Interpretability and the arithmetized completeness theorem - PowerPoint PPT Presentation

Interpretability The arithmetized completeness theorem An application More investigations Countable completeness theorem { θ n } n ∈ ω : p.r. enumeration of all L C -sentences. X 0 := T + Z ; � X n ∪ { θ n } if X n ∪ { θ n } is consistent; X n +1 := X n ∪ {¬ θ n } otherwise . X := � n ∈ ω X n . X is Henkin complete. Define an equivalence relation ∼ on C by c ∼ d : ⇔ c = d ∈ X . Define a structure M by |M| := C/ ∼ and R M ([ c 0 ] , . . . , [ c n ]) : ⇔ R ( c 0 , . . . , c n ) ∈ X etc...

Interpretability The arithmetized completeness theorem An application More investigations Countable completeness theorem { θ n } n ∈ ω : p.r. enumeration of all L C -sentences. X 0 := T + Z ; � X n ∪ { θ n } if X n ∪ { θ n } is consistent; X n +1 := X n ∪ {¬ θ n } otherwise . X := � n ∈ ω X n . X is Henkin complete. Define an equivalence relation ∼ on C by c ∼ d : ⇔ c = d ∈ X . Define a structure M by |M| := C/ ∼ and R M ([ c 0 ] , . . . , [ c n ]) : ⇔ R ( c 0 , . . . , c n ) ∈ X etc... ∀ ϕ : L C -sentence, ( M | = ϕ ⇔ ϕ ∈ X ). M is a model of T .

Interpretability The arithmetized completeness theorem An application More investigations Countable completeness theorem { θ n } n ∈ ω : p.r. enumeration of all L C -sentences. X 0 := T + Z ; � X n ∪ { θ n } if X n ∪ { θ n } is consistent; X n +1 := X n ∪ {¬ θ n } otherwise . X := � n ∈ ω X n . X is Henkin complete. Define an equivalence relation ∼ on C by c ∼ d : ⇔ c = d ∈ X . Define a structure M by |M| := C/ ∼ and R M ([ c 0 ] , . . . , [ c n ]) : ⇔ R ( c 0 , . . . , c n ) ∈ X etc... ∀ ϕ : L C -sentence, ( M | = ϕ ⇔ ϕ ∈ X ). M is a model of T .

Interpretability The arithmetized completeness theorem An application More investigations The arithmetized completeness theorem First, we arithmetize the notion of the provability.

Interpretability The arithmetized completeness theorem An application More investigations The arithmetized completeness theorem First, we arithmetize the notion of the provability. S : r.e. L -theory, T : L A -theory. ( L : countable)

Interpretability The arithmetized completeness theorem An application More investigations The arithmetized completeness theorem First, we arithmetize the notion of the provability. S : r.e. L -theory, T : L A -theory. ( L : countable) ∃ σ ( x ) : Σ 1 formula s.t. ∀ ϕ : L -sentence, ϕ ∈ S ⇔ T ⊢ σ ( � ϕ � ) . σ ( x ) is called a numeration of S in T .

Interpretability The arithmetized completeness theorem An application More investigations The arithmetized completeness theorem First, we arithmetize the notion of the provability. S : r.e. L -theory, T : L A -theory. ( L : countable) ∃ σ ( x ) : Σ 1 formula s.t. ∀ ϕ : L -sentence, ϕ ∈ S ⇔ T ⊢ σ ( � ϕ � ) . σ ( x ) is called a numeration of S in T . For Σ 1 numeration σ ( x ) of S in T , we can construct a Σ 1 formula Pr σ ( x ) s.t. ∀ ϕ : L -sentence, S ⊢ ϕ ⇔ T ⊢ Pr σ ( � ϕ � ) . Pr σ ( x ) is called the provability predicate of σ ( x ) .

Interpretability The arithmetized completeness theorem An application More investigations The arithmetized completeness theorem σ ( x ) , σ ′ ( x ) : numerations of S and S ′ in T respectively. Define ( σ | n )( x ) := σ ( x ) ∧ x ≤ ¯ n . ( σ ∨ σ ′ )( x ) := σ ( x ) ∨ σ ′ ( x ) .

Interpretability The arithmetized completeness theorem An application More investigations The arithmetized completeness theorem σ ( x ) , σ ′ ( x ) : numerations of S and S ′ in T respectively. Define ( σ | n )( x ) := σ ( x ) ∧ x ≤ ¯ n . ( σ ∨ σ ′ )( x ) := σ ( x ) ∨ σ ′ ( x ) . ( σ | n )( x ) is a numeration of { ϕ ∈ S | � ϕ � ≤ n } .

