Second-order arithmetic

The framework of reverse mathematics: the two-sorted language of second-order arithmetic, its axioms and Henkin models, ω-models and the absoluteness of arithmetic formulas, the base theory RCA₀ and its characterization by Turing ideals, and the coding of real numbers and continuous functions, with the intermediate value theorem in RCA₀ and Specker sequences.

Contents

1. 1

   The language and models of second-order arithmetic

   The two-sorted language of second-order arithmetic, its formula classes, the axioms of Z₂, Henkin structures and ω-models, absoluteness of arithmetic formulas, and the program of reverse mathematics.

   1. 1.1 The language
   2. 1.2 The axioms
   3. 1.3 Structures and ω-models
   4. 1.4 The program
   5. 1.5 Exercises
   6. 1.6 Notes and references
   35 min
2. 2

   The base theory RCA₀

   Δ⁰₁ comprehension and Σ⁰₁ induction; consequences of Σ⁰₁ induction; the ω-models of RCA₀ are the Turing ideals; the computable sets form the least ω-model, and non-provability in RCA₀ follows from computability-theoretic constructions.

   1. 2.1 The system
   2. 2.2 ω-models and Turing ideals
   3. 2.3 The role of induction
   4. 2.4 Exercises
   5. 2.5 Notes and references
   35 min
3. 3

   Mathematics in RCA₀

   Coding real numbers and continuous functions in second-order arithmetic; RCA₀ proves the completeness of the reals for sequences with a modulus of convergence and the intermediate value theorem; Specker sequences show that RCA₀ does not prove the monotone convergence theorem.

   1. 3.1 Real numbers
   2. 3.2 Continuous functions
   3. 3.3 The intermediate value theorem
   4. 3.4 Specker sequences
   5. 3.5 Exercises
   6. 3.6 Notes and references
   35 min