Large cardinals in AECs: back to the road.

(This was originally posted by Andrés Villaveces in avn va – עולם)

The connections between Model Theory and Large Cardinals have recently been given a very interesting boost (and twist) in the work of Will Boney, a graduate student at Carnegie Mellon University.

Boney builds his results on a line originally opened in the papers by Makkai and Shelah ([MaSh:285] Categoricity of theories in , with a compact cardinal — Annals Pure and Applied Logic 47 (1990) 41-97) and Kolman and Shelah ([KoSh:362] Categoricity of Theories in , when is a measurable cardinal. Part 1 — Fundamenta Math 151 (1996) 209-240) and a follow-up by Shelah ([Sh:472] Categoricity of Theories in , when is a measurable cardinal. Part II — Fundamenta Math 170 (2001) 165-196): use of strongly compact ultrafilters to get relatively strong “compactness-like” properties, uses of measurable embeddings to get reasonable independence notions.

But Boney seems to go much further: by taking seriously the consequences of the presence of the embeddings and ultrafilters, he provides

• a Łoś Theorem for Abstract Elementary Classes – under strongly compact cardinals : closure under (-complete) ultrapowers of models in the AEC, connections between realization of types inside monster models of the class, in both the ultrapower and the “approximations”
• a duality (under categoricity at some – no large card. hypotheses here) between tameness and type shortness. Tameness can be regarded as a strong coherence property over domains, for types, type shortness is the analog but switching the coherence from domains of the types to realizations of the type
• under a proper class of strongly compact cardinals, nothing less than a version of the Shelah Conjecture (eventual Categoricity Transfer from a Successor, for AECs) – Boney essentially gets tameness and type shortness of such classes; the meat of the proof really is there
• under smaller large cardinals (measurables, weakly compacts, etc.), he gets different, weaker results, by using ultrapower techniques allowing him to gain control of properties of the class – reflection properties become crucial for model theoretic properties other than categoricity (stability, amalgamation, uniqueness of limit models!)
• finally, his results open up many questions that puzzle one: what is the actual strength of Shelah’s Conjecture? where can the counterexamples started by Hart and Shelah (and refined by Baldwin and Kolesnikov) be pushed?

Here is also a skeleton for a Seminar Lecture in our Logic Seminar in Bogotá (in Spanish).