One natural notion of "random (countably infinite) L-structure" is a probability measure on the space of L-structures with domain omega which is invariant and ergodic for the natural action of the symmetric group Sym(omega) on this space. We call such a measure an ergodic structure. The most famous example of an ergodic structure is the Erdos-Renyi random graph model on domain omega, which gives measure 1 to the isomorphism type of the Rado graph. Ergodic structures also arise naturally as limits of sequences of finite structures which are convergent in the appropriate sense, generalizing the graph limits of Lovasz and Szegedy. Some ergodic structures (like the Erdos-Renyi random graph model) are almost surely isomorphic to a single countable structure (like the Rado graph), and the countable structures which arise in this way have been completely characterized by Ackerman, Freer, and Patel. In this talk, we will consider properly ergodic structures, those which do not give measure 1 to any single isomorphism type. What do properly ergodic models "look like"? To address this question, we develop an analogue of the Scott rank for ergodic structures, which leads to a precise characterization of those sentences of the infinitary logic L_{omega_1,omega} which admit properly ergodic models. This is joint work with Ackerman, Freer, and Patel.

https://us06web.zoom.us/j/87929792112?pwd=WWNnTkEwOFQwRDdOVXNaK1UzZHBFZz09
Meeting ID: 879 2979 2112
Passcode: 496589