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 , we denote:
Further, for a set of ordinals, we denote:
Definition: Let be a regular cardinal, and let be a sequence of elements of . Given a limit ordinal , we say that is approachable with respect to if there is an unbounded of order type such that every initial segment of is enumerated prior to stage . More precisely:
For every , there exists a such that .
Definition: Let be a regular cardinal and define to be the collection of such that there is a sequence of elements of and a club such that every is singular and approachable with respect to .
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 is in if almost every (modulo clubs) ordinal in is uniformly approachable, and this uniformity is captured by a single sequence.
Proposition: is a (possible improper) normal ideal over .
One thing to note is that, if , then there is a club of singular ordinals which are all approachable by way of a single sequence . So one can imagine that if this is indeed possible, then 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 makes this more evident.
Theorem (Shelah): Let be a regular cardinal. Then for , we have if and only if there is a sequence and a club such that:
- Each is a closed (but not necessarily unbounded) subset of ;
- if then ;
- If , then is singular, and is a club subset of of order type .
With this in hand, we now note that is actually quite large.
Theorem (Shelah): Suppose that for regular cardinals . Then there is a stationary in such that is stationary for stationarily-many .
Corollary: Suppose that for regular . Then there is a stationary such that .
Thus, we see that a continuity condition is just a witness that these particular stationary sets live in . Beyond giving us a link between squares and scales (which I want to fill out in the next post), 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 and its applications.