The Spencinar has been continuing as usual since the Thanksgiving break. Two weeks ago, Spencer talked about diamond principles. I'm working on these notes still. Last week, I talked about precipitousness of the nonstationary ideal (the subject of the learning project #1).
I'm about to retire this old seminar notebook, so I thought I would record here a Spencinar that I gave over the summer about the approachability property. This will also provide some useful background for future posts.
This material comes mostly from Eisworth's friendly handbook chapter, where I was originally inspired to pursue the line of research that I'm on. I'm going to focus on the following result of Shelah's (interpreted through Eisworth's chapter):
Main Theorem: If $\aleph_\omega$ is a strong limit, then $\rm{Refl}(\aleph_{\omega+1})$ implies the approachability property at $\aleph_\omega$.
This result is unexpected since approachability is a rather weak "non-reflection" principle. It's interesting since Magidor's model of $\rm{Refl}(\aleph_{\omega+1})$ satisfies approachability property at $\aleph_\omega$, and in fact uses this to show that stationarity is preserved after forcing. It's also interesting since the proof seems to use special properties of $\aleph_\omega$, giving an example of the phenomenon that this singular cardinal in particular seems to have a lot of ZFC structure. This is nice to understand, because there is an important open question asking whether the failure of singular cardinals hypothesis at $\aleph_\omega$ implies approachability (and this is consistently false for larger singular cardinals).
I. $\mathrm{AP}_\mu$: review.
Shelah defined the ideal $I[\lambda]$ for a regular cardinal $\lambda$ by:
$S\subseteq I[\lambda]$ iff there is
- a sequence $\langle a_\alpha:\alpha<\lambda\rangle$ of bounded subsets of $\lambda$ and
- $C\subseteq \lambda$ club,
so that for every $\delta\in S\cap C$, there is $A_\delta\subseteq \delta$ unbounded of order-type $\rm{cf}(\delta)$ so that
- $\{A\cap \beta:\beta<\delta\}\subseteq \{a_\beta:\beta<\delta\}$ (or in words, all initial segments of $A_\delta$ are enumerated in $\bar{a}$ at a stage before $\delta$.
Such $\delta$ is said to be
approachable with respect to $\bar{a}$. It can be shown (cf. Eisworth's handbook chapter) that $I[\lambda]$ is a normal ideal on $\lambda$.
For $\mu$ a singular cardinal, the
approachability property $\rm{AP}_\mu$ is the statement $\mu^+\in I[\mu^+]$. (Some other authors denote this by $\rm{AP}_{\mu^+}$. Restating the definitions, this means there is $\bar{a}$ and a club of $\delta<\mu^+$ of points which are approachable with respect to $\bar{a}$.
2. Elementary substructures
We will define a $\lambda$-approximating sequence to be a $\subseteq$-increasing sequence $\mathcal{M}=\langle M_\alpha:\alpha<\lambda\rangle$ of elementary substructures of $(H(\theta); \in, \triangleleft)$ for sufficiently large regular $\theta$ so that $\lambda\in M_0$ and for all $\alpha$, $|M_\alpha|<\lambda$ and $M_\alpha\cap \lambda\in \lambda$. Most importantly, we require that $\langle M_\alpha:\alpha\le \beta\rangle\in M_{\beta+1}$. This is a slight strengthening of the concept of "IA sequence" which we saw in the learning project.
If $\mathcal{M}$ is a $\lambda$-approximating sequence, then define $S[\mathcal{M}]$ to be the set of $\delta<\lambda$ such that $M_\delta\cap\lambda=\delta$, there is a cofinal $a\subseteq \delta$ of order-type $\rm{cf}(\delta)$ such that every inital segment of $a$ is in $M_\delta$.
This gives a useful characterization of approachability. $I[\lambda]$ is just ideal generated by nonstationary sets together with sets of the form $S[\mathcal{M}]$. For a given $\bar{a}$, let $\mathcal{M}$ be a $\lambda$-approximating sequence with $\bar{a}\in M_0$ and $E\subset\lambda$ be club so that $M_\alpha\cap\lambda=\alpha$ for all $\alpha\in E$. In this club, the set of points approachable with respect to $\bar{a}$ is contained in $S[\mathcal{M}]\cap E$.
Conversely, given a sequence $\mathcal{M}$, let $\bar{a}$ be the sequence of bounded subsets of $\lambda$ in $\bigcup_\alpha M_\alpha$, and $E$ be the club of $\delta$ so that $\{a_\alpha:\alpha<\delta\}$ is an enumeration of the bounded subsets of $M_\delta$.
