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) $E_{\omega }^{\lambda }\in I[\lambda ]$;
2) if ${\mu }^{<\kappa }<\lambda$ for all $\mu <\lambda$, then $E_{\kappa }^{\lambda }\in I[\lambda ]$;
3) if $S\subseteq \lambda$ is nonstationary, then $S\in I[\lambda ]$;
4) if ${\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}}_{\lambda }$ holds, then $I[{\lambda }^{+}]=\mathcal{P}({\lambda }^{+})$. (the definition of ${\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}}_{\lambda }$ 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 $⟨C_{\alpha }\mid \alpha <{\lambda }^{+}⟩$ is a ${\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}}_{\lambda }$-sequence, then simply let ${\mathcal{D}}_{\alpha }:=\{\text{acc}(C_{\alpha })\}\cup [C_{\alpha }{]}^{<\omega }$ for all $\alpha <{\lambda }^{+}$, and restrict your attention to $E:=\text{acc}({\lambda }^{+})$. $\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}$

So, $I[{\lambda }^{+}]$ is rather trivial in the presence of ${\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}}_{\lambda }$. 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, $I[{\mathrm{\aleph }}_2]$ has the property that $S\cap E_{{\mathrm{\aleph }}_1}^{{\mathrm{\aleph }}_2}$ is nonstationary for every $S\in I[{\mathrm{\aleph }}_2]$. $\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}$

In other words, $I[{\kappa }^{++}]$ restricted to cofinality ${\kappa }^{+}$ may coincide with nonstationary ideal $N{S}_{{\kappa }^{+}}^{{\kappa }^{++}}$. 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 $\lambda >{\mathrm{\aleph }}_0$. For every ordinal $\alpha <{\lambda }^{+}$, fix an injection $d_{\alpha }:\alpha \to \lambda$. Notice that for every $\alpha <\delta <{\lambda }^{+}$:

• $\mathrm{\forall }i<\lambda \mathrm{\exists }j\in [i,\lambda )$ such that $d_{\delta }^{-1}[i]\cap \alpha \subseteq d_{\alpha }^{-1}[j]$;
• $\mathrm{\forall }i<\lambda \mathrm{\exists }j\in [i,\lambda )$ 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 $C:=\bigcap _{\alpha \in u}{C}_{\alpha }^{\delta }$ is a club in $\lambda$. Put $i:=min(C\setminus sup(d_{\delta }[u])+1)$, and $A_{\delta }:=d_{\delta }^{-1}[i]$. 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 $i\in C_{\alpha }^{\delta }$, 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. $\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}$

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 $E_{\kappa }^{\lambda }$.
Proof. We already know that $E_{\omega }^{\lambda }\in I[\lambda ]$. We also know that $E_{\kappa }^{\lambda }\in I[\lambda ]$ in the case $\lambda ={\kappa }^{++}$. Thus, we shall assume that ${\mathrm{\aleph }}_0<\kappa <{\kappa }^{++}<\lambda$.
In an earlier post, we proved that $E_{{\mathrm{\aleph }}_1}^{{\mathrm{\aleph }}_3}$ 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 $\overrightarrow{C}=⟨C_{\beta }\mid \beta \in E_{\kappa }^{{\kappa }^{++}}⟩$. Next, fix a large enough regular cardinal $\theta$, and let $\overrightarrow{M}=⟨M_{\alpha }\mid \alpha <\lambda ⟩$ 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 $I[\lambda ]$,$$S:=\{\delta \in E_{\kappa }^{\lambda }\mid \mathrm{\exists }{A}_{\delta }\subseteq \delta (\text{otp}(A_{\delta })<\delta =sup(A_{\delta }),\mathrm{\forall }\gamma <\delta \mathrm{\exists }\alpha <\delta [A_{\delta }\cap \gamma \in {\mathcal{D}}_{\alpha }])\}.$$

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 $\overrightarrow{N}:=⟨N_i\mid i<{\kappa }^{++}⟩$ be a sequence of elementary submodels of $({\mathcal{H}}_{\theta },\in )$ such that for all $i<{\kappa }^{++}$:

• $|N_i|={\kappa }^{++}$, with ${\kappa }^{++}\subseteq N_i$;
• $⟨N_j\mid j\le i⟩\in N_{i+1}$;
• $\overrightarrow{M},\overrightarrow{C},D,\lambda \in N_i$;
• $N_i=\bigcup _{j<i}{N}_j$ 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 $f:{\kappa }^{++}\to D$ by letting: $$f(i):=sup(N_i\cap \lambda ),(i<{\kappa }^{++}.)$$
By the second and fourth defining properties of $\overrightarrow{N}$, we get that $f$ is increasing and continuous, and that $f\upharpoonright \tau \in N_{\tau +1}$ for all $\tau <{\kappa }^{++}$. Put $ϵ:=sup(f[{\kappa }^{++}])$. Since ${\kappa }^{++}+ϵ+1\subset M_{ϵ+1}$, we get (by elementarity) the existence of some strictly-increasing and cofinal function $g:{\kappa }^{++}\to ϵ$ that lies in $M_{ϵ+1}$.
Since $f$ and $g$ are both continuous and cofinal in $ϵ$, a standard back-and-fourth argument yields the existence of a club $c\subseteq {\kappa }^{++}$ for which $f\upharpoonright c=g\upharpoonright c$. 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 $\delta \in D\cap E_{\kappa }^{\lambda }$, 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 $M_{ϵ+1}$, since $g,\overrightarrow{C},\beta ,\tau \in M_{ϵ+1}$ ;
• $A_{\delta }\cap \gamma =(f\upharpoonright \tau )[c_{\beta }]$, and the latter belongs to $N_{\tau +1}$ since $f\upharpoonright \tau ,\overrightarrow{C},\beta \in N_{\tau +1}$.

Denote $A:=A_{\delta }\cap \gamma$. Then $A\in M_{ϵ+1}$, and $A,\overrightarrow{M}\in N_{\tau +1}$. Consequently: $$N_{\tau +1}\models \mathrm{\exists }\alpha <\lambda (A\in M_{\alpha }).$$ 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, $A_{\delta }\cap \gamma \in \bigcup _{\alpha <\delta }{\mathcal{D}}_{\alpha }$. $\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}$

So, $S$ is an example of a stationary subset of $E_{\kappa }^{\lambda }$ that belongs to $I[\lambda ]$. $\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}$

1. Notice that in the above proof, we could have replaced ${\kappa }^{++}$ with any regular cardinal $\mu <\lambda$ for which $E_{\kappa }^{\mu }$ 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 $S\subseteq E_{\kappa }^{\lambda }$ such that:

• $S\in I[\lambda ]$, and $S$ reflects in the following sense:
• $\{\delta \in E_{\mu }^{\lambda }\mid S\cap \delta \text{ is stationary}\}$ is stationary in $\lambda$. $\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}$

2. The proof may be adapted to show that if $\kappa <{\kappa }^{+}<\lambda$ are regular, then there exists a sequence $⟨C_{\delta }\mid \delta \in E_{\kappa }^{\lambda }⟩$, 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 $C_{\delta }\cap \gamma \in {\mathcal{D}}_{\alpha }$.