Degrees of unsolvability of constructible sets of integers

  title={Degrees of unsolvability of constructible sets of integers},
  author={George Boolos and Hilary Putnam},
  journal={Journal of Symbolic Logic},
  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… 

Analytic sets having incomparable kleene degrees

One concern of descriptive set theory is the classification of definable sets of reals. Taken loosely ‘definable’ can mean anything from projective to formally describable in the language of

The limits of E-recursive enumerability

  • G. Sacks
  • Mathematics, Computer Science
    Ann. Pure Appl. Log.
  • 1986

High and low Kleene degrees of coanalytic sets

We say that a set A of reals is recursive in a real y together with a set B of reals if one can imagine a computing machine with an ability to perform a countably infinite sequence of program steps

Systems of notations and the ramified analytical hierarchy

It is shown that arithmetically minimal systems of notations can be constructed which provide notations for all ramified analytical ordinals (all the ordinals in the minimum β -model for analysis).

A Recursion‐Theoretic Characterization of Constructible Reals

Let Ly denote Godel's constructive universe up to level y. A countable ordinal y is said to be an index if Ly+1 contains a real not in L r The notion was introduced by Boolos and Putnam [1] who also

The Ramified analytical Hierarchy using Extended Logics

  • P. Welch
  • Computer Science
    Bull. Symb. Log.
  • 2018
The use of Extended Logics to replace ordinary second order definability in Kleene’s Ramified Analytical Hierarchy is investigated and it is found that a wide spectrum of models can be so generated from abstract definability notions.


  • Liang Yu
  • Mathematics
    The Bulletin of Symbolic Logic
  • 2020
A characterization of the Borel generated $\sigma $-ideals having approximation property under the assumption that every real is constructible is given, answering Mauldin’s question raised in [15].

Thin Maximal Antichains in the Turing Degrees

A corollary of the main result gives a negative solution to a question of Jockusch under the assumption that every real is constructible.



A note on constructible sets of integers

  • H. Putnam
  • Mathematics
    Notre Dame J. Formal Log.
  • 1963
Is there always a constructible set of integers of order a + 1 which is not of order α, when a is less than ω1 ? 2 I shall answer this question in the negative. It is clear that there is always a

Hierarchies of number-theoretic predicates

The existence of hierarchies of point sets in analysis has long been familiar from the work of Borel and Lusin. The study of the hierarchies in number theory which we consider here began with a

A minimal model for set theory

In the proof of the consistency of the Continuum Hypothesis and the Axiom of Choice with the other axioms of set theory, Godel [ l ] introduced the notion of a constructible set and showed that the

A note on function quantification

In [7, p. 211] the question was raised whether, for k>0, each predicate expressible in both the ^-fl-function-quantifier forms is hyperarithmetical in predicates expressible in the

Arithmetical predicates and function quantifiers

The extended class of number-theoretic predicates obtained by applying the two-sorted predicate calculus to recursive predicates of number variables and variables for one-place number-thoretic functions is considered, and it is shown that similarly these fall into a hierarchy according to the sequences of alternating function quantifiers by which they can be defined from arithmetical predicates.

Arithmetical extensions of relational systems

© Foundation Compositio Mathematica, 1956-1958, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: // implique l’accord avec les

Quantification of number-theoretic functions

© Foundation Compositio Mathematica, 1959-1960, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: // implique l’accord avec les

Computability and Unsolvability

  • Martin D. Davis
  • Computer Science
    McGraw-Hill Series in Information Processing and Computers
  • 1958
Only for you today! Discover your favourite computability and unsolvability book right here by downloading and getting the soft file of the book. This is not your time to traditionally go to the book

The hierarchy of constrmctible sets of integers

  • Ph.D. thesis, M.I.T. Humanities Department,
  • 1966

Quantification of number-theoretic predicates

  • Compositio mathematica, vol
  • 1959