Logic Seminar
Scott Mutchnik
UIC
Non-integral classification theory and classical classification theory
Abstract: Traditionally, we’ve thought of Shelah’s n-strict order property (or NSOP_n) hierarchy as being defined for positive integer values of n. However, this all changed recently due to the observation that the properties NSOP_r make sense even for non-integer values of r, including all real values of r not less than 3. It is open whether any of the properties NSOP_r for r real are distinct from all of Shelah’s original properties NSOP_n for integer values of n, raising the troubling, yet tantalizing, possibility that the classical classification-theoretic hierarchy is incomplete. Recent work demonstrates the complexity of this situation even at the purely combinatorial level.
In this talk, we discuss the implications of these new, real-valued strict order properties, and the questions surrounding them, for the more longstanding problems on the status of the classical classification-theoretic properties.
We first address the question of whether NSOP_2 is equal to NSOP_3. We discuss how the real-valued NSOP_r hierarchy reveals fine structure in between NSOP_2 and NSOP_3. Specifically, we show that it even makes sense to extend the definition of NSOP_r to r strictly in between 2 and 3, in the sense that NSOP_2 will still imply NSOP_r for r greater than 2. This observation applies an earlier result of the speaker than NSOP_1 is equal to NSOP_2.
We then turn to the question of whether every NTP_2, NSOP_(n + 1) theory is NSOP_n for n an integer at least 3. We give a precise sense in which, if the real-valued NSOP_r and integer-valued NSOP_n hierarchies coincide on sufficiently general grounds, this identity involving NTP_2 must be true. To a first approximation, this gives an apparent dichotomy between the possibility that the properties NSOP_r for real-valued r really are new properties of theories, and the resolution of main cases of the NTP_2-NSOP_n problem. If time permits, we will discuss even weaker, asymptotic conditions for the resolution of this problem, as well as the implications of the techniques involved for NTP_2 graph theory.
Tuesday January 20, 2026 at 3:00 PM in 636 SEO