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An undecidable linear order that is $$n$$-decidable for all $$n$$. (English) Zbl 0966.03043
Summary: 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 $$\varphi (\overline{x})$$ and a sequence $$\overline{a}$$ of elements of the linear order, will determine whether or not $$\varphi (\overline{a})$$ 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 M. Moses [in: J. N. Crossley et al. (eds.), Logical methods. In honor of A. Nerode’s 60th birthday. Basel: Birkhäuser. Prog. Comput. Sci. Appl. Log. 12, 572-592 (1993; Zbl 0824.03020)]. Here we answer the latter, also positively.

##### MSC:
 03D45 Theory of numerations, effectively presented structures 06A05 Total orders 03D35 Undecidability and degrees of sets of sentences 03B25 Decidability of theories and sets of sentences
##### Keywords:
computable linear order; $$n$$-decidable linear orders
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##### References:
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