Interpretability The arithmetized completeness theorem An application More investigations The arithmetized completeness theorem σ ( x ) , σ ′ ( x ) : numerations of S and S ′ in T respectively. Define ( σ | n )( x ) := σ ( x ) ∧ x ≤ ¯ n . ( σ ∨ σ ′ )( x ) := σ ( x ) ∨ σ ′ ( x ) . ( σ | n )( x ) is a numeration of { ϕ ∈ S | � ϕ � ≤ n } . Con σ : ≡ ¬ Pr σ ( � 0 = ¯ 1 � ) .

Interpretability The arithmetized completeness theorem An application More investigations The arithmetized completeness theorem σ ( x ) , σ ′ ( x ) : numerations of S and S ′ in T respectively. Define ( σ | n )( x ) := σ ( x ) ∧ x ≤ ¯ n . ( σ ∨ σ ′ )( x ) := σ ( x ) ∨ σ ′ ( x ) . ( σ | n )( x ) is a numeration of { ϕ ∈ S | � ϕ � ≤ n } . Con σ : ≡ ¬ Pr σ ( � 0 = ¯ 1 � ) . Theorem T : consistent r.e. extension of PA, σ ( x ) : numeration of T in T . Then

Interpretability The arithmetized completeness theorem An application More investigations The arithmetized completeness theorem σ ( x ) , σ ′ ( x ) : numerations of S and S ′ in T respectively. Define ( σ | n )( x ) := σ ( x ) ∧ x ≤ ¯ n . ( σ ∨ σ ′ )( x ) := σ ( x ) ∨ σ ′ ( x ) . ( σ | n )( x ) is a numeration of { ϕ ∈ S | � ϕ � ≤ n } . Con σ : ≡ ¬ Pr σ ( � 0 = ¯ 1 � ) . Theorem T : consistent r.e. extension of PA, σ ( x ) : numeration of T in T . Then 1 (G¨ odel, Feferman) If σ is Σ 1 , then T � Con σ .

Interpretability The arithmetized completeness theorem An application More investigations The arithmetized completeness theorem σ ( x ) , σ ′ ( x ) : numerations of S and S ′ in T respectively. Define ( σ | n )( x ) := σ ( x ) ∧ x ≤ ¯ n . ( σ ∨ σ ′ )( x ) := σ ( x ) ∨ σ ′ ( x ) . ( σ | n )( x ) is a numeration of { ϕ ∈ S | � ϕ � ≤ n } . Con σ : ≡ ¬ Pr σ ( � 0 = ¯ 1 � ) . Theorem T : consistent r.e. extension of PA, σ ( x ) : numeration of T in T . Then 1 (G¨ odel, Feferman) If σ is Σ 1 , then T � Con σ . 2 (Mostowski) ∀ n ∈ ω , T ⊢ Con σ | n .

Interpretability The arithmetized completeness theorem An application More investigations The arithmetized completeness theorem Next we arithmetize the above construction of Henkin extension T + Z .

Interpretability The arithmetized completeness theorem An application More investigations The arithmetized completeness theorem Next we arithmetize the above construction of Henkin extension T + Z . C := { c n | n ∈ ω } : set of new constants. L C := L ∪ C .

Interpretability The arithmetized completeness theorem An application More investigations The arithmetized completeness theorem Next we arithmetize the above construction of Henkin extension T + Z . C := { c n | n ∈ ω } : set of new constants. L C := L ∪ C . odel numbering of L -formulas to L C -formulas. Extend G¨

Interpretability The arithmetized completeness theorem An application More investigations The arithmetized completeness theorem Next we arithmetize the above construction of Henkin extension T + Z . C := { c n | n ∈ ω } : set of new constants. L C := L ∪ C . odel numbering of L -formulas to L C -formulas. Extend G¨ Fml C ( x ) · · · “ x is an L C -formula”. C ( x ) · · · “ x is a new constant”.

Interpretability The arithmetized completeness theorem An application More investigations The arithmetized completeness theorem Next we arithmetize the above construction of Henkin extension T + Z . C := { c n | n ∈ ω } : set of new constants. L C := L ∪ C . odel numbering of L -formulas to L C -formulas. Extend G¨ Fml C ( x ) · · · “ x is an L C -formula”. C ( x ) · · · “ x is a new constant”. Define the p.r. set Z as above. Let ζ ( x ) be a suitable numeration of Z s.t. ∀ ϕ , PA ⊢ Pr σ ∨ ζ ( � ϕ � ) → Pr σ ( � ϕ � ) .

