Scales from I[\lambda]

I want to make good on the promise of linking up scales to square-like principles that I made a couple of posts ago. In particular I want to sketch how one produces scales using the machinery of I[\lambda]. If we believe that I[\lambda] contains a lot of stationary sets, then most of the work was actually done by working through Claim 2.6A of Chapter 1 from Cardinal Arithmetic:

Theorem: Let I be an ideal on a set A of regular cardinals with \kappa>|A|^+ regular. Assume that:

  1. \lambda>\kappa^{++} is regular such that there is some stationary S\subseteq S^\lambda_{\kappa^+} which has a continuity condition \bar C;
  2. \vec f=\langle f_\alpha : \alpha<\lambda\rangle is a sequence of functions from A to the ordinals;
  3. \vec f obeys \bar C.

Then \vec f has a \leq_I-exact upper bound.

We just have to show that it’s we can construct sequences which obey continuity conditions, and then there’s a relatively standard argument which allows us to move immediately from an exact upper bound for an appropriate sequence to a scale. Let’s briefly recall what continuity conditions are, and how they relate to I[\lambda]:

Theorem (Shelah): Let \lambda be a regular cardinal. Then for S\subseteq \lambda, we have S\in I[\lambda] if and only if there is a sequence \bar C= \langle C_\alpha : \alpha<\lambda\rangle and a club E\subseteq\lambda such that:

  1. Each C_\alpha is a closed (but not necessarily unbounded) subset of \alpha;
  2. if \beta\in nacc(C_\alpha) then C_\beta=\alpha\cap C_\alpha;
  3. If \delta\in S\cap E, then \delta is singular, and C_\delta is a club subset of \delta of order type cf(\delta).

The sequence \bar C is called a continuity condition for S, and functions somewhat like a square sequence over S. The major difference is that we only have coherency on the non-accumulation points which is a significant weakening, but still allows them to be useful enough. So the fact that I[\lambda] contains a stationary subset of S^\lambda_\kappa for every regular\kappa such that \kappa^{++}<\lambda can be regarded as a weak fragment of square which is true in ZFC.

Since we want to produce scales, our focus will be on I[\mu^+] for \mu singular. In particular, we have that for every regular \kappa<\mu, there is a stationary S\subseteq S^{\mu^+}_{\kappa} such that S\in I[\mu^+]. Further, we also have that if \mu is strong limit, then S^{\mu^+}_{\leq cf(\mu)}\in I[\mu^+], though this won’t be particularly important to us (a proof of this can be found in Todd Eisworth’s Handbook chapter for the interested).

First, we show how to produce sequences that obey continuity conditions. So, fix a set of regular cardinals A\subseteq \mu cofinal in \mu with ot(A)=cf(\mu) and such that |A|^+<\min A. Following standard notation, we will let J^{bd}[A] denote the ideal of bounded subsets of A.

We first show that \prod A/J^{bd}[A] is \mu^+ directed. To see this, note that is suffices to show that \prod A/J^{bd}[A] is \mu-directed. For any set F\subseteq \prod A/J^{bd}[A] such that |F|=\mu, then we can rewrite F=\bigcup_{i<cf(\mu)}F_i where |F_i|<\mu. From there, we use \mu-directedness to produce bounds f_i\in\prod A/J^{bd}[A] for each F_i, and then bound \{f_i : i<cf(\mu)\} by f\in \prod A/J^{bd}[A]. For \mu-directedness, let F\subseteq \prod A/J^{bd}[A] be such that |F|<\mu. Then let f be defined by f(a)=\sup \{g(a) : g\in F\}, and note that f is defined almost everywhere since each a\in A is regular and A is cofinal in \mu. Clearly then f+1 is a <_{J^{bd}[A]}-upper bound for F.

