## Notre Dame Journal of Formal Logic

- Notre Dame J. Formal Logic
- Volume 58, Number 1 (2017), 97-105.

### Strange Structures from Computable Model Theory

#### Abstract

Let $L$ be a countable language, let $\mathcal{I}$ be an isomorphism-type of countable $L$-structures, and let $a\in {2}^{\omega}$. We say that $\mathcal{I}$ is $a$-*strange* if it contains a computable-from-$a$ structure and its Scott rank is exactly ${\omega}_{1}^{a}$. For all $a$, $a$-strange structures exist. Theorem (AD): If $\mathcal{C}$ is a collection of ${\aleph}_{1}$ isomorphism-types of countable structures, then for a Turing cone of $a$’s, no member of $\mathcal{C}$ is $a$-strange.

#### Article information

**Source**

Notre Dame J. Formal Logic, Volume 58, Number 1 (2017), 97-105.

**Dates**

Received: 21 July 2013

Accepted: 23 March 2014

First available in Project Euclid: 17 November 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.ndjfl/1479351686

**Digital Object Identifier**

doi:10.1215/00294527-3767941

**Mathematical Reviews number (MathSciNet)**

MR3595343

**Zentralblatt MATH identifier**

06686419

**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] 03E60: Determinacy principles

**Keywords**

Scott rank computable structures axiom of determinacy

#### Citation

Becker, Howard. Strange Structures from Computable Model Theory. Notre Dame J. Formal Logic 58 (2017), no. 1, 97--105. doi:10.1215/00294527-3767941. https://projecteuclid.org/euclid.ndjfl/1479351686