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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
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[1] Downey, R., “Computability theory and linear orderings,” pp. 823–976 in Handbook of Recursive Mathematics , vol. 2, edited by Yu. L. Ershov, S. S. Goncharov, A. Nerode, J. B. Remmel, and V. W. Marek, Elsevier, New York, 1998. · Zbl 0941.03045
[2] Ehrenfeucht, A., “An application of games to the completeness problem for formalized theories,” Fundamenta Mathematicæ , vol. 49 (1961), pp. 129–41. · Zbl 0096.24303
[3] Fraï”ssé, R., “Sur quelques classifications des systèmes de relations,” Publications in Science and Universal Algebra , series A, vol. 1 (1954), pp. 35–182. · Zbl 0068.24302
[4] Fröhlich, A., and J. C. Shepherdson, “Effective procedures in field theory,” Philosophical Transactions of the Royal Society of London , series A, vol. 248 (1955), pp. 407–32. JSTOR: · Zbl 0070.03502
[5] Moses, M., “Relations intrinsically recursive in linear orders,” Zeitschrift für mathematische Logik und Grundlagen der Mathematik , vol. 32 (1986), pp. 467–72. · Zbl 0596.03038
[6] Moses, M., “\(n\)–Recursive linear orders without \((n+1)\)-recursive copies,” pp. 572–92 in Logical Methods: in Honor of Anil Nerode’s Sixtieth Birthday , edited by J. N. Crossley, J. B. Remmel, R. A. Shore, and M. E. Sweedler, Birkhäuser, Boston, 1993. · Zbl 0824.03020
[7] Rabin, M., “Computable algebra, general theory and theory of computable fields,” Transactions of the American Mathematical Society , vol. 95 (1960), pp. 341–60. · Zbl 0156.01201
[8] Rosenstein, J. G., Linear Orderings , Academic Press, New York, 1982. · Zbl 0488.04002
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