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Intrinsic bounds on complexity and definability at limit levels

  • John Chisholm (a1), Ekaterina B. Fokina (a2), Sergey S. Goncharov (a3), Valentina S. Harizanov (a4), Julia F. Knight (a5) and Sara Quinn (a6)...
Abstract
Abstract

We show that for every computable limit ordinal α, there is a computable structure that is categorical, but not relatively categorical (equivalently, it does not have a formally Scott family). We also show that for every computable limit ordinal α, there is a computable structure with an additional relation R that is intrinsically on , but not relatively intrinsically on (equivalently, it is not definable by a computable Σα formula with finitely many parameters). Earlier results in [7], [10], and [8] establish the same facts for computable successor ordinals α.

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[2] C. J. Ash , C. G. Jockusch Jr., and J. F. Knight , Jumps of orderings, Transactions of the American Mathematical Society, vol. 319 (1990), pp. 573599.

[4] C. J. Ash , J. Knight , M. Manasse , and T. Slaman , Generic copies of countable structures, Annals of Pure and Applied Logic, vol. 42 (1989), pp. 195205.

[5] S. A. Badaev , Computable enumerations of families of general recursive functions, Algebra and Logic, vol. 16 (1977), pp. 129148 (Russian), 83–98 (English translation).

[8] S. S. Goncharov , V. Harizanov , J. Knight , C. McCoy , R. Miller , and R. Solomon , Enumerations in computable structure theory, Annals of Pure and Applied Logic, vol. 136 (2005), pp. 219246.

[12] V. L. Selivanov , The numerations of families of general recursive functions, Algebra and Logic, vol. 15 (1976), pp. 205226 (Russian), 128–141 (English translation).

[13] I. N. Soskov , Intrinsically hyperarithmetical sets, Mathematical Logic Quarterly, vol. 42 (1996), no. 4, pp. 469480.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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