# 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.