Continuity Conditions and I[\lambda]

In the previous post, I worked through a result of Shelah’s that allows us to produce exact upper bounds from continuity conditions. I want to use this post to briefly talk about where these things are coming from. As usual, for regular \kappa<\lambda, we denote:

S^\lambda_\kappa=\{\alpha<\lambda : cf(\alpha)=\kappa\}

Further, for a set C of ordinals, we denote:

acc(C)=\{\alpha\in C : \alpha=\sup (\alpha\cap C)\}

nacc(C)=C\setminus acc(C).

Definition: Let \lambda be a regular cardinal, and let \vec a=\langle a_\alpha : \alpha<\lambda\rangle be a sequence of elements of [\lambda]^{<\lambda}. Given a limit ordinal \delta<\lambda, we say that \delta is approachable with respect to \vec a if there is an unbounded A\subseteq \delta of order type cf(\delta) such that every initial segment of A is enumerated prior to stage \delta. More precisely:

For every \alpha<\delta, there exists a \beta<\delta such that A\cap\alpha=a_\beta.

Definition: Let \lambda be a regular cardinal and define I[\lambda] to be the collection of S\subseteq\lambda such that there is a sequence \vec a=\langle a_\alpha :\alpha<\lambda of elements of [\lambda]^{<\lambda} and a club E\subseteq\lambda such that every \delta\in E\cap S is singular and approachable with respect to \vec a.

So the idea is that an ordinal is approachable with respect to some sequence above if there is some unbounded set whose initial segments get captured in a timely manner. A set of ordinals S is in I[\lambda] if almost every (modulo clubs) ordinal in S is uniformly approachable, and this uniformity is captured by a single sequence.

Proposition: I[\lambda] is a (possible improper) normal ideal over \lambda.

One thing to note is that, if \lambda\in I[\lambda], then there is a club of singular ordinals which are all approachable by way of a single sequence \vec a. So one can imagine that if this is indeed possible, then \lambda must have some nice combinatorial structure. It turns out that this is indeed possible, and this yields a square-like principle. The following alternative characterization of I[\lambda] makes this more evident.

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).

With this in hand, we now note that I[\lambda] is actually quite large.

Theorem (Shelah): Suppose that \kappa^+<\sigma<\lambda for regular cardinals \kappa,\sigma,\lambda. Then there is a stationary S\subseteq S^\lambda_\kappa in I[\lambda] such that S\cap \theta is stationary for stationarily-many \theta\in S^\sigma_\kappa.

Corollary: Suppose that \kappa^{++}<\lambda for regular \lambda,\kappa. Then there is a stationary S \subseteq S^\lambda_\kappa such that S\in I[\lambda].

Thus, we see that a continuity condition is just a witness that these particular stationary sets live in I[\lambda]. Beyond giving us a link between squares and scales (which I want to fill out in the next post), I[\lambda] is interesting in its own right. I won’t get into it much for now, but Todd Eisworth’s handbook chapter has a nice exposition on I[\lambda] and its applications.


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