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Uncountable master codes and the jump hierarchy

Published online by Cambridge University Press:  12 March 2014

Robert S. Lubarsky
Robert S. Lubarsky*
Affiliation:
Department of Mathematics, Cornell University, Ithaca, New York 14853
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The Turing jump can easily be iterated any finite number of times. The challenge is posed by by transfinite iterations. a(ω) has an intuitively compelling definition: {‹m, n›: n ∈ a(m)}. Starting with Putnam et al. [1], [2], [9] and culminating in Jockusch-Simpson [8] and Hodes [6], recent work has justified this ω-jump and extended it through .

Of great use are the master codes of Jensen [3], [7]. Briefly, a Δn(Lβ)-master code is a complete Δn(Lβ) set of ordinals, with supremum as small as possible. Connections with classical recursion theory are tight. When the Δn(Lβ) master codes are reals, they are Turing equivalent. If a is a Δn(Lβ) MC, then a′ is a Δn + 1(Lβ) MC. Intuitively satisfying transfinite jumps, such as the ω-jump above, produce a Turing jump hierarchy equal to an initial segment of the master codes.

Hodes capitalized on these facts by defining 0α, α < , set-theoretically as the degree of the αth MC which is a real, and then proving the equivalence of a jump-theoretic definition of 0α. The successor codes are jumps of their predecessors, and for a limit λ, 0λ is the minimum of a set of degrees associated with {0α: α: < λ}.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 52 , Issue 4 , December 1987 , pp. 952 - 958
Copyright
Copyright © Association for Symbolic Logic 1987

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References

REFERENCES

[0]Barwise, J., Admissible sets and structures, Springer-Verlag, Berlin, 1975.CrossRefGoogle Scholar
[1]Boolos, G. and Putnam, H., Degrees of unsolvability of constructible sets of integers, this Journal, vol. 33 (1968), pp. 497–513.Google Scholar
[2]Boyd, R., Hensel, G., and Putnam, H., A recursion-theoretic characterization of the ramified analytical hierarchy, Transactions of the American Mathematical Society, vol. 141 (1969), pp. 47–62.CrossRefGoogle Scholar
[3]Devlin, K. J., Aspects of constructibility, Lecture Notes in Mathematics, vol. 354, Springer-Verlag, Berlin, 1973.CrossRefGoogle Scholar
[4]Friedman, S. D., Negative solutions of Post's problem. II, Annals of Mathematics, ser. 2, vol. 113 (1981), pp. 25–43.CrossRefGoogle Scholar
[5]Friedman, S. D., Uncountable admissibles. II: Compactness, Israel Journal of Mathematics, vol. 40 (1981), pp. 129–149.CrossRefGoogle Scholar
[6]Hodes, H., Jumping through the transfinite, this Journal, vol. 45 (1980), pp. 204–220.Google Scholar
[7]Jensen, R. B., The fine structure of the constructible hierarchy, Annals of Mathematical Logic, vol. 8 (1972), pp. 229–308.CrossRefGoogle Scholar
[8]Jockusch, C. and Simpson, S., A degree-theoretic characterization of the ramified analytical hierarchy, Annals of Mathematical Logic, vol. 10 (1976), pp. 1–32.CrossRefGoogle Scholar
[9]Leeds, S. and Putnam, H., An intrinsic characterization of the hierarchy of constructible sets of integers, Logic Colloquium '69, North-Holland, Amsterdam, 1971, pp. 311–350.CrossRefGoogle Scholar
[10]Marek, W. and Srebrny, M., Gaps in the constructible universe, Annals of Mathematical Logic, vol. 6 (1974), pp. 359–394.CrossRefGoogle Scholar
[11]Sacks, G., Forcing with perfect closed sets, Axiomatic set theory, Proceedings of Symposia in Pure Mathematics, vol. 13, part 1, American Mathematical Society, Providence, Rhode Island, 1971, pp. 331–355.CrossRefGoogle Scholar
[12]Steel, J., Forcing with tagged trees, Annals of Mathematical Logic, vol. 15 (1978), pp. 55–74.CrossRefGoogle Scholar