E^\lambda_\kappa\in I[\lambda]$ in the case $\lambda=\kappa^{++}$. Thus, we shall assume that $\aleph_0<\kappa<\kappa^{++}<\lambda$.
In an earlier post, we proved that $E^{\aleph_3}_{\aleph_1}$ carries a club guessing sequence, and since here $\kappa$ is regular and uncountable, the same argument shows that $E^{\kappa^{++}}_\kappa$ carries a club-guessing sequence. Thus, let us fix such a club-guessing sequence $\overrightarrow C=\langle C_\beta\mid \beta\in E^{\kappa^{++}}_\kappa\rangle$. Next, fix a large enough regular cardinal $\theta$, and let $\overrightarrow M=\langle M_\alpha\mid \alpha<\lambda\rangle$ be an increasing sequence of elementary submodels of $(\mathcal H_\theta,\in)$ such that for all $\alpha<\lambda$:

  • $|M_\alpha|<\lambda$;
  • $\kappa^{++}+\alpha+1\subseteq M_{\alpha+1}$;
  • $\overrightarrow C,\lambda\in M_\alpha$.

We shall now make a somewhat naive move, and simply let $\mathcal D_\alpha:=M_\alpha\cap\mathcal P(\lambda)$ for all $\alpha<\lambda$. Since $\mathcal D_\alpha\in[\mathcal P(\lambda)]^{<\lambda}$ for all $\alpha<\lambda$, the following set $S$ is obviously in $I[\lambda]$,$$S:=\{ \delta\in E^\lambda_\kappa\mid \exists A_\delta\subseteq\delta(\text{otp}(A_\delta)<\delta=\sup(A_\delta),\forall\gamma<\delta\exists\alpha<\delta[A_\delta\cap\gamma\in\mathcal D_\alpha])\}.$$

On the other hand, the next statement is not entirely obvious.

Subclaim. $S$ is stationary.
Proof. Given a club $D\subseteq\lambda$, we shall seek some $\delta\in S\cap D$. Wlog, $\min(D)>\kappa$.
Let $\overrightarrow N:=\langle N_i\mid i<\kappa^{++}\rangle$ be a sequence of elementary submodels of $(\mathcal H_\theta,\in)$ such that for all $i<\kappa^{++}$:

  • $|N_i|=\kappa^{++}$, with $\kappa^{++}\subseteq N_{i}$;
  • $\langle N_j\mid j\le i\rangle\in N_{i+1}$;
  • $\overrightarrow M, \overrightarrow C,D,\lambda\in N_i$;
  • $N_i=\bigcup_{j<i}N_j$ whenever $i$ is a limit ordinal.

For all $i<\kappa^{++}$, as $D,\lambda\in N_i$ and $\sup(D)=\lambda$, we get by elementarity that $\sup(N_i\cap D)=\sup(N_i\cap\lambda)$. Recalling that $D$ is closed, we infer that $\sup(N_i\cap\lambda)\in D$ for all $i<\kappa^{++}$. Thus, it make sense to define a function $f:\kappa^{++}\rightarrow D$ by letting: $$f(i):=\sup(N_i\cap\lambda),\quad(i<\kappa^{++}.)$$
By the second and fourth defining properties of $\overrightarrow N$, we get that $f$ is increasing and continuous, and that $f\restriction\tau\in N_{\tau+1}$ for all $\tau<\kappa^{++}$. Put $\epsilon:=\sup(f[\kappa^{++}])$. Since $\kappa^{++}+\epsilon+1\subset M_{\epsilon+1}$, we get (by elementarity) the existence of some strictly-increasing and cofinal function $g:\kappa^{++}\rightarrow\epsilon$ that lies in $M_{\epsilon+1}$.
Since $f$ and $g$ are both continuous and cofinal in $\epsilon$, a standard back-and-fourth argument yields the existence of a club $c\subseteq\kappa^{++}$ for which $f\restriction c=g\restriction c$. Since $\overrightarrow C$ is a club guessing sequence, we may find some $\beta\in E^{\kappa^{++}}_\kappa$ such that $C_\beta\subseteq c$. Put $\delta:=f(\beta)$ and $A_\delta:=f[c_\beta]$. Then $\delta\in D\cap E^\lambda_\kappa$, and $A_\delta$ is a cofinal subset of $\delta$ of order-type $\kappa<\min(D)\le\delta$. Thus, we are left with showing that $A_\delta\cap\gamma\in\bigcup_{\alpha<\delta}\mathcal D_\alpha$ for all $\gamma\in A_\delta$.
Fix $\gamma\in A_\delta$, and let $\tau:=f^{-1}(\gamma)$. Then:

  • $\tau+1<\beta$;
  • $A_\delta\cap\gamma=g[c_\beta\cap\tau]$, and the latter belongs to $M_{\epsilon+1}$, since $g,\overrightarrow C,\beta,\tau\in M_{\epsilon+1}$ ;
  • $A_\delta\cap\gamma=(f\restriction\tau)[c_\beta]$, and the latter belongs to $N_{\tau+1}$ since $f\restriction\tau,\overrightarrow C,\beta\in N_{\tau+1}$.

Denote $A:=A_\delta\cap\gamma$. Then $A\in M_{\epsilon+1}$, and $A,\overrightarrow M\in N_{\tau+1}$. Consequently: $$N_{\tau+1}\models \exists\alpha<\lambda(A\in M_\alpha).$$ It follows that there exists some $\alpha<\sup(N_{\tau+1}\cap\lambda)=f(\tau+1)<f(\beta)=\delta$ such that $A\in M_\alpha$. In particular, $A_\delta\cap\gamma\in\bigcup_{\alpha<\delta}\mathcal D_\alpha$. $\square$

So, $S$ is an example of a stationary subset of $E^\lambda_\kappa$ that belongs to $I[\lambda]$. $\square$

1. Notice that in the above proof, we could have replaced $\kappa^{++}$ with any regular cardinal $\mu<\lambda$ for which $E^\mu_\kappa$ carries a club-guessing sequence. In particular, the above proof shows:

Theorem (Shelah). If $\kappa<\kappa^+<\mu<\lambda$ are all regular cardinals, then there exists a subset $S\subseteq E^\lambda_\kappa$ such that:

  • $S\in I[\lambda]$, and $S$ reflects in the following sense:
  • $\{\delta\in E^\lambda_\mu\mid S\cap\delta\text{ is stationary}\}$ is stationary in $\lambda$. $\square$

2. The proof may be adapted to show that if $\kappa<\kappa^+<\lambda$ are regular, then there exists a sequence $\langle C_\delta\mid \delta\in E^\lambda_\kappa\rangle$, and an enumeration $\{ \mathcal D_\alpha\mid\alpha<\lambda\}\subseteq[\mathcal P(\lambda)]^{<\lambda}$, such that for every club $D\subseteq\lambda$, there exists $\delta\in\text{acc}(D)$ with:

  • $\text{otp}(C_\delta)=\text{cf}(\delta)=\kappa$;
  • $C_\delta$ is a club subset of $D\cap\delta$;
  • for every $\gamma<\delta$, there exists $\alpha<\delta$ such that $C_\delta\cap\gamma\in\mathcal D_\alpha$.

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