## Notre Dame Journal of Formal Logic

- Notre Dame J. Formal Logic
- Volume 59, Number 1 (2018), 35-59.

### Classifications of Computable Structures

Karen Lange, Russell Miller, and Rebecca M. Steiner

#### Abstract

Let $\mathcal{K}$ be a family of structures, closed under isomorphism, in a fixed computable language. We consider effective lists of structures from $\mathcal{K}$ such that every structure in $\mathcal{K}$ is isomorphic to exactly one structure on the list. Such a list is called a *computable classification* of $\mathcal{K}$, up to isomorphism. Using the technique of Friedberg enumeration, we show that there is a computable classification of the family of computable algebraic fields and that with a $\mathbf{0\text{'}}$-oracle, we can obtain similar classifications of the families of computable equivalence structures and of computable finite-branching trees. However, there is no computable classification of the latter, nor of the family of computable torsion-free abelian groups of rank $1$, even though these families are both closely allied with computable algebraic fields.

#### Article information

**Source**

Notre Dame J. Formal Logic, Volume 59, Number 1 (2018), 35-59.

**Dates**

Received: 27 June 2014

Accepted: 31 March 2015

First available in Project Euclid: 30 June 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.ndjfl/1498788255

**Digital Object Identifier**

doi:10.1215/00294527-2017-0015

**Mathematical Reviews number (MathSciNet)**

MR3744350

**Zentralblatt MATH identifier**

06848190

**Subjects**

Primary: 03C57: Effective and recursion-theoretic model theory [See also 03D45]

Secondary: 03D45: Theory of numerations, effectively presented structures [See also 03C57; for intuitionistic and similar approaches see 03F55]

**Keywords**

Friedberg enumeration computable structure theory equivalence structure effective classification

#### Citation

Lange, Karen; Miller, Russell; Steiner, Rebecca M. Classifications of Computable Structures. Notre Dame J. Formal Logic 59 (2018), no. 1, 35--59. doi:10.1215/00294527-2017-0015. https://projecteuclid.org/euclid.ndjfl/1498788255