## Notre Dame Journal of Formal Logic

- Notre Dame J. Formal Logic
- Volume 39, Number 4 (1998), 519-526.

### An Undecidable Linear Order That Is $n$-Decidable for All $n$

John Chisholm and Michael Moses

#### Abstract

A linear order is $n$-*decidable* if its universe is $\mathbb{N}$ and the relations defined by $\Sigma_{n}$ formulas are uniformly computable. This means that there is a computable procedure which, when applied to a $\Sigma_{n}$ formula $\varphi(\bar{x})$ and a sequence $\bar{a}$ of elements of the linear order, will determine whether or not $\varphi(\bar{a})$ is true in the structure. A linear order is *decidable* if the relations defined by *all* formulas are uniformly computable.

These definitions suggest two questions. Are there, for each $n$, $n$-decidable linear orders that are not $(n+1)$-decidable? Are there linear orders that are $n$-decidable for all $n$ but not decidable? The former was answered in the positive by Moses in 1993. Here we answer the latter, also positively.

#### Article information

**Source**

Notre Dame J. Formal Logic, Volume 39, Number 4 (1998), 519-526.

**Dates**

First available in Project Euclid: 5 December 2002

**Permanent link to this document**

https://projecteuclid.org/euclid.ndjfl/1039118866

**Digital Object Identifier**

doi:10.1305/ndjfl/1039118866

**Mathematical Reviews number (MathSciNet)**

MR1776223

**Zentralblatt MATH identifier**

0966.03043

**Subjects**

Primary: 03D35: Undecidability and degrees of sets of sentences

Secondary: 03B25: Decidability of theories and sets of sentences [See also 11U05, 12L05, 20F10]

#### Citation

Chisholm, John; Moses, Michael. An Undecidable Linear Order That Is $n$-Decidable for All $n$. Notre Dame J. Formal Logic 39 (1998), no. 4, 519--526. doi:10.1305/ndjfl/1039118866. https://projecteuclid.org/euclid.ndjfl/1039118866