3. Colorings
We can also rephrase notions of approachability in terms of colorings, given some extra assumptions. Here $S^\lambda_\kappa$ denotes the set of points of cofinality $\kappa$ below $\lambda$.
Theorem 1: Suppose $\kappa<\lambda$ be regular with $2^{<\kappa}<\lambda$. If $d:[\lambda]^2\rightarrow \omega$ and $\mathcal{M}$ is a $\lambda$-approximating sequence containing $\{\kappa, d\}$, then for every $\delta\in S[\mathcal{M}]\cap S^\lambda_\kappa$, there is a cofinal $H\subseteq \delta$ homogeneous for d.
Proof: If $\delta\in S[\mathcal{M}]\cap S^\lambda_\kappa$, let $\{\alpha_i:i<\kappa\}$ cofinal in $\delta$ such that $\{a_i:i<\zeta\}\in M_\delta$ for all $\zeta<\kappa$. By induction on $i<\kappa$, we can define $\epsilon_i,f_i$ so that:
- $\epsilon_0=\alpha_0$.
- $f_i:i\rightarrow \theta$, $f_i(j)=d(\epsilon_j,\delta)$.
- $\epsilon_i$ is the least $\alpha$ so that for all $j<i$, $\alpha>\alpha_j,\epsilon_j$ and $d(\epsilon_j,\alpha)=d(\epsilon_j,\delta)$ for all $j<i$, if such exists. The construction ends if no such exists.
This last condition says that the function $d(\epsilon_i,\epsilon_j)$ depends only on $i$, where $i<j$, and uses $\delta$ as a "reference point". Let $i^*$ be the length of the construction.
We show the construction goes up to $\delta$. By part of the definition of $S[\mathcal{M}]$, $M_\delta\cap\lambda=\delta$. Since $\kappa\in M_\delta$, $M_\delta$ must be $<\kappa$-closed, so it contains $f_i$ for all $i<i^*$ (this uses $2^{<\kappa}<\lambda$). This means the construction above can be done in $M_\delta$ and therefore all $\epsilon_i<\delta$, $\delta$ witnesses condition in 3.
The proof concludes by finding $\xi$ so that $\{i<\kappa:d(\epsilon_i,\delta)=\xi\}$ is unbounded, giving a homogeneous set. $\Box$
This motivates yet another conception of "approachability".
Definition: For $d:[\lambda]^2\rightarrow\chi$ (some $\chi$), define $S(d)$ to be the set of $\delta<\lambda$ for which there is a cofinal $H_\delta\subseteq \delta$ homogeneous for $d$, and $S^*(d)=\lambda-S(d)$.
So $S(d)$ is club for every $d$ if the approachability property holds at the predecessor of $\lambda$ (under the hypotheses of Theorem 1).
Actually, under those hypotheses, the approachability ideal is generated by a single set over the nonstationary ideal. We can see this result in a way by using colorings. It turns out there is a certain class of colorings whose members have "all of the complexity" of the general case.
Definition: $d:[\lambda]^2\rightarrow\omega$ is a coloring as above with $\lambda=\mu^+$. We say $d$ is
normal if $\{\beta<\alpha:d(\beta,\alpha)<i\}|<\mu$ for all $i<\rm{cf}(\mu)$. We say $d$ is
transitive if $d(\alpha,\gamma)\le \mathrm{max}\{d(\alpha,\beta),d(\beta,\gamma)\}$ for all $\alpha<\beta<\gamma<\lambda$.
Normal transitive colorings are not difficult (but a little tricky) to construct. We'll just take this as an unproven fact (this construction was used, for example, in John's Spencinar talk about Martin's Maximum).
Proposition: If $\mu$ is strong limit and $d$ is a normal, transitive coloring of $\mu^+$, then there is $\bar{a}$ so that almost every point of $S(d)$ is approachable with respect to $\bar{a}$.
