2019/08/29 by Roland Walker, Walker, Roland
Economics, Econometrics and Finance · Mathematics · #03C45 (Primary) 03C52 (Secondary) #Advanced Topology and Set Theory #Combinatorics #Complex Systems and Time Series Analysis #Computer science #Economic theories and models #Epistemology #FOS: Mathematics #Generalization #Geometry #Invariant (physics) #Logic (math.LO) #Mathematical analysis #Mathematical physics #Mathematics #NIP #Orthogonality #Philosophy #Property (philosophy) #Pure mathematics #Rank (graph theory) #Triviality #math.LO #msc:03C45 #msc:03C52
paper · pdf · doi:10.48550/arxiv.1908.11400
32 pages
openalex publication_date 2019/08/29 · arxiv created 2021/10/25 · arxiv updated 2021/10/26 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
Building on Pierre Simon's notion of distality, we introduce distality rank as a property of first-order theories and give examples for each rank m such that 1≤ m ≤ ω. For NIP theories, we show that distality rank is invariant under base change. We also define a generalization of type orthogonality called m-determinacy and show that theories of distality rank m require certain products to be m-determined. Furthermore, for NIP theories, this behavior characterizes m-distality. If we narrow the scope to stable theories, we observe that m-distality can be characterized by the maximum cycle size found in the forking "geometry," so it coincides with (m-1)-triviality. On a broader scale, we see that m-distality is a strengthening of Saharon Shelah's notion of m-dependence.