Interpretability The arithmetized completeness theorem An application More investigations The arithmetized completeness theorem Next we arithmetize the above construction of Henkin extension T + Z . C := { c n | n ∈ ω } : set of new constants. L C := L ∪ C . odel numbering of L -formulas to L C -formulas. Extend G¨ Fml C ( x ) · · · “ x is an L C -formula”. C ( x ) · · · “ x is a new constant”. Define the p.r. set Z as above. Let ζ ( x ) be a suitable numeration of Z s.t. ∀ ϕ , PA ⊢ Pr σ ∨ ζ ( � ϕ � ) → Pr σ ( � ϕ � ) . PA ⊢ Con σ → Con σ ∨ ζ .

Interpretability The arithmetized completeness theorem An application More investigations The arithmetized completeness theorem Lastly, we arithmetize the Henkin completeness.

Interpretability The arithmetized completeness theorem An application More investigations The arithmetized completeness theorem Lastly, we arithmetize the Henkin completeness. For any L A -formula ξ ( x ) , define Hcm ξ to be the conjunction of the following L A -sentences: ∀ x (Fml C ( x ) → ( ξ ( ¬ x ) ↔ ¬ ξ ( x ))) ; ∀ x, y (Fml C ( x ) ∧ Fml C ( y ) → ( ξ ( x ∨ y ) ↔ ( ξ ( x ) ∨ ξ ( y )))) ; · · · ; ∀ x, y (Fml C ( x ) → ( ξ ( ∃ ux ) ↔ ∃ v ( C ( v ) ∧ ξ ( x [ v/u ])))) .

Interpretability The arithmetized completeness theorem An application More investigations The arithmetized completeness theorem Lastly, we arithmetize the Henkin completeness. For any L A -formula ξ ( x ) , define Hcm ξ to be the conjunction of the following L A -sentences: ∀ x (Fml C ( x ) → ( ξ ( ¬ x ) ↔ ¬ ξ ( x ))) ; ∀ x, y (Fml C ( x ) ∧ Fml C ( y ) → ( ξ ( x ∨ y ) ↔ ( ξ ( x ) ∨ ξ ( y )))) ; · · · ; ∀ x, y (Fml C ( x ) → ( ξ ( ∃ ux ) ↔ ∃ v ( C ( v ) ∧ ξ ( x [ v/u ])))) . Hcm ξ states that the set defined by ξ ( x ) is Henkin complete.

Interpretability The arithmetized completeness theorem An application More investigations The arithmetized completeness theorem Lastly, we arithmetize the Henkin completeness. For any L A -formula ξ ( x ) , define Hcm ξ to be the conjunction of the following L A -sentences: ∀ x (Fml C ( x ) → ( ξ ( ¬ x ) ↔ ¬ ξ ( x ))) ; ∀ x, y (Fml C ( x ) ∧ Fml C ( y ) → ( ξ ( x ∨ y ) ↔ ( ξ ( x ) ∨ ξ ( y )))) ; · · · ; ∀ x, y (Fml C ( x ) → ( ξ ( ∃ ux ) ↔ ∃ v ( C ( v ) ∧ ξ ( x [ v/u ])))) . Hcm ξ states that the set defined by ξ ( x ) is Henkin complete. The arithmetized completeness theorem ∀ σ ( x ) : numeration of T , ∃ ξ ( x ) : L A -formula s.t. 1 PA ⊢ Con σ → Hcm ξ and 2 PA ⊢ ∀ x (Pr σ ( x ) → ξ ( x )) .

Interpretability The arithmetized completeness theorem An application More investigations The arithmetized completeness theorem Proof. { θ n } n ∈ ω : p.r. enumeration of all L C -sentences. ξ ( x ) · · · “ x is contained in the leftmost consistent path”. PA ⊢ ∀ x (Pr σ ( x ) → ξ ( x )) . PA ⊢ Con σ ∨ ζ → Hcm ξ .

Interpretability The arithmetized completeness theorem An application More investigations The arithmetized completeness theorem Theorem(Feferman, 1960) T : extension of PA. σ : numeration of S in T . Then S ≤ T + Con σ .

