Math 290D: Proof Theory
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