Shelah Cardinals and Modern Set Theory

Question

Do Shelah cardinals play an essential role in any modern set theory results or was the concept basically made obsolete by Woodin cardinals?

Answer 1

I do not think Shelah cardinals are well understood yet, and they do not seem to have been studied much. They are beyond Woodin cardinals, so they are beyond the current threshold of "true understanding" provided by inner model theory. For this reason, currently there can be no known results for which Shelah cardinals are optimal consistencywise. But the situation is a bit worse:

There has been a lot of work understanding Woodin cardinals. Some proofs would simplify from the use of Shelah cardinals in the assumptions, but at the loss of technical insights, so this direction has not really been explored. (For example, Shelah's proper forcing book has a chapter on strong properties of ideals on $\omega_1$. Some of the results there use the existence of Shelah cardinals. In the ones I've studied, the "right" assumption is the existence of Woodin cardinals, although usually some not entirely routine work is involved in replacing the hypotheses. Nowadays, we directly work from the Woodin cardinals rather than from less than optimal assumptions.) And the truth is we do not have good candidate statements that we expect are equiconsistent with Shelah cardinals.

This is very different from how the situation was with Woodin cardinals for years. For example, the existence of a saturated ideal on $\omega_1$ was (correctly) expected to be equiconsistent with the existence of one Woodin cardinal, but the proof took about 20+ years to arrive from the moment we were in a position to expect something like this.

Shelah cardinals appear in some proofs, of course. But not as an essential tool. The examples I know in inner model theory are all of the form: We want to show some situation cannot happen in a certain scenario where we have some anti-large cardinal assumptions in play. As part of the argument, we produce some extenders and show that they witness some cardinals are Shelah, thus being beyond the anti-large cardinal framework we were working with. Unfortunately, these arguments are embedded in a technical setting that makes it unlikely they will survive once we develop inner model theory well enough to reach Shelah cardinals. (What I mean is, the arguments use assumptions about comparison arguments, about sequences of extenders, and so on, that are only true if there are no Woodin cardinals, or something similar).

There do not seem to be too many papers exploring the combinatorics of Shelah cardinals yet. The following are about the only I know:

1. MR1913018 (2003e:03104): Ernest Schimmerling. "Woodin cardinals, Shelah cardinals, and the Mitchell-Steel core model", Proc. Amer. Math. Soc. 130 (2002), no. 11, 3385–3391.

2. MR1269896 (95d:03095): Toshio Suzuki. "Witnessing numbers of Shelah cardinals", Math. Logic Quart. 39 (1993), no. 1, 62–66.

Of course, I expect the situation will change as we understand better inner model theory.