# Sh:800

- Shelah, S.
*On complicated models and compact quantifiers*. Preprint. -
Abstract:

What we do can be looked at as:finding and classifying compact second order logic quantifiers on automorphisms of definable models of \psi which are already definable,

building a model M such that if we define in M a model N = N_{M, \bar \psi} of \psi, then any automorphism of N is inner (that is, first order definable in M) at least in some respect.

This can be looked at as classifying the \psi-s; so for more complicated \psi-s we have fewer such automorphisms.

As a test case, we consider the specific examples of “the model completion of the theory of triangle-free graphs."

More elaborately, we look here again at building models M with second order properties. In particular, M such that every isomorphism between two interpretations of a theory t in M is definable in M or at least is “somewhat" definable (e.g. having a dense linear order, saying this holds for a dense family of intervals). For transparency we can concentrate on t-s of finite vocabulary. If we restrict ourselves to finite t-s, this implies that we get a compact logic when we add to first order logic the second order quantifiers on isomorphisms from one interpretation. We already know this in some instances (e.g. t the theory of Boolean Algebras or the theory of ordered fields) but here we try to analyze a general t. Hence, at least for the time being, we try to sort out what we can get by forcing rather than really proving it (in ZFC).

We may consider the question: for a given T if there is an \kappa-iso-rigid-model of T (so \kappa-full), then our constructions give one. For more details, see the introduction to [Sh:384].

- Version 2023-09-10 (51p)

@article{Sh:800, author = {Shelah, Saharon}, title = {{On complicated models and compact quantifiers}} }