Logic Seminar
David Gonzalez
Notre Dame
Parameters Matter
Abstract: Scott rank measures the descriptive complexity of a countable structure. It has been precisely defined in many non-equivalent ways over the past several decades. Montalbán gave a definition of Scott rank about 10 years ago that has become standard in the literature because it is equivalent to many interesting measurements coming from various areas of logic. Later, he also defined the so-called parameterized Scott rank, which is his equivalent to unparameterized Scott rank after adding a parameter. This notion is equally robust. A lighthearted debate emerged about which rank is better, usually with the underlying assumption that it does not really matter which notion is used. This talk challenges this underlying assumption. In particular, we demonstrate the asymmetry of these notions in Ehrenfeucht theories and discuss how a counterexample to Vaught's conjecture would behave quite differently depending on which notion is preferred. We also discuss how to use our analysis to produce models where the unparameterized and parameterized Scott rank differ, addressing a question of Alvir, Csima, and MacLean. This is joint work with Dino Rossegger and Dan Turetsky.
Tuesday March 31, 2026 at 3:00 PM in 636 SEO