The Big Five

The systems of the Big Five beyond the base theory: arithmetic comprehension and its equivalents (Turing jumps, ranges of injections, monotone convergence, König's lemma), weak König's lemma with Scott ideals, Σ⁰₁ separation and the Heine–Borel theorem, and a survey of arithmetic transfinite recursion and Π¹₁ comprehension.

Contents

1. 1

   Arithmetic comprehension

   The system ACA₀; it is equivalent to Σ⁰₁ comprehension, to the existence of Turing jumps, and to the existence of ranges of injections; its ω-models are the jump ideals; the monotone convergence theorem and König's lemma are equivalent to ACA₀.

   1. 1.1 The system and its equivalent forms
   2. 1.2 Monotone convergence
   3. 1.3 König's lemma
   4. 1.4 Exercises
   5. 1.5 Notes and references
   40 min
2. 2

   Weak König's lemma

   The system WKL₀; its ω-models are the Scott ideals; there is a Scott ideal of low sets, so WKL₀ does not prove ACA₀; WKL₀ is equivalent to Σ⁰₁ separation and to the Heine–Borel covering theorem for [0, 1].

   1. 2.1 The system and its ω-models
   2. 2.2 Σ⁰₁ separation
   3. 2.3 The Heine–Borel theorem
   4. 2.4 Conservation and other equivalents
   5. 2.5 Exercises
   6. 2.6 Notes and references
   40 min
3. 3

   Transfinite recursion and Π¹₁ comprehension

   Countable well-orderings in second-order arithmetic; the systems ATR₀ and Π¹₁-CA₀; Π¹₁-CA₀ is equivalent to the existence of the set of ill-founded trees in a sequence; a survey of the theorems equivalent to ATR₀ and Π¹₁-CA₀.

   1. 3.1 Well-orderings
   2. 3.2 Π¹₁ comprehension and trees
   3. 3.3 A survey of equivalents
   4. 3.4 Exercises
   5. 3.5 Notes and references
   35 min