Proof: Let $\langle a_\xi:\xi<\lambda\rangle$ enumerate all subsets of the form $\{\beta<\alpha:d(\beta,\alpha)<i\}$ in some order. This is fine since $\lambda^{<\lambda}=\lambda$ ($\mu$ strong limit). Then for any $\delta\in S(d)$ of uncountable cofinality there exists a cofinal $H_\delta\subseteq\delta$ homogeneous for $d$. Actually, $d(\beta,\delta)$ is bounded in $\omega$: by transitivity, if $\beta_1<\beta_2<\delta$, then $d(\beta_1,\delta)\le \max\{i,d(\beta_2,\delta)\}$, and since $\delta$ has uncountable cofinality and $d(\beta,\delta)$ is increasing in $\beta\in H_\delta$ (if above $i$), then $d(\beta,\delta)$ is bounded. So almost every point of $S(d)$ is approachable. $\Box$
So the proposition gives a really nice characterization of approachability property under the hypotheses of Theorem 1: $\mathrm{AP}_\mu$ holds iff there exists (equivalently for all) normal transitive colorings, $S(d)$ contains a club of points of uncountable cofinality.
4. Proof of Main Theorem
Now we start working towards the proof of the theorem. The first result is that $S^*(d)$ can only reflect in itself.
Lemma 2: If $S^*(d)\cap \alpha$ is stationary in $\alpha$, then $\alpha\in S^*(d)$.
Proof: Otherwise $\alpha\in S(d)$, so let $H\subset \alpha$ be cofinal in $\alpha$ and homogeneous for $d$. Then every limit point of $H$ must be in $S(d)$ by the definition of $S(d)$, so $S^*(d)$ is disjoint from a club in $\alpha$. $\Box$
So if the approachability property fails, then there is a coloring $d$ (namely, any normal transitive one) for which $S^*(d)\cap \mathrm{cof}(>\omega)$ is stationary. Take $n<\omega$ such that $S=S^*(d)\cap S^{\aleph_{\omega+1}}_{\aleph_{n+1}}$ is stationary.
The next claim is the crucial fact giving a non-reflecting stationary set.
Claim: $S$ cannot reflect at points of cofinality $>2^{\aleph_n}$.
Proof of Claim: Suppose $S$ reflects at $\tau$. If $\tau>2^{\aleph_n}$, then $\tau^{<\aleph_{n+1}}=\tau$. So we can construct a $\lambda$-approximating sequence containing all the relevant parameters which is $<\aleph_{n+1}$ closed. Then for any $\delta\in S^\tau_{\aleph_{n+1}}$ with $M_\delta=\delta$, any $A\subseteq \delta$ of order-type ${\aleph_{n+1}}$ has all of its initial segments in $M_\delta$. This shows that $S^\tau_{\aleph_{n+1}}\in I[\tau]$. This is impossible by Theorem 1, which says almost all points below $\tau$ are in $S(d)$, so $S$ cannot reflect at $\tau$. $\Box$
Now fix $k<\omega$ so that $2^{\aleph_n}=\aleph_{n+k}$. Such exists by $\mu$ strong limit. Define $S_0=S$, and $S_{i+1}$ to be the reflection points of $S_i$. Observe that if $\delta\in S_i$, then $\mathrm{cf}(\delta)\ge \aleph_{n+1+i}$, since the set of points below an ordinal with cofinality less than that ordinal is club. This implies that $S_k$ is empty, so let $i^*<\omega$ be maximal so that $S_{i^*}$ is stationary. Then $S_{i^*}$ does not reflect stationarily often. Hence we have proven the main theorem.
5. Normal scales?
Let's try to apply some of these ideas. This section requires a basic knowledge of pcf theory. Let $\langle f_\alpha:\alpha<\mu^+\rangle$ be a scale.
The "difference function" of the scale is a transitive function $d:[\mu^+]^2\rightarrow\omega$ defined so that $d(\alpha,\beta)$ for $\alpha<\beta$ is the supremum of all $n$ so that $f_\alpha(n)\ge f_\beta(n)$. In light of the previous explorations on approachability, it is natural to ask if it can ever be normal.
Proposition: Normal scales don't exist.
Proof: Let $\vec{f}$ be a scale on $\langle \mu_i:i<\omega\rangle$. Pick $\gamma_1<\mu_1$ so that $B_1=\{\alpha:f_\alpha(1)<\gamma_1\}$ has size $\kappa^+$. Let $A_0$ be a set of $\mu_0$ many $\alpha<\mu^+$ in $B_1$.
If $n>0$, then pick $\gamma_{n+1}$ s.t. $\gamma_{n+1}>\sup_{\alpha\in A_{n-1}} f_\alpha(n+1)$ and $B_{n+1}=\{\alpha\in B_n: f_\alpha(n+1)<\gamma_{n+1}\}$ has size $\mu^+$. Let $A_n$ be a set of $\mu_{n}$ many from $B_{n+1}$.
Then $\bigcup_n A_n$ has size $\kappa$ and is bounded by $n\mapsto \gamma_n$.
