Sealing is a type of generic absoluteness condition introduced by Woodin that asserts in strong terms that the theory of the universally Baire sets cannot be changed by forcing. generic-LSA is the statement that the Largest Suslin Axiom (LSA) holds in all generic extensions. Under a mild large cardinal hypothesis, we show that Sealing is equiconsistent with generic-LSA. As a consequence, Sealing is weaker than the theory ``ZFC+there is a Woodin cardinal which is a limit of Woodin cardinals". Sealing's consistency being weak represents an obstruction to the current program of descriptive inner model theory. Going beyond this bound in core model induction applications seems challenging and requires us to construct third order objects (subsets of the universally Baire sets). We will state the precise theorems and discuss their impact on current developments of inner model theory. Time allowed, we will talk a bit about how to overcome the the obstructions imposed by Sealing. This is joint work with G. Sargsyan.
The logic of metric structures was introduced by Ben Yaacov, Berenstein , Henson and Usvyatsov. It is a version of continuous logic which allows fruitful model theory for many kinds of metric structures .There are many aspects of this logic which make it similar to first order logic, like compactness, a complete proof system, an omitting types theorem for complete types etc. But when one tries to generalize the omitting type criteria to general (non-complete) types the problem turns out to be essentially more difficult than the first order situation.For instance one can have two types (in a complete theory) that each one can be omitted , but they can not be omitted simultaneously.
The precise structural understanding of uncountably categorical theories given by the Baldwin-Lachlan theorem is known to fail in continuous logic in the context of inseparably categorical theories. The primary obstacle is the absence of strongly minimal sets in some inseparably categorical theories. We will develop the concept of strongly minimal sets in continuous logic and discuss some common conditions under which they are present in an ω-stable metric theory. Finally we will examine the extent to which we recover the Baldwin-Lachlan theorem in the presence of strongly minimal sets.
Metric structures are like first-order structures except that the formulas take truth values in the unit interval, and instead of equality there is a distance predicate with respect to which every function and predicate is uniformly continuous. Pre-metric structures are similar the distance predicate is only a pseudo-metric. In recent years the model theory of metric and pre-metric structures has been successfully developed in a way that is closely parallel to first order model theory, with many applications to analysis.
We consider general structures, where formulas still have truth values in the unit interval, but the predicates and functions need not be continuous with respect to a distance predicate. It is shown that every general structure can be expanded to a pre-metric structure by adding a distance predicate that is a uniform limit of formulas. Moreover, any two such expansions have the same notion of uniform convergence. This can be used to extend almost all of the model theory of metric structures to general structures in a precise way. For instance, the notion of a stable theory extends in a natural way to general structures, and the main results carry over.