Logic Workshop

November 22
Alex Kruckman, Wesleyan University
The complexity of ages admitting a universal limit structure

An age is a hereditary class of finitely generated structures with the joint embedding property which is countable up to isomorphism. If $K$ is an age, a $K$-limit is a countable structure $M$ such that every finitely generated substructure of $M$ is in $K$. A $K$-limit $U$ is universal if every $K$-limit embeds in $U$. It is well-known that $K$ has the amalgamation property (AP) if and only if $K$ admits a homogeneous limit (the Fraïssé limit), which is always universal. But not every age with a universal limit has AP. We show that, while the existence of a universal limit can be characterized by the well-definedness of a certain ordinal-valued rank on structures in $K$, it is not equivalent to any finitary diagrammatic property like AP. More precisely, we show that for ages in a fixed sufficiently rich language $L$, the property of admitting a universal limit is complete coanalytic. This is joint work with Aristotelis Panagiotopoulos.