Why are stationary limits so ubiquitous when studying large cardinals?

Question

While studying large cardinals, I have frequently noticed the following phenomenon: If X and Y are two different types of large cardinals, then every cardinal of type X is a stationary limit of cardinals of type Y.

So, Mahlo cardinals are stationary limits of inaccessible cardinals by definition. But weakly compact cardinals are stationary limits of Mahlos, measurable cardinals are stationary limits of loads of large cardinals (Mahlo, weakly compact, Ramsey etc), supercompact cardinals are stationary limits of measurables and so on.

I find it somewhat surprising that this kind of "stationary limit" relationship occurs so often between different types of large cardinal. Is there some theorem which is being implicitly used when comparing different types of large cardinal ?

Answer 1

Strictly speaking, it isn't a universal phenomenon. For example, stationary limits do not arise in the hyperinaccessibility hierarchy between inaccessible and Mahlo cardinals, nor in the hierarchy of worldy cardinals nor in hyperworldy nor hyper-otherwordly.

A cardinal is $\alpha$-inaccessible if it is inaccessible and for every $\beta<\alpha$ it is a limit of $\beta$-inaccessible cardinals. Thus, the $0$-inaccessible cardinals are exactly the inaccessible cardinals, and the $1$-inaccessibles are the inaccessible limits of inaccessibles, and the $2$-inaccessibles are the inaccessible limits of inaccessible limits of inaccessible cardinals, and so on.

Meanwhile, for $\kappa$ to be hyperinaccessible means that $\kappa$ is $\kappa$-inaccessible, and then one defines $\alpha$-hyperinaccessible and so on. One proceeds to the hyper-hyper-inaccessibles and the hyper${}^\alpha$-inaccessible and eventually to the richly inaccessible cardinals, and the utterly inaccessible cardinals and the deeply inaccessible cardinals, and so on.

For $\kappa$ to be a stationary limit of inaccessible cardinals would imply that $\kappa$ is hyperinaccessible and richly so, and deeply so, and utterly so, and so on, already swamping this entire hierarchy. Thus, the hierarchy of inaccessibility is a little more refined and slower than the hierarchy requiring stationary limits and stationary reflection.

A similar distinction plays out lower down with the worldy cardinals, and the hierarchy of hyperworldliness and otherwordliness and hyperotherwordliness. To insist upon stationary limits here would be to jump immediately up to Mahloness.

In summary, my answer is that stationary limits are a convenient way of jumping higher in the large cardinal hierarchy, but there are also many well-studied ways of jumping not quite so high, and there is a rich hierarchy of lesser notions falling short of stationary limits.

Answer 2

In many cases where $\kappa$ having large cardinal property X implies the existence of a cardinal $\bar{\kappa}< \kappa$ having large cardinal property Y, this can be proved by the following type of argument:

There is an elementary embedding $j$ from $V$ (or something smaller than $V$, e.g. in the case of weakly compact or strongly unfoldable cardinals) to a transitive model $M$ such that $j$ has critical point $\kappa$ and $M$ satisfies "$\kappa$ has large cardinal property Y."

The existence of such a cardinal less than $j(\kappa)$ then reflects downward by the elementarity of $j$ to give a cardinal $\bar{\kappa} < \kappa$ having large cardinal property Y in $V$.

This kind of argument can always be extended to show that the set of such $\bar{\kappa}$ is stationary in $\kappa$, using the fact that for any club $C \subset \kappa$, we have $\kappa \in j(C)$.