Now let \kappa<\mu be regular, and let S\subseteq S^{\mu^+}_\kappa be such that S\in I[\lambda] with \bar C=\langle C_\alpha : \alpha<\lambda\rangle a witnessing continuity condition. We first recall what it means for a sequence \vec f=\langle f_\alpha : \alpha<\mu^+\rangle to obey \bar C:

Definition: We say \vec f weakly obeys \bar C if:

If \alpha<\lambda is such that ot(C_\alpha)\leq \kappa, then for each \beta\in nacc(C_\alpha), we have f_\beta(i)<f_\alpha(i) for each i<\kappa.

This definition looks like a weakening of the one originally given, but it’s all that was required for the proof of Claim 2.6A to go through. Now we inductively define a sequence \vec f which obeys \bar C as follows. We first let f_0 be any function in \prod A/J^{bd}[A]. At stage \alpha, we suppose that f_\beta has been defined for each \beta<\alpha. We let f_\alpha' be a <_{J^{bd}[A]}-upper bound for \{f_\beta : \beta<\alpha\} as guaranteed by \mu^+-directedness. If C_\alpha is empty or ot(C_\alpha)>\kappa, then we just set f_\alpha=f_\alpha'. Otherwise, we let f_\alpha be defined by setting f_\alpha(a)=\max\{f_\alpha'(a), \sup_{\beta\in C_\alpha}f_\beta(a)\}+1. Note that since \kappa<\mu, we know that f_\alpha(a) is defined almost everywhere. It is also clear by construction that \langle f_\alpha : \alpha<\mu^+\rangle is a <_{J^{bd}[A]}-increasing sequence which weakly obeys \bar C and so we are done.

I also want to note that we could have started with a fixed sequence \vec g=\langle g_\alpha : \alpha<\mu^+\rangle, and asked that not only \vec f weakly obey \bar C, but also that g_\alpha<f_{\alpha+1} for each \alpha<\mu^+. So for example if \vec g weakly obeyed some other continuity condition \bar D, then the resulting \vec f would weakly obey both \bar D and \bar C. Further, if \vec f obeys a continuity condition for a stationary subset of S^{\mu^+}_\kappa, then one can show that the exact upper bound f produced by Claim 2.6A satisfies:

\{\ a\in A : cf(f(a))<\kappa\}\in J^{bd}[A].

 Okay, with all of this in hand, we can produce a <_{J^{bd}[A]}-increasing sequence of functions \vec f=\langle f_\alpha : \alpha<\mu^+\rangle in \prod A/J^{bd}[A] with the following properties:

  1. \vec f has an exact upper bound f;
  2. For every regular \kappa with \min(A)\leq \kappa<\mu, the set \{a\in A : cf(f(a))<\kappa\}\in J^{bd}[A].

So, we then have that sequence \vec f witnesses that \prod_{a\in A} f(a)/J^{bd}[A] has true cofinality \mu^+. Now, by possibly altering f on a null set we may assume \min \{f(a):a\in A\}>|A|. Let B=\{cf(f(a)) : a\in A\}, and note by condition 2, that B is cofinal in \mu and has order type cf(\mu). A relatively standard argument then allows us to conclude that tcf (\prod B/J^{bd}[B])=\mu^+, and letting the witnessing sequence be \vec h=\langle h_\alpha : \alpha<\lambda\rangle, we get that (B,\vec h) is a scale on \mu.

Honestly, parts of this sketch are pretty bare-bones, but the idea was to show that Claim 2.6A (once appropriately modified) is the only really difficult part behind producing scales. In fact, that claim plays the same role that the trichotomy theorem does for the theory of exact upper bounds. In particular, it shows us that, provided we can construct certain sorts of sequences, we can then get nice exact upper bounds. It just turns out that these sequences, once we have enough of the I[\lambda] combinatorics in hand, are relatively easy to produce. From there, it’s just standard arguments showing that we really only need exact upper bounds to do a lot of the things we want. An alternative approach to exact upper bounds (outside of I[\lambda] or trichotomy) is also furnished through what Abraham and Magidor call (*)_\kappa. It turns out that (*)_\kappa is incredibly similar to having continuity conditions for a stationary subset of S^\lambda_\kappa lying around.

Overall though, all three of these approaches are doing roughly the same thing.

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