April 21
James Hanson,
University of Maryland
How bad could it be? The semilattice of definable sets in continuous logic
Continuous first-order logic is a generalization of discrete first-order logic suited for studying structures with natural underlying metrics, such as operator algebras and $\mathbb{R}$-trees. While many things from discrete model theory generalize directly to continuous model theory, there are also new subtleties, such as the correct notion of 'definability' for subsets of a structure. Definable sets are conventionally taken to be those that admit relative quantification in an appropriate sense. An easy argument then establishes that the union of definable sets is definable, but in general the intersection of definable sets may fail to be. This raises the question of which semilattices arise as the partial order of definable sets in a continuous theory.
After giving an overview of the basic properties of definable sets in continuous logic, we will give a largely visual proof that any finite semilattice (and therefore any finite lattice) is the partial order of definable sets in some superstable continuous first-order theory. We will then discuss a partial extension of this to certain infinite semilattices.