Degrees of unsolvability of constructible sets of integers

@article{Boolos1969DegreesOU,
  title={Degrees of unsolvability of constructible sets of integers},
  author={George Boolos and Hilary Putnam},
  journal={Journal of Symbolic Logic},
  year={1969},
  volume={33},
  pages={497 - 513}
}
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… 

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