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Computable trees of Scott rank ω1CK, and computable approximation

  • Wesley Calvert (a1), Julia F. Knight (a2) and Jessica Millar (a3)

Makkai [10] produced an arithmetical structure of Scott rank ω1CK. In [9], Makkai's example is made computable. Here we show that there are computable trees of Scott rank ω1CK. We introduce a notion of “rank homogeneity”. In rank homogeneous trees, orbits of tuples can be understood relatively easily. By using these trees, we avoid the need to pass to the more complicated “group trees” of [10] and [9], Using the same kind of trees, we obtain one of rank ω1CK that is “strongly computably approximable”. We also develop some technology that may yield further results of this kind.

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[1] C. J. Ash and J. F. Knight , Pairs of recursive structures, Annals of Pure and Applied Logic, vol. 46 (1990), pp. 211234.

[3] C. J. Ash and J. F. Knight , Ramified systems, Annals of Pure and Applied Logic, vol. 70 (1994), pp. 205221.

[8] J. Harrison , Recursive pseudo well-orderings, Transactions of the American Mathematical Society, vol. 131 (1968), pp. 526543.

[12] M. Nadel , Scott sentences and admissible sets, Annals of Mathematical Logic, vol. 7 (1974), pp. 267294.

[15] G. E. Sacks , Higher Recursion Theory, Springer-Verlag, 1990.

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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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