Recursive Axiomatizability A first order theory T is recursively axiomatizable iff • there is Γ such that • Γ ⊢ ϕ ⇐ ⇒ T ⊢ ϕ for any ϕ ∈ L T and • { � � | ϕ ∈ Γ } is decidable; ϕ – p. 14
Recursive Axiomatizability A first order theory T is recursively axiomatizable iff • there is Γ such that • Γ ⊢ ϕ ⇐ ⇒ T ⊢ ϕ for any ϕ ∈ L T and • { � � | ϕ ∈ Γ } is decidable; ϕ and/or • there is Γ such that • Γ ⊢ ϕ ⇐ ⇒ T ⊢ ϕ for any ϕ ∈ L T and • { � � | ϕ ∈ Γ } is semi-decidable; ϕ – p. 14
Recursive Axiomatizability A first order theory T is recursively axiomatizable iff • there is Γ such that • Γ ⊢ ϕ ⇐ ⇒ T ⊢ ϕ for any ϕ ∈ L T and • { � � | ϕ ∈ Γ } is decidable; ϕ and/or • there is Γ such that • Γ ⊢ ϕ ⇐ ⇒ T ⊢ ϕ for any ϕ ∈ L T and • { � � | ϕ ∈ Γ } is semi-decidable; ϕ and/or • { � � | T ⊢ ϕ } is semi-decidable. ϕ – p. 14
Recursive Axiomatizability A first order theory T is recursively axiomatizable iff • there is Γ such that • Γ ⊢ ϕ ⇐ ⇒ T ⊢ ϕ for any ϕ ∈ L T and • { � � | ϕ ∈ Γ } is decidable; ϕ and/or • there is Γ such that • Γ ⊢ ϕ ⇐ ⇒ T ⊢ ϕ for any ϕ ∈ L T and • { � � | ϕ ∈ Γ } is semi-decidable; ϕ and/or • { � � | T ⊢ ϕ } is semi-decidable. ϕ Th( N ) = { ϕ ∈ L PA | N | = ϕ } is negation complete. – p. 14
Craig’s Theorem If { � � | T ⊢ ϕ } is semi-decidable, ϕ then there is Γ such that • Γ ⊢ ϕ ⇐ ⇒ T ⊢ ϕ for any ϕ ∈ L T and • { � � | ϕ ∈ Γ } is decidable ϕ – p. 15
Craig’s Theorem If { � � | T ⊢ ϕ } is semi-decidable, ϕ then there is Γ such that • Γ ⊢ ϕ ⇐ ⇒ T ⊢ ϕ for any ϕ ∈ L T and • { � � | ϕ ∈ Γ } is decidable ϕ (Proof) Take a recursive predicate R such that T ⊢ ϕ ⇐ ⇒ ∃ nR ( � � , n ) for any ϕ ∈ L T . ϕ – p. 15
Craig’s Theorem If { � � | T ⊢ ϕ } is semi-decidable, ϕ then there is Γ such that • Γ ⊢ ϕ ⇐ ⇒ T ⊢ ϕ for any ϕ ∈ L T and • { � � | ϕ ∈ Γ } is decidable ϕ (Proof) Take a recursive predicate R such that T ⊢ ϕ ⇐ ⇒ ∃ nR ( � � , n ) for any ϕ ∈ L T . ϕ Define the following recursive set of axioms Γ = { ψ | ( ∃ n, � ψ � )( R ( � � , n ) & ψ ≡ ϕ ∧ ... ∧ ϕ ) } . ϕ � < � ϕ – p. 15
Craig’s Theorem If { � � | T ⊢ ϕ } is semi-decidable, ϕ then there is Γ such that • Γ ⊢ ϕ ⇐ ⇒ T ⊢ ϕ for any ϕ ∈ L T and • { � � | ϕ ∈ Γ } is decidable ϕ (Proof) Take a recursive predicate R such that T ⊢ ϕ ⇐ ⇒ ∃ nR ( � � , n ) for any ϕ ∈ L T . ϕ Define the following recursive set of axioms Γ = { ψ | ( ∃ n, � ψ � )( R ( � � , n ) & ψ ≡ ϕ ∧ ... ∧ ϕ ) } . ϕ � < � ϕ • ψ ∈ Γ ⇒ T ⊢ ϕ & ψ ≡ ϕ ∧ ... ∧ ϕ ⇒ T ⊢ ψ ; • T ⊢ ϕ ⇒ ∃ nR ( � � , n ) ⇒ ϕ ∧ ... ∧ ϕ ∈ Γ ⇒ Γ ⊢ ϕ . ϕ � �� � n +1 – p. 15
Henkin Construction Henkin’s Lemma : If Γ �⊢ ⊥ then there is maximal consistent ∆ ⊇ Γ . – p. 16
Henkin Construction Henkin’s Lemma : If Γ �⊢ ⊥ then there is maximal consistent ∆ ⊇ Γ . (Proof) Let ϕ n ’s enumerate all L formulae. Define � Γ n if Γ n ∪ { ϕ n } ⊢ ⊥ Γ n +1 := Γ n ∪ { ϕ n } if Γ n ∪ { ϕ n } �⊢ ⊥ . starting from Γ 0 := Γ . Take ∆ := � n ∈ ω Γ n . � – p. 16
