English

Visceral theories without assumptions

Logic 2026-05-06 v3

Abstract

Let TT be a theory with a definable topology. TT is t-minimal in the sense of Mathews if every definable set in one variable has finite boundary. If TT is t-minimal, we show that there is a good dimension theory for definable sets, satisfying properties similar to dp-rank in dp-minimal theories, with one key exception: the dimension of dom(f)\operatorname{dom}(f) can be less than the dimension of im(f)\operatorname{im}(f) for a definable function ff. Using the dimension theory, we show that any definable field in a t-minimal theory is perfect. We then specialize to the case where TT is visceral in the sense of Dolich and Goodrick, meaning that TT is t-minimal and the definable topology comes from a definable uniformity (i.e., a definable uniform structure). We show that almost all of Dolich and Goodrick's tame topology theorems for visceral theories hold without their additional assumptions of definable finite choice (DFC) and no space-filling functions (NSFF). Lastly, we produce an example of a visceral theory with a space-filling curve, answering a question of Dolich and Goodrick.

Keywords

Cite

@article{arxiv.2404.11453,
  title  = {Visceral theories without assumptions},
  author = {Will Johnson},
  journal= {arXiv preprint arXiv:2404.11453},
  year   = {2026}
}

Comments

60 pages, fix typos and add a couple remarks

R2 v1 2026-06-28T15:57:25.835Z