Naturality and Definability III
Mohsen Asgharzadeh, Mohammad Golshani, Saharon Shelah
TL;DR
This work probes the relationship between category-theoretic naturality and model-theoretic definability through the uni-construction framework. It shows that under mild hypotheses naturality can imply definability, and uses forcing with reverse Easton iterations to obtain a global model where every uniformisable uni-construction problem becomes weakly natural, thereby extending Hodges–Shelah's results beyond cardinality and parameter constraints. A key culmination is a uniformity theorem: if a natural construction on a definable class of two-sorted models is representable by a formula, then there exists a uniformly definable class function yielding the corresponding expansion without extra parameters. Collectively, the paper advances the Hodges–Shelah program by bridging multi-sorted model theory, forcing techniques, and automorphism-based analyses to clarify when naturality entails definability and when uniformity can be achieved.
Abstract
In this paper, we deal with the notions of naturality from category theory and definablity from model theory and their interactions. In this regard, we present three results. First, we show, under some mild conditions, that naturality implies definablity. Second, by using the reverse Easton iteration of Cohen forcing notions, we construct a transitive model of ZFC in which every uniformisable construction is weakly natural. Finally, we show that if F is a natural construction on a class K of structures which is represented by some formula, then it is uniformly definable without any extra parameters. Our results answer some questions by Hodges and Shelah.