The range of possible ordinal values for the Cantor–Bendixson rank of an effectively closed set in Cantor space, and the relationship between the ranks and the Turing degrees of its members, have been closely studied. The wide array of results on this topic have often made use of useful topological features of Cantor space (notably, compactness) and of key properties of computable well-orders. In this paper, a number of these results are extended to the case of Cantor space on the first uncountable ordinal, in which these properties do not hold, using admissible recursion theory.
D.Cenzer, P.Clote, R.L.Smith, R.I.Soare and S.S.Wainer, Members of countable classes, Annals of Pure and Applied Logic31 (1986), 145–163. doi:10.1016/0168-0072(86)90067-9.
2.
C.-T.Chong, Techniques of Admissible Recursion Theory, Springer, Berlin, 1984.
3.
N.Greenberg and J.Knight, Computable structure theory using admissible recursion theory on ω1, in: Effective Mathematics of the Uncountable, N.Greenberg, D.Hirschfeldt, J.D.Hamkins and R.Miller, eds, ASL-Cambridge, Cambridge, UK, 2013, pp. 50–80. doi:10.1017/CBO9781139028592.005.
4.
R.Johnston, Computability in uncountable binary trees, The Journal of Symbolic Logic84(3) (2019), 1049–1098. doi:10.1017/jsl.2019.5.
5.
A.S.Kechris, Classical Descriptive Set Theory, Springer, Berlin, 2012.
6.
S.C.Kleene, On notation for ordinal numbers, The Journal of Symbolic Logic3(04) (1938), 150–155. doi:10.2307/2267778.
7.
S.Kripke, Transfinite recursion on admissible ordinals, Journal of Symbolic Logic29 (1964), 161–162.
R.A.Shore, On the jump of an α-recursively enumerable set, Transactions of the American Mathematical Society217 (1976), 351–363. doi:10.1090/S0002-9947-1976-0424544-2.
10.
R.A.Shore, α-recursion theory, Handbook of Mathematical Logic (1977), 653–680. doi:10.1016/S0049-237X(08)71118-2.
