Notre Dame Journal of Formal Logic

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

John Chisholm and Michael Moses


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

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

First available in Project Euclid: 5 December 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03D35: Undecidability and degrees of sets of sentences
Secondary: 03B25: Decidability of theories and sets of sentences [See also 11U05, 12L05, 20F10]


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.

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