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Degrees of unsolvability of constructible sets of integers

  • George Boolos (a1) (a2) and Hilary Putnam (a2)

Why the Post-Kleene arithmetical hierarchy of degrees of (recursive) unsolvability was extended into the transfinite is not clear. Perhaps it was thought that if a hierarchy of sufficiently fine structure could be described that would include all sets of integers, some light might be thrown on the Continuum Hypothesis, and its truth or falsity possibly even ascertained. There is also some evidence in the 1955 papers of Kleene (cf. Kleene [2], [3], [4]) that it was once hoped that a theorem for the analytical hierarchy analogous to the result of Post and Kleene

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J. W. Addison and S. C. Kleene , [1] A note on function quantification, Proceedings of the American Mathematical Society, vol. 8 (1957), pp. 10021006.

P. J. Cohen , [1] A minimal model for set theory, Bulletin of the American Mathematical Society, vol. 69 (1963), pp. 537540.

K. Gödel , [1] Consistency-proof for the generalized continuum hypothesis, Proceedings of the National Academy of Sciences of the United States of America, vol. 25 (1939), pp. 220224.

S. C. Kleene , [2] Arithmetical predicates and function quantifiers, Transactions of the American Mathematical Society, vol. 79 (1955), pp. 312340.

S. C. Kleene , [3] On the forms of predicates in the theory of constructive ordinals. II, American journal of mathematics, vol. 77 (1955), pp. 405428.

S. C. Kleene , [4] Hierarchies of number-theoretic predicates, Bulletin of the American Mathematical Society, vol. 61 (1955), pp. 193213.

H. Putnam , [1] A note on constructible sets of Integers, Notre Dame journal of formal logic, vol. 4 (1963), pp. 270273.

H. Wang , [1] A survey of mathematical logic, North Holland, Amsterdam, 1963.

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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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