Cut-elimination for Intuitionistic Provability Logic Iris van der - PowerPoint PPT Presentation

Introduction to provability logic Cut-elimination Cut-elimination for Intuitionistic Provability Logic Iris van der Giessen Utrecht University i.vandergiessen@uu.nl April 26, 2019 Iris van der Giessen Cut-elimination for Intuitionistic Provability Logic

Introduction to provability logic Cut-elimination Provability in Peano Arithmetic PA ◮ What can Peano arithmetic say about its own provability? ◮ Provability predicate Prov( � A � ): ‘There exists a G¨ odel number p coding a correct proof from the axioms of PA of formula A (coded by G¨ odel number � A � ).’ Example (Formal second incompleteness theorem) PA ⊢ ¬ Prov( ⊥ ) → ¬ Prov( ¬ Prov( ⊥ )) ◮ Normal modal logic GL is given by Hilbert system: ◮ Propositional tautologies ◮ K-axiom: � ( A → B ) → � A → � B ◮ G¨ odel-L¨ ob’s axiom: � ( � A → A ) → � A ◮ closed under Modus Ponens, Substitution and Necessitation. ◮ Solovays completeness theorem Iris van der Giessen Cut-elimination for Intuitionistic Provability Logic

Introduction to provability logic Cut-elimination Provability in Peano Arithmetic PA ◮ What can Peano arithmetic say about its own provability? ◮ Provability predicate Prov( � A � ): ‘There exists a G¨ odel number p coding a correct proof from the axioms of PA of formula A (coded by G¨ odel number � A � ).’ Example (Formal second incompleteness theorem) PA ⊢ ¬ Prov( ⊥ ) → ¬ Prov( ¬ Prov( ⊥ )) ◮ Normal modal logic GL is given by Hilbert system: ◮ Propositional tautologies ◮ K-axiom: � ( A → B ) → � A → � B ◮ G¨ odel-L¨ ob’s axiom: � ( � A → A ) → � A ◮ closed under Modus Ponens, Substitution and Necessitation. ◮ Solovays completeness theorem Iris van der Giessen Cut-elimination for Intuitionistic Provability Logic

Introduction to provability logic Cut-elimination Provability in Peano Arithmetic PA ◮ What can Peano arithmetic say about its own provability? ◮ Provability predicate Prov( � A � ): ‘There exists a G¨ odel number p coding a correct proof from the axioms of PA of formula A (coded by G¨ odel number � A � ).’ Example (Formal second incompleteness theorem) PA ⊢ ¬ Prov( ⊥ ) → ¬ Prov( ¬ Prov( ⊥ )) ◮ Normal modal logic GL is given by Hilbert system: ◮ Propositional tautologies ◮ K-axiom: � ( A → B ) → � A → � B ◮ G¨ odel-L¨ ob’s axiom: � ( � A → A ) → � A ◮ closed under Modus Ponens, Substitution and Necessitation. ◮ Solovays completeness theorem Iris van der Giessen Cut-elimination for Intuitionistic Provability Logic

Introduction to provability logic Cut-elimination Provability in Peano Arithmetic PA ◮ What can Peano arithmetic say about its own provability? ◮ Provability predicate Prov( � A � ): ‘There exists a G¨ odel number p coding a correct proof from the axioms of PA of formula A (coded by G¨ odel number � A � ).’ Example (Formal second incompleteness theorem) GL ⊢ ¬ � ⊥ → ¬ � ( ¬ � ⊥ ) ◮ Normal modal logic GL is given by Hilbert system: ◮ Propositional tautologies ◮ K-axiom: � ( A → B ) → � A → � B ◮ G¨ odel-L¨ ob’s axiom: � ( � A → A ) → � A ◮ closed under Modus Ponens, Substitution and Necessitation. ◮ Solovays completeness theorem Iris van der Giessen Cut-elimination for Intuitionistic Provability Logic

