Within mathematical logic and model theory, classification theory originates from Shelah’s program for counting the number of models of a first-order theory with a given cardinality. Modern classification theory has evolved into a startlingly rich system of interactions between the syntactic properties of theories, which describe combinatorial patterns within formulas, and the structural properties of models, which generalize geometric phenomena from throughout mathematics. These interactions have allowed us to understand first-order theories at a far higher level of complexity than originally expected. Since Džamonja and Shelah first defined their classical hierarchy of syntactic properties of theories, one of the most troubling problems in classification theory remains to understand the most complex levels of this hierarchy, especially the nth strict order property or SOP_n; while these properties (and their negations) often have deep structural consequences for models of first-order theories, we are still trying to make sense of the full picture.

The project I’ll talk about starts with the observation that the properties SOP_n, originally defined for integer values of n, could just as well have been defined for non-integer values. This prompts the question of whether the SOP_r hierarchy for real values of r is distinct from the SOP_n hierarchy for integer values of n, or in other words, whether there are any properties "missing" from the original integer-valued hierarchy. As I’ll demonstrate in my talk, examining the fine structure in between the levels of this original hierarchy reveals intricate interactions between classification theory and combinatorics, and I’ll discuss the problem of distinctness of the real-valued and integer-valued hierarchies both at the level of logic and at a more combinatorial level.