Hyperarithmetical Sets

1. Preamble: Kleene [1943], Post [1944] and Mostowski [1947] . . . . . . . . . . . 2 1A. Post’s degrees of unsolvability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1B. Kleene’s arithmetical hierarchy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1C. Kleene [1943] vs. Post [1944] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1D. Mostowski [1947] and the analogies . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2. On into the transfinite! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2A. Notations for ordinals, S1 and O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2B. The Ha-sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2C. Myhill [1955] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2D. Effective grounded recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3. The basic facts about HYP (1950 – 1960) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3A. Codings and uniformities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3B. HYP as effective Borel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3C. Lebesgue [1905] and Mostowski [1951] . . . . . . . . . . . . . . . . . . . . . . . . 14 3D. The analytical hierarchy; HYP ⊆ ∆1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3E. Kleene’s Theorem, HYP = ∆1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3F. Addison [1959] and the revised analogies . . . . . . . . . . . . . . . . . . . . . . 20 3G. Relativization and the Kreisel Uniformization Theorem . . . . . . . . 21 3H. HYP-quantification and the Spector-Gandy Theorem . . . . . . . . . . 23 3I. The Kleene [1959a] HYP hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3J. Inductive definability on N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3K. HYP as recursive in 2E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4A. IND and HYP on abstract structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4B. Effective descriptive set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5. Appendix: some basic facts and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

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