Interpretability The arithmetized completeness theorem An application More investigations The arithmetized completeness theorem Theorem(Feferman, 1960) T : extension of PA. σ : numeration of S in T . Then S ≤ T + Con σ . Proof. ξ ( x ) : as in the arithmetized completeness theorem.

Interpretability The arithmetized completeness theorem An application More investigations The arithmetized completeness theorem Theorem(Feferman, 1960) T : extension of PA. σ : numeration of S in T . Then S ≤ T + Con σ . Proof. ξ ( x ) : as in the arithmetized completeness theorem. d ( x ) : ≡ x = x , η c ( x ) : ≡ C ( x ) ∧ ξ ( � c = ˙ x � ) , · · ·

Interpretability The arithmetized completeness theorem An application More investigations The arithmetized completeness theorem Theorem(Feferman, 1960) T : extension of PA. σ : numeration of S in T . Then S ≤ T + Con σ . Proof. ξ ( x ) : as in the arithmetized completeness theorem. d ( x ) : ≡ x = x , η c ( x ) : ≡ C ( x ) ∧ ξ ( � c = ˙ x � ) , · · · By induction, ∀ ϕ , PA + Hcm ξ ⊢ t ( ϕ ) ↔ ξ ( � ϕ � ) .

Interpretability The arithmetized completeness theorem An application More investigations The arithmetized completeness theorem Theorem(Feferman, 1960) T : extension of PA. σ : numeration of S in T . Then S ≤ T + Con σ . Proof. ξ ( x ) : as in the arithmetized completeness theorem. d ( x ) : ≡ x = x , η c ( x ) : ≡ C ( x ) ∧ ξ ( � c = ˙ x � ) , · · · By induction, ∀ ϕ , PA + Hcm ξ ⊢ t ( ϕ ) ↔ ξ ( � ϕ � ) . ∀ ϕ , T + Con σ ⊢ t ( ϕ ) ↔ ξ ( � ϕ � ) .

Interpretability The arithmetized completeness theorem An application More investigations The arithmetized completeness theorem Theorem(Feferman, 1960) T : extension of PA. σ : numeration of S in T . Then S ≤ T + Con σ . Proof. ξ ( x ) : as in the arithmetized completeness theorem. d ( x ) : ≡ x = x , η c ( x ) : ≡ C ( x ) ∧ ξ ( � c = ˙ x � ) , · · · By induction, ∀ ϕ , PA + Hcm ξ ⊢ t ( ϕ ) ↔ ξ ( � ϕ � ) . ∀ ϕ , T + Con σ ⊢ t ( ϕ ) ↔ ξ ( � ϕ � ) . t is a translation of L S into L T + Con σ .

Interpretability The arithmetized completeness theorem An application More investigations The arithmetized completeness theorem Theorem(Feferman, 1960) T : extension of PA. σ : numeration of S in T . Then S ≤ T + Con σ . Proof. ξ ( x ) : as in the arithmetized completeness theorem. d ( x ) : ≡ x = x , η c ( x ) : ≡ C ( x ) ∧ ξ ( � c = ˙ x � ) , · · · By induction, ∀ ϕ , PA + Hcm ξ ⊢ t ( ϕ ) ↔ ξ ( � ϕ � ) . ∀ ϕ , T + Con σ ⊢ t ( ϕ ) ↔ ξ ( � ϕ � ) . t is a translation of L S into L T + Con σ . ϕ : L S -sentence s.t. S ⊢ ϕ . T ⊢ Pr σ ( � ϕ � ) .

Interpretability The arithmetized completeness theorem An application More investigations The arithmetized completeness theorem Theorem(Feferman, 1960) T : extension of PA. σ : numeration of S in T . Then S ≤ T + Con σ . Proof. ξ ( x ) : as in the arithmetized completeness theorem. d ( x ) : ≡ x = x , η c ( x ) : ≡ C ( x ) ∧ ξ ( � c = ˙ x � ) , · · · By induction, ∀ ϕ , PA + Hcm ξ ⊢ t ( ϕ ) ↔ ξ ( � ϕ � ) . ∀ ϕ , T + Con σ ⊢ t ( ϕ ) ↔ ξ ( � ϕ � ) . t is a translation of L S into L T + Con σ . ϕ : L S -sentence s.t. S ⊢ ϕ . T ⊢ Pr σ ( � ϕ � ) . T + Con σ ⊢ ξ ( � ϕ � ) . T + Con σ ⊢ t ( ϕ ) .

