Hyperarithmetic sets

The hyperarithmetic sets, obtained by iterating the Turing jump along Kleene's notations: the H-sets and Spector's uniqueness theorem, the uniform Δ¹₁ definitions of the H-sets, Kleene's theorem that the hyperarithmetic sets are exactly the Δ¹₁ sets, computable trees without hyperarithmetic paths, the Kleene basis theorem, and Harrison's pseudo-well-orderings.

Contents

1. 1

   Iterating the jump along notations

   The H-sets, iterating the Turing jump along Kleene's notations; their basic reductions; Spector's uniqueness theorem; the hyperarithmetic sets and degrees.

   1. 1.1 The H-sets
   2. 1.2 Spector's uniqueness theorem
   3. 1.3 Hyperarithmetic sets
   4. 1.4 Exercises
   5. 1.5 Notes and references
   35 min
2. 2

   The hyperarithmetic sets are the Δ¹₁ sets

   The H-sets are uniformly Δ¹₁ along O; quantification over hyperarithmetic sets is Π¹₁; Kleene's theorem that a set is hyperarithmetic if and only if it is Δ¹₁, through Spector's boundedness theorem.

   1. 2.1 The H-sets are Δ¹₁
   2. 2.2 Kleene's theorem
   3. 2.3 Exercises
   4. 2.4 Notes and references
   35 min
3. 3

   Σ¹₁ classes and pseudo-well-orderings

   Σ¹₁ classes as projections of closed sets of Baire space; a computable tree with paths but no hyperarithmetic path; the Kleene basis theorem; Harrison's computable linear orders that are ill-founded but have no hyperarithmetic descending sequence; consequences for reverse mathematics.

   1. 3.1 Σ¹₁ classes
   2. 3.2 Trees without hyperarithmetic paths
   3. 3.3 Pseudo-well-orderings
   4. 3.4 Consequences for reverse mathematics
   5. 3.5 Exercises
   6. 3.6 Notes and references
   35 min