Introduction to provability logic Cut-elimination Provability in Peano Arithmetic PA ◮ What can Peano arithmetic say about its own provability? ◮ Provability predicate Prov( � A � ): ‘There exists a G¨ odel number p coding a correct proof from the axioms of PA of formula A (coded by G¨ odel number � A � ).’ Example (Formal second incompleteness theorem) GL ⊢ ¬ � ⊥ → ¬ � ( ¬ � ⊥ ) ◮ Normal modal logic GL is given by Hilbert system: ◮ Propositional tautologies ◮ K-axiom: � ( A → B ) → � A → � B ◮ G¨ odel-L¨ ob’s axiom: � ( � A → A ) → � A ◮ closed under Modus Ponens, Substitution and Necessitation. ◮ Solovays completeness theorem Iris van der Giessen Cut-elimination for Intuitionistic Provability Logic

Introduction to provability logic Cut-elimination Provability in Peano Arithmetic PA ◮ What can Peano arithmetic say about its own provability? ◮ Provability predicate Prov( � A � ): ‘There exists a G¨ odel number p coding a correct proof from the axioms of PA of formula A (coded by G¨ odel number � A � ).’ Example (Formal second incompleteness theorem) GL ⊢ � ( ¬ � ⊥ ) → � ⊥ ◮ Normal modal logic GL is given by Hilbert system: ◮ Propositional tautologies ◮ K-axiom: � ( A → B ) → � A → � B ◮ G¨ odel-L¨ ob’s axiom: � ( � A → A ) → � A ◮ closed under Modus Ponens, Substitution and Necessitation. ◮ Solovays completeness theorem Iris van der Giessen Cut-elimination for Intuitionistic Provability Logic

Introduction to provability logic Cut-elimination Provability in Peano Arithmetic PA ◮ What can Peano arithmetic say about its own provability? ◮ Provability predicate Prov( � A � ): ‘There exists a G¨ odel number p coding a correct proof from the axioms of PA of formula A (coded by G¨ odel number � A � ).’ Example (Formal second incompleteness theorem) GL ⊢ � ( � ⊥ → ⊥ ) → � ⊥ ◮ Normal modal logic GL is given by Hilbert system: ◮ Propositional tautologies ◮ K-axiom: � ( A → B ) → � A → � B ◮ G¨ odel-L¨ ob’s axiom: � ( � A → A ) → � A ◮ closed under Modus Ponens, Substitution and Necessitation. ◮ Solovays completeness theorem Iris van der Giessen Cut-elimination for Intuitionistic Provability Logic

Introduction to provability logic Cut-elimination Provability in Peano Arithmetic PA ◮ What can Peano arithmetic say about its own provability? ◮ Provability predicate Prov( � A � ): ‘There exists a G¨ odel number p coding a correct proof from the axioms of PA of formula A (coded by G¨ odel number � A � ).’ Example (Formal second incompleteness theorem) GL ⊢ � ( � ⊥ → ⊥ ) → � ⊥ ◮ Normal modal logic GL is given by Hilbert system: ◮ Propositional tautologies ◮ K-axiom: � ( A → B ) → � A → � B ◮ G¨ odel-L¨ ob’s axiom: � ( � A → A ) → � A ◮ closed under Modus Ponens, Substitution and Necessitation. ◮ Solovays completeness theorem Iris van der Giessen Cut-elimination for Intuitionistic Provability Logic

Introduction to provability logic Cut-elimination Intuitionistic G¨ odel-L¨ ob logic ◮ Intuitionistic provability logic is given by Hilbert system GLi: ◮ Intuitionistic propositional tautologies ◮ K-axiom: � ( A → B ) → � A → � B ◮ G¨ odel-L¨ ob’s axiom: � ( � A → A ) → � A ◮ closed under Modus Ponens, Substitution and Necessitation. Fact Intuitionistic provability logic is not the logic of the provability theory in Heyting Arithmetic! Iris van der Giessen Cut-elimination for Intuitionistic Provability Logic