Interpretability The arithmetized completeness theorem An application More investigations 1 Interpretability 2 The arithmetized completeness theorem 3 An application 4 More investigations

Interpretability The arithmetized completeness theorem An application More investigations An application We can construct a model by interpretation.

Interpretability The arithmetized completeness theorem An application More investigations An application We can construct a model by interpretation. t : interpretation of S in T , M : model of T .

Interpretability The arithmetized completeness theorem An application More investigations An application We can construct a model by interpretation. t : interpretation of S in T , M : model of T . Define an L S -structure N as follows: |N | := { a ∈ |M| : M | = d ( a ) } ; For c ∈ L S : constant, c N := the unique a ∈ |M| s.t. M | = d ( a ) ∧ η c ( a ) ; · · · .

Interpretability The arithmetized completeness theorem An application More investigations An application We can construct a model by interpretation. t : interpretation of S in T , M : model of T . Define an L S -structure N as follows: |N | := { a ∈ |M| : M | = d ( a ) } ; For c ∈ L S : constant, c N := the unique a ∈ |M| s.t. M | = d ( a ) ∧ η c ( a ) ; · · · . By induction, ∀ ϕ : L S -sentence, M | = t ( ϕ ) ⇔ N | = ϕ .

Interpretability The arithmetized completeness theorem An application More investigations An application We can construct a model by interpretation. t : interpretation of S in T , M : model of T . Define an L S -structure N as follows: |N | := { a ∈ |M| : M | = d ( a ) } ; For c ∈ L S : constant, c N := the unique a ∈ |M| s.t. M | = d ( a ) ∧ η c ( a ) ; · · · . By induction, ∀ ϕ : L S -sentence, M | = t ( ϕ ) ⇔ N | = ϕ . Suppose S ⊢ ϕ .

Interpretability The arithmetized completeness theorem An application More investigations An application We can construct a model by interpretation. t : interpretation of S in T , M : model of T . Define an L S -structure N as follows: |N | := { a ∈ |M| : M | = d ( a ) } ; For c ∈ L S : constant, c N := the unique a ∈ |M| s.t. M | = d ( a ) ∧ η c ( a ) ; · · · . By induction, ∀ ϕ : L S -sentence, M | = t ( ϕ ) ⇔ N | = ϕ . Suppose S ⊢ ϕ . Since S ≤ T , T ⊢ t ( ϕ ) .

Interpretability The arithmetized completeness theorem An application More investigations An application We can construct a model by interpretation. t : interpretation of S in T , M : model of T . Define an L S -structure N as follows: |N | := { a ∈ |M| : M | = d ( a ) } ; For c ∈ L S : constant, c N := the unique a ∈ |M| s.t. M | = d ( a ) ∧ η c ( a ) ; · · · . By induction, ∀ ϕ : L S -sentence, M | = t ( ϕ ) ⇔ N | = ϕ . Suppose S ⊢ ϕ . Since S ≤ T , T ⊢ t ( ϕ ) . M | = t ( ϕ ) , so N | = ϕ .

Interpretability The arithmetized completeness theorem An application More investigations An application We can construct a model by interpretation. t : interpretation of S in T , M : model of T . Define an L S -structure N as follows: |N | := { a ∈ |M| : M | = d ( a ) } ; For c ∈ L S : constant, c N := the unique a ∈ |M| s.t. M | = d ( a ) ∧ η c ( a ) ; · · · . By induction, ∀ ϕ : L S -sentence, M | = t ( ϕ ) ⇔ N | = ϕ . Suppose S ⊢ ϕ . Since S ≤ T , T ⊢ t ( ϕ ) . M | = t ( ϕ ) , so N | = ϕ . N is a model of S .

Interpretability The arithmetized completeness theorem An application More investigations An application Definition M , N : models of arithmetic. def . M is an initial segment of N ( M ⊆ e N ) ⇔ 1 |M| ⊆ |N | and 2 ∀ a ∈ |M|∀ b ∈ |N | , ( N | = b < a ⇒ b ∈ |M| ).

Interpretability The arithmetized completeness theorem An application More investigations An application Definition M , N : models of arithmetic. def . M is an initial segment of N ( M ⊆ e N ) ⇔ 1 |M| ⊆ |N | and 2 ∀ a ∈ |M|∀ b ∈ |N | , ( N | = b < a ⇒ b ∈ |M| ). Theorem(Orey(1961), H´ ajek (1971,1972)) For any consistent r.e. extensions S, T of PA, T.F.A.E.: (i) S ≤ T . (ii) ∀M | = T ∃N | = S s.t. M ⊆ e N . (iii) ∀ θ : Π 1 sentence, S ⊢ θ ⇒ T ⊢ θ . (iv) ∀ σ ( x ) : Σ 1 numeration of S ∀ n ∈ ω , T ⊢ Con σ | n .

