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# Π11 relations and paths through

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When bounds on complexity of some aspect of a structure are preserved under isomorphism, we refer to them as intrinsic. Here, building on work of Soskov [34], [33], we give syntactical conditions necessary and sufficient for a relation to be intrinsically on a structure. We consider some examples of computable structures and intrinsically relations R. We also consider a general family of examples of intrinsically relations arising in computable structures of maximum Scott rank.

For three of the examples, the maximal well-ordered initial segment in a Harrison ordering, the superatomic part of a Harrison Boolean algebra, and the height-possessing part of a Harrison p-group, we show that the Turing degrees of images of the relation in computable copies of the structure are the same as the Turing degrees of paths through Kleene's . With this as motivation, we investigate the possible degrees of these paths. We show that there is a path in which ∅′ is not computable. In fact, there is one in which no noncomputable hyperarithmetical set is computable. There are paths that are Turing incomparable, or Turing incomparable over a given hyperarithmetical set. There is a pair of paths whose degrees form a minimal pair. However, there is no path of minimal degree.

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[1]Ash, C. J. and Knight, J. F., Possible degrees in recursive copies, Annals of Pure and Applied Logic, vol. 75 (1995), pp. 215221.
[2]Ash, C. J. and Knight, J. F., Possible degrees in recursive copies II, Annals of Pure and Applied Logic, vol. 87 (1997), pp. 151165.
[3]Ash, C. J. and Knight, J. F., Computable structures and the hyperarithmetical hierarchy, Elsevier, 2000.
[4]Ash, C. J., Knight, J. F., Manasse, M., and Slaman, T., Generic copies of countable structures, Annals of Pure and Applied Logic, vol. 42 (1989), pp. 195205.
[5]Ash, C. J. and Nerode, A., Intrinsically recursive relations, Aspects of effective algebra (Crossley, J. N., editor), Steel's Creek, Australia, Upside Down A Book Co., 1981, pp. 2641.
[6]Barker, E., Intrinsically relations, Annals of Pure and Applied Logic, vol. 39 (1988), pp. 105130.
[7]Barwise, J., Infinitary logic and admissible sets, this Journal, vol. 34 (1969), pp. 226252.
[8]Chisholm, J., Effective model theory vs. recursive model theory, this Journal, vol. 55 (1990), pp. 11681191.
[9]Downey, R., Jockusch, C. G. Jr., and Stob, M., Array nonrecursive degrees and genericity, Computability, enumerability, unsolvability, London Mathematical Society Lecture Notes Series, vol. 224, Cambridge University Press, 1996, pp. 93104.
[10]Friedman, H., One hundred and two problems in mathematical logic, this Journal, vol. 40 (1975), pp. 113129.
[11]Friedman, H., Recursiveness in paths through , Proceedings of the American Mathematical Society, vol. 54 (1976), pp. 311315.
[12]Goncharov, S. S., The quantity of nonautoequivalent constructivizations, Algebra and Logic, vol. 16 (1977), pp. 169185, English translation.
[13]Goncharov, S. S., Harizanov, V. S., Knight, J. F., McCoy, C., Miller, R. G., and Solomon, R., Enumerations in computable structure theory, submited.
[14]Goncharov, S. S. and Knight, J. F., Computable structure and non-structure theorems, Algebra and Logic, vol. 41 (2002), pp. 351373, English translation.
[15]Harizanov, V. S., Some effects of Ash-Nerode and other decidability conditions on degree spectra, Annals of Pure and Applied Logic, vol. 55 (1991), pp. 5165.
[16]Harrington, L., McLaughlin's conjecture, handwritten notes, 1976.
[17]Harrison, J., Recursive pseudo-well-orderings, Transactions of the American Mathematical Society, vol. 131 (1968), pp. 526543.
[18]Hirschfeldt, D. R., Khoussainov, B., Shore, R. A., and Slinko, A. M., Degree spectra and computable dimensions in algebraic structures, Annals of Pure and Applied Logic, vol. 115 (2002), pp. 71113.
[19]Jockusch, C. G. Jr., Recursiveness of initial segments of Kleene's , Fundamenta Mathematical vol. 87 (1975), pp. 161167.
[20]Jockusch, C. G. Jr., and McLaughlin, T. G., Countable retracing functions and predicates, Pacific Journal of Mathematics, vol. 30 (1969), pp. 6793.
[21]Kaplansky, I., Infinite abelian groups, University of Michigan Press, 1954.
[22]Khousainov, B. and Shore, R. A., Computable isomorphisms, degree spectra of relations and Scott families, Annals of Pure and Applied Logic, vol. 93 (1998), pp. 153193.
[23]Kreisel, G., Set theoretic problems suggested by the notion of potential totality, Infinitistic methods, Proceedings of the Symposium on Foundations of Mathematics, Warsaw, 1959, Pergamon, 1961, pp. 103140.
[24]Lachlan, A. H., Solution to a problem of Spector, Canadian Journal of Mathematics, vol. 23 (1971), pp. 247256.
[25]Lerman, M., Degrees of unsolvability, Springer-Verlag, Berlin, 1983.
[26]Manasse, M. S., Techniques and counterexamples in almost categorical recursive model theory, Ph.D. thesis, University of Wisconsin-Madison, 1982.
[27]Parikh, R., A note on paths through , Proceedings of the American Mathematical Society, vol. 39 (1973), pp. 178180.
[28]Ressayre, J. P., Models with compactness properties relative to an admissible language, Annals of Mathematical Logic, vol. 11 (1977), pp. 3155.
[29]Rogers, H. Jr., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.
[30]Sacks, G. E., On the number of countable models, Southeast Asian Conference on Logic (Singapore, 1981) (Chong, C. T. and Wicks, M. J., editors), North-Holland, 1983, pp. 185195.
[31]Sacks, G. E., Higher recursion theory, Springer-Verlag, 1990.
[32]Scott, D., Logic with denumerably long formulas and finite strings of quantifiers, The theory of models (Addison, J. W., Henkin, L., and Tarski, A., editors), Proc. 1963 Internat. Sympos. Berkeley, North-Holland, 1965, pp. 329341.
[33]Soskov, I. N., Intrinsically hyperarithmetical sets, Mathematical Logic Quarterly, vol. 42 (1996), pp. 469480.
[34]Soskov, I. N., Intrinsically relations, Mathematical Logic Quarterly, vol. 42 (1996), pp. 109126.
[35]Spector, C., Recursive well-orderings, this Journal, vol, 20 (1955), pp. 151163.
[36]Steel, J. R., Forcing with tagged trees, Annals of Mathematical Logic, vol. 15 (1978), pp. 5574.
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