Introduction to provability logic Cut-elimination Intuitionistic G¨ odel-L¨ ob logic ◮ Intuitionistic provability logic is given by Hilbert system GLi: ◮ Intuitionistic propositional tautologies ◮ K-axiom: � ( A → B ) → � A → � B ◮ G¨ odel-L¨ ob’s axiom: � ( � A → A ) → � A ◮ closed under Modus Ponens, Substitution and Necessitation. Fact Intuitionistic provability logic is not the logic of the provability theory in Heyting Arithmetic! Iris van der Giessen Cut-elimination for Intuitionistic Provability Logic

Introduction to provability logic Cut-elimination Gentzen style G3ip+GLR ◮ A := p | ⊥ | A ∨ A | A ∧ A | A → A | � A ◮ Define ¬ A = A → ⊥ and ⊡ Γ = Γ ∪ � Γ p , Γ ⇒ p , p atomic (At) Γ , ⊥ ⇒ C (L ⊥ ) Γ , A , B ⇒ C Γ ⇒ A Γ ⇒ B R ∧ L ∧ Γ ⇒ A ∧ B Γ , A ∧ B ⇒ C Γ , A ⇒ C Γ , B ⇒ C Γ ⇒ A i R ∨ i L ∨ Γ ⇒ A 1 ∨ A 2 Γ , A ∨ B ⇒ C Γ , A ⇒ B Γ , A → B ⇒ A Γ , B ⇒ C R → L → Γ ⇒ A → B Γ , A → B ⇒ C ⊡ Γ , � A ⇒ A GLR Π , � Γ ⇒ � A տ Π can also contain boxed formulas. Iris van der Giessen Cut-elimination for Intuitionistic Provability Logic

Introduction to provability logic Cut-elimination Gentzen style G3ip+GLR ◮ A := p | ⊥ | A ∨ A | A ∧ A | A → A | � A ◮ Define ¬ A = A → ⊥ and ⊡ Γ = Γ ∪ � Γ p , Γ ⇒ p , p atomic (At) Γ , ⊥ ⇒ C (L ⊥ ) Γ , A , B ⇒ C Γ ⇒ A Γ ⇒ B R ∧ L ∧ Γ ⇒ A ∧ B Γ , A ∧ B ⇒ C Γ , A ⇒ C Γ , B ⇒ C Γ ⇒ A i R ∨ i L ∨ Γ ⇒ A 1 ∨ A 2 Γ , A ∨ B ⇒ C Γ , A ⇒ B Γ , A → B ⇒ A Γ , B ⇒ C R → L → Γ ⇒ A → B Γ , A → B ⇒ C ⊡ Γ , � A ⇒ A GLR Π , � Γ ⇒ � A տ Π can also contain boxed formulas. Iris van der Giessen Cut-elimination for Intuitionistic Provability Logic

Introduction to provability logic Cut-elimination Gentzen style G3ip+GLR ◮ A := p | ⊥ | A ∨ A | A ∧ A | A → A | � A ◮ Define ¬ A = A → ⊥ and ⊡ Γ = Γ ∪ � Γ p , Γ ⇒ p , p atomic (At) Γ , ⊥ ⇒ C (L ⊥ ) Γ , A , B ⇒ C Γ ⇒ A Γ ⇒ B R ∧ L ∧ Γ ⇒ A ∧ B Γ , A ∧ B ⇒ C Γ , A ⇒ C Γ , B ⇒ C Γ ⇒ A i R ∨ i L ∨ Γ ⇒ A 1 ∨ A 2 Γ , A ∨ B ⇒ C Γ , A ⇒ B Γ , A → B ⇒ A Γ , B ⇒ C R → L → Γ ⇒ A → B Γ , A → B ⇒ C ⊡ Γ , � A ⇒ A GLR Π , � Γ ⇒ � A տ Π can also contain boxed formulas. Iris van der Giessen Cut-elimination for Intuitionistic Provability Logic