Interpretability The arithmetized completeness theorem An application More investigations An application (i) S ≤ T . (ii) ∀M | = T ∃N | = S s.t. M ⊆ e N . (i) ⇒ (ii) Let θ be an interpretation of S in T . Let M be any model of T .

Interpretability The arithmetized completeness theorem An application More investigations An application (i) S ≤ T . (ii) ∀M | = T ∃N | = S s.t. M ⊆ e N . (i) ⇒ (ii) Let θ be an interpretation of S in T . Let M be any model of T . Let N be a model of S defined by t and M .

Interpretability The arithmetized completeness theorem An application More investigations An application (i) S ≤ T . (ii) ∀M | = T ∃N | = S s.t. M ⊆ e N . (i) ⇒ (ii) Let θ be an interpretation of S in T . Let M be any model of T . Let N be a model of S defined by t and M . Define a function f in M satisfying f (0 M ) = 0 N and f ( S M ( a )) = S N ( f ( a )) .

Interpretability The arithmetized completeness theorem An application More investigations An application (i) S ≤ T . (ii) ∀M | = T ∃N | = S s.t. M ⊆ e N . (i) ⇒ (ii) Let θ be an interpretation of S in T . Let M be any model of T . Let N be a model of S defined by t and M . Define a function f in M satisfying f (0 M ) = 0 N and f ( S M ( a )) = S N ( f ( a )) . Then f is an isomorphism of an initial segment of N .

Interpretability The arithmetized completeness theorem An application More investigations An application (ii) ∀M | = T ∃N | = S s.t. M ⊆ e N . (iii) ∀ θ : Π 1 sentence, S ⊢ θ ⇒ T ⊢ θ . (ii) ⇒ (iii) Let θ be any Π 1 sentence s.t. S ⊢ θ . Let M be any model of T .

Interpretability The arithmetized completeness theorem An application More investigations An application (ii) ∀M | = T ∃N | = S s.t. M ⊆ e N . (iii) ∀ θ : Π 1 sentence, S ⊢ θ ⇒ T ⊢ θ . (ii) ⇒ (iii) Let θ be any Π 1 sentence s.t. S ⊢ θ . Let M be any model of T . By (ii), ∃N | = S s.t. M | = N .

Interpretability The arithmetized completeness theorem An application More investigations An application (ii) ∀M | = T ∃N | = S s.t. M ⊆ e N . (iii) ∀ θ : Π 1 sentence, S ⊢ θ ⇒ T ⊢ θ . (ii) ⇒ (iii) Let θ be any Π 1 sentence s.t. S ⊢ θ . Let M be any model of T . By (ii), ∃N | = S s.t. M | = N . N | = θ .

Interpretability The arithmetized completeness theorem An application More investigations An application (ii) ∀M | = T ∃N | = S s.t. M ⊆ e N . (iii) ∀ θ : Π 1 sentence, S ⊢ θ ⇒ T ⊢ θ . (ii) ⇒ (iii) Let θ be any Π 1 sentence s.t. S ⊢ θ . Let M be any model of T . By (ii), ∃N | = S s.t. M | = N . N | = θ . Since θ is Π 1 , M | = θ .

Interpretability The arithmetized completeness theorem An application More investigations An application (ii) ∀M | = T ∃N | = S s.t. M ⊆ e N . (iii) ∀ θ : Π 1 sentence, S ⊢ θ ⇒ T ⊢ θ . (ii) ⇒ (iii) Let θ be any Π 1 sentence s.t. S ⊢ θ . Let M be any model of T . By (ii), ∃N | = S s.t. M | = N . N | = θ . Since θ is Π 1 , M | = θ . By completeness theorem, T ⊢ θ .

Interpretability The arithmetized completeness theorem An application More investigations An application (i) S ≤ T . (iii) ∀ θ : Π 1 sentence, S ⊢ θ ⇒ T ⊢ θ . (iv) ∀ σ ( x ) : Σ 1 numeration of S ∀ n ∈ ω , T ⊢ Con σ | n . (iii) ⇒ (iv) By Mostowski’s theorem, ∀ n ∈ ω , S ⊢ Con σ | n .

