Possible topics (*'d topics are high priority, others will be covered depending on interest):
- *Proof systems (natural deduction and sequent calculi, classical and intuitionistic) (references: chapters 2 and 5 from Girard and Avigad)
- The double-negation and Friedman-Dragalin translations (see Avigad and Friedman)
- Lambda calculus and the Curry-Howard isomorphism (see chapter 3 from Girard)
- Semantics of intuitionistic logic (see Troelstra and Van Dalan)
- Properties of first-order logic: Herbrand's Theorem, the interpolation theorem, etc (see Avigad)
- Theories of arithmetic weaker than Peano arithmetic:
- Primitive recursive arithmetic (see Avigad and Troelstra)
- Weak theories of arithmetic (bounded induction and collection, Parson's theorem) (see Buss)
- Bounded arithmetic (see Buss and Buss)
- Basics of reverse mathematics (the "big 5", some examples) (see Simpson)
- *Godel's Dialectica interpretation (see Avigad and Feferman)
- The Paris-Harrington Theorem (see Paris and Harrington and Kaye)
- *Cut-elimination for first-order logic (see Avigad)
- *Cut-elimination (and ordinal analysis) for Peano arithmetic (see Avigad and chapter 7 of Pohlers)
- Linear logic
- Upper bounds on cut-elimination