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Fall 1998 An Undecidable Linear Order That Is $n$-Decidable for All $n$
John Chisholm, Michael Moses
Notre Dame J. Formal Logic 39(4): 519-526 (Fall 1998). DOI: 10.1305/ndjfl/1039118866

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.

Citation

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John Chisholm. Michael Moses. "An Undecidable Linear Order That Is $n$-Decidable for All $n$." Notre Dame J. Formal Logic 39 (4) 519 - 526, Fall 1998. https://doi.org/10.1305/ndjfl/1039118866

Information

Published: Fall 1998
First available in Project Euclid: 5 December 2002

zbMATH: 0966.03043
MathSciNet: MR1776223
Digital Object Identifier: 10.1305/ndjfl/1039118866

Subjects:
Primary: 03D35
Secondary: 03B25

Rights: Copyright © 1998 University of Notre Dame

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Vol.39 • No. 4 • Fall 1998
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