Interpretability The arithmetized completeness theorem An application More investigations An application (i) S ≤ T . (iii) ∀ θ : Π 1 sentence, S ⊢ θ ⇒ T ⊢ θ . (iv) ∀ σ ( x ) : Σ 1 numeration of S ∀ n ∈ ω , T ⊢ Con σ | n . (iii) ⇒ (iv) By Mostowski’s theorem, ∀ n ∈ ω , S ⊢ Con σ | n . By (iii), ∀ n ∈ ω , T ⊢ Con σ | n .

Interpretability The arithmetized completeness theorem An application More investigations An application (i) S ≤ T . (iii) ∀ θ : Π 1 sentence, S ⊢ θ ⇒ T ⊢ θ . (iv) ∀ σ ( x ) : Σ 1 numeration of S ∀ n ∈ ω , T ⊢ Con σ | n . (iii) ⇒ (iv) By Mostowski’s theorem, ∀ n ∈ ω , S ⊢ Con σ | n . By (iii), ∀ n ∈ ω , T ⊢ Con σ | n . (iv) ⇒ (i). Let σ ∗ ( x ) : ≡ σ ( x ) ∧ Con σ | x .

Interpretability The arithmetized completeness theorem An application More investigations An application (i) S ≤ T . (iii) ∀ θ : Π 1 sentence, S ⊢ θ ⇒ T ⊢ θ . (iv) ∀ σ ( x ) : Σ 1 numeration of S ∀ n ∈ ω , T ⊢ Con σ | n . (iii) ⇒ (iv) By Mostowski’s theorem, ∀ n ∈ ω , S ⊢ Con σ | n . By (iii), ∀ n ∈ ω , T ⊢ Con σ | n . (iv) ⇒ (i). Let σ ∗ ( x ) : ≡ σ ( x ) ∧ Con σ | x . Then σ ∗ ( x ) numerates S in T and PA ⊢ Con σ ∗ .

Interpretability The arithmetized completeness theorem An application More investigations An application (i) S ≤ T . (iii) ∀ θ : Π 1 sentence, S ⊢ θ ⇒ T ⊢ θ . (iv) ∀ σ ( x ) : Σ 1 numeration of S ∀ n ∈ ω , T ⊢ Con σ | n . (iii) ⇒ (iv) By Mostowski’s theorem, ∀ n ∈ ω , S ⊢ Con σ | n . By (iii), ∀ n ∈ ω , T ⊢ Con σ | n . (iv) ⇒ (i). Let σ ∗ ( x ) : ≡ σ ( x ) ∧ Con σ | x . Then σ ∗ ( x ) numerates S in T and PA ⊢ Con σ ∗ . By Feferman’s theorem, S ≤ T + Con σ ∗ .

Interpretability The arithmetized completeness theorem An application More investigations An application (i) S ≤ T . (iii) ∀ θ : Π 1 sentence, S ⊢ θ ⇒ T ⊢ θ . (iv) ∀ σ ( x ) : Σ 1 numeration of S ∀ n ∈ ω , T ⊢ Con σ | n . (iii) ⇒ (iv) By Mostowski’s theorem, ∀ n ∈ ω , S ⊢ Con σ | n . By (iii), ∀ n ∈ ω , T ⊢ Con σ | n . (iv) ⇒ (i). Let σ ∗ ( x ) : ≡ σ ( x ) ∧ Con σ | x . Then σ ∗ ( x ) numerates S in T and PA ⊢ Con σ ∗ . By Feferman’s theorem, S ≤ T + Con σ ∗ . S ≤ T .

Interpretability The arithmetized completeness theorem An application More investigations 1 Interpretability 2 The arithmetized completeness theorem 3 An application 4 More investigations

Interpretability The arithmetized completeness theorem An application More investigations Incompleteness Model theoretic proof of the second incompleteness theorem (Kreisel)(Kikuchi,1994)

Interpretability The arithmetized completeness theorem An application More investigations Incompleteness Model theoretic proof of the second incompleteness theorem (Kreisel)(Kikuchi,1994) Theorem T, S : consistent r.e. extensions of PA. M : models of T . If M | = Con S , then ∃N | = S s.t. M ⊆ e N and ∃ ξ ( x ) : L A -formula s.t. ∀ ϕ , ( M | = Pr S ( � ϕ � ) ⇒ N | = ϕ ), ∀ ϕ , ( M | = ξ ( � ϕ � ) ⇔ N | = ϕ ).

