The theory of the Ziegler degrees

Rodney G. Downey; Steffen Lempp; Isabella Scott

The theory of the Ziegler degrees

Logic 2026-07-03 v1

Authors: Rodney G. Downey , Steffen Lempp , Isabella Scott

Abstract

The Ziegler degrees were introduced to characterize definability in group theory. In [JLSta], the authors show that the first order-theory of the Ziegler degrees (as a partial order) is undecidable. We improve this result, showing that the theory of the Ziegler degrees is bi-interpretable with true second-order arithmetic.

Keywords

Ziegler degrees model theory undecidability bi-interpretability second-order arithmetic definability

Cite

@article{arxiv.2607.02870,
  title  = {The theory of the Ziegler degrees},
  author = {Rodney G. Downey and Steffen Lempp and Isabella Scott},
  journal= {arXiv preprint arXiv:2607.02870},
  year   = {2026}
}

Comments

13 pages; comments welcome!