Infinite injury priority arguments

Finite injury arguments fail when a requirement must act infinitely often. The course presents the infinite injury method on two theorems: the minimal pair theorem of Lachlan and Yates, with Posner's trick and the preservation of agreement between expansionary stages, and Sacks' theorem that there is a high incomplete c.e. degree, in which the infinitary strategies enumerate infinitely many numbers. Both constructions are organized on a tree of strategies and verified along the true path.

Contents

1. 1

   Minimal pairs and the preservation of agreement

   Minimal pairs of degrees; Posner's trick, which reduces the meet requirements to one functional; expansionary stages and the lemma that agreement is preserved when only one side changes between them; why the resulting strategy causes infinite injury.

   1. 1.1 Minimal pairs
   2. 1.2 Length of agreement and expansionary stages
   3. 1.3 The preservation lemma
   4. 1.4 The strategy and its two outcomes
   5. 1.5 Exercises
   6. 1.6 Notes and references
   35 min
2. 2

   Trees of strategies

   The tree of strategies of the minimal pair construction; the priority order, initialization to the right of the current path, and the true path; the verification of the minimal pair theorem.

   1. 2.1 The tree and the priority order
   2. 2.2 The construction
   3. 2.3 The true path
   4. 2.4 Verification
   5. 2.5 Exercises
   6. 2.6 Notes and references
   45 min
3. 3

   A high incomplete c.e. degree

   Sacks' theorem that some c.e. degree is high but incomplete, by a tree construction in which the infinitary strategies enumerate infinitely many numbers; the columns of the set code which c.e. sets are infinite; further results on the c.e. degrees.

   1. 3.1 The requirements
   2. 3.2 The construction
   3. 3.3 Verification
   4. 3.4 Further results
   5. 3.5 Exercises
   6. 3.6 Notes and references
   40 min