Interpretability The arithmetized completeness theorem An application More investigations Incompleteness Model theoretic proof of the second incompleteness theorem (Kreisel)(Kikuchi,1994) Theorem T, S : consistent r.e. extensions of PA. M : models of T . If M | = Con S , then ∃N | = S s.t. M ⊆ e N and ∃ ξ ( x ) : L A -formula s.t. ∀ ϕ , ( M | = Pr S ( � ϕ � ) ⇒ N | = ϕ ), ∀ ϕ , ( M | = ξ ( � ϕ � ) ⇔ N | = ϕ ). Suppose ∀M | = T , M | = Con T . By using Theorem, lead a contradiction.

Interpretability The arithmetized completeness theorem An application More investigations Faithful interpretability Faithful interpretability (Feferman, Kreisel, Orey, 1960)(Lindstr¨ om, 1984)

Interpretability The arithmetized completeness theorem An application More investigations Faithful interpretability Faithful interpretability (Feferman, Kreisel, Orey, 1960)(Lindstr¨ om, 1984) Definition A interpretation t of S in T is faithful def . ⇔ ∀ ϕ , ( T ⊢ t ( ϕ ) ⇒ S ⊢ ϕ ).

Interpretability The arithmetized completeness theorem An application More investigations Faithful interpretability Faithful interpretability (Feferman, Kreisel, Orey, 1960)(Lindstr¨ om, 1984) Definition A interpretation t of S in T is faithful def . ⇔ ∀ ϕ , ( T ⊢ t ( ϕ ) ⇒ S ⊢ ϕ ). S is faithful interpretable in T def . ⇔ ∃ t : faithful interpretation of S in T .

Interpretability The arithmetized completeness theorem An application More investigations Faithful interpretability Faithful interpretability (Feferman, Kreisel, Orey, 1960)(Lindstr¨ om, 1984) Definition A interpretation t of S in T is faithful def . ⇔ ∀ ϕ , ( T ⊢ t ( ϕ ) ⇒ S ⊢ ϕ ). S is faithful interpretable in T def . ⇔ ∃ t : faithful interpretation of S in T . Theorem(Lindstr¨ om) S, T : r.e. extensions of PA. T.F.A.E.: S is faithful interpretable in T . 1 S ≤ T and ∀ ϕ , ( T ⊢ Pr φ ( � ϕ � ) ⇒ S ⊢ ϕ ). 2 ∀ θ : Π 1 sentence, ( S ⊢ θ ⇒ T ⊢ θ ) and 3 ∀ σ : Σ 1 sentence, ( T ⊢ σ ⇒ S ⊢ θ ) and

Interpretability The arithmetized completeness theorem An application More investigations Degrees of interpretability Degrees of interpretability of extensions of PA (Lindstr¨ om, 1979)

Interpretability The arithmetized completeness theorem An application More investigations Degrees of interpretability Degrees of interpretability of extensions of PA (Lindstr¨ om, 1979) Definition S, T : extensions of PA. def . S ≡ T ⇔ S ≤ T & T ≤ S .

Interpretability The arithmetized completeness theorem An application More investigations Degrees of interpretability Degrees of interpretability of extensions of PA (Lindstr¨ om, 1979) Definition S, T : extensions of PA. def . S ≡ T ⇔ S ≤ T & T ≤ S . ≡ is an equivalence relation on extensions of PA.

Interpretability The arithmetized completeness theorem An application More investigations Degrees of interpretability Degrees of interpretability of extensions of PA (Lindstr¨ om, 1979) Definition S, T : extensions of PA. def . S ≡ T ⇔ S ≤ T & T ≤ S . ≡ is an equivalence relation on extensions of PA. Equivalence classes are called degrees of interpretability.

Interpretability The arithmetized completeness theorem An application More investigations Degrees of interpretability Degrees of interpretability of extensions of PA (Lindstr¨ om, 1979) Definition S, T : extensions of PA. def . S ≡ T ⇔ S ≤ T & T ≤ S . ≡ is an equivalence relation on extensions of PA. Equivalence classes are called degrees of interpretability. D T := the set of degrees of extensions of T .

Interpretability The arithmetized completeness theorem An application More investigations Degrees of interpretability Degrees of interpretability of extensions of PA (Lindstr¨ om, 1979) Definition S, T : extensions of PA. def . S ≡ T ⇔ S ≤ T & T ≤ S . ≡ is an equivalence relation on extensions of PA. Equivalence classes are called degrees of interpretability. D T := the set of degrees of extensions of T . d ( S ) := degree of S .