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.
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.
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.
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