Notre Dame Journal of Formal Logic

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


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