Henkin Construction Henkin’s Lemma : If Γ �⊢ ⊥ then there is maximal consistent ∆ ⊇ Γ . (Proof) Let ϕ n ’s enumerate all L formulae. Define � Γ n if Γ n ∪ { ϕ n } ⊢ ⊥ Γ n +1 := Γ n ∪ { ϕ n } if Γ n ∪ { ϕ n } �⊢ ⊥ . starting from Γ 0 := Γ . Take ∆ := � n ∈ ω Γ n . � Note : The theory generated by ∆ is negation complete: either ϕ ∈ ∆ or ¬ ϕ ∈ ∆ holds for any ϕ ∈ L . – p. 16
The Statement (4) If a first order theory T satisfies the following: • T is consistent; • T is recursively axiomatizable; • T essentially contains Robinson Arithmetic Q , then the following hold: 1st incompleteness: T is not complete. – p. 17
Robinson Arithmetic Q Language (function) 0 ; S ( - ) ; + , · ; (relation) < . Axioms 1. ¬ ( S ( x ) = 0) ; 2. S ( x ) = S ( y ) → x = y ; 3. x = 0 ∨ ∃ y ( x = S ( y )) ; 4. x +0 = x ; and x + S ( y ) = S ( x + y ) ; 5. x · 0 = 0 ; and x · S ( y ) = ( x · y )+ x .; 6. x < y ↔ ∃ z ( x + S ( z ) = y ) . – p. 18
Robinson Arithmetic Q Language (function) 0 ; S ( - ) ; + , · ; (relation) < . Axioms 1. ¬ ( S ( x ) = 0) ; 2. S ( x ) = S ( y ) → x = y ; 3. x = 0 ∨ ∃ y ( x = S ( y )) ; 4. x +0 = x ; and x + S ( y ) = S ( x + y ) ; 5. x · 0 = 0 ; and x · S ( y ) = ( x · y )+ x .; 6. x < y ↔ ∃ z ( x + S ( z ) = y ) . Remarks • first introduced by R. M. Robison in 1950 w/o < ; • has no induction axiom (schema). – p. 18
Theories containing Q • PA extends Q by induction scheme: ϕ (0) ∧∀ x ( ϕ ( x ) → ϕ ( S ( x )) → ∀ xϕ ( x ) for ϕ ∈ L Q . – p. 19
Theories containing Q • PA extends Q by induction scheme: ϕ (0) ∧∀ x ( ϕ ( x ) → ϕ ( S ( x )) → ∀ xϕ ( x ) for ϕ ∈ L Q . • IΣ n extends Q by induction for Σ 0 n formulae: 1. Σ 0 x ) | ϕ ∈ ∆ 0 n = {∃ x n ∀ x n − 1 ...Qx 1 ϕ ( � 0 } and 2. ϕ ∈ ∆ 0 0 iff all quantifiers in ϕ are bounded (i.e., of the forms ∀ x < t and ∃ x < t ). – p. 19
Theories containing Q • PA extends Q by induction scheme: ϕ (0) ∧∀ x ( ϕ ( x ) → ϕ ( S ( x )) → ∀ xϕ ( x ) for ϕ ∈ L Q . • IΣ n extends Q by induction for Σ 0 n formulae: 1. Σ 0 x ) | ϕ ∈ ∆ 0 n = {∃ x n ∀ x n − 1 ...Qx 1 ϕ ( � 0 } and 2. ϕ ∈ ∆ 0 0 iff all quantifiers in ϕ are bounded (i.e., of the forms ∀ x < t and ∃ x < t ). • PRA extends Q by 1. L PRA := L Q ∪ {F | F ∈ PrimRec } ; 2. induction for quantifier-free L PRA formulae. – p. 19
Theories containing Q • PA extends Q by induction scheme: ϕ (0) ∧∀ x ( ϕ ( x ) → ϕ ( S ( x )) → ∀ xϕ ( x ) for ϕ ∈ L Q . • IΣ n extends Q by induction for Σ 0 n formulae: 1. Σ 0 x ) | ϕ ∈ ∆ 0 n = {∃ x n ∀ x n − 1 ...Qx 1 ϕ ( � 0 } and 2. ϕ ∈ ∆ 0 0 iff all quantifiers in ϕ are bounded (i.e., of the forms ∀ x < t and ∃ x < t ). • PRA extends Q by 1. L PRA := L Q ∪ {F | F ∈ PrimRec } ; 2. induction for quantifier-free L PRA formulae. • ZFC extends Q ... – p. 19
Theories containing Q • PA extends Q by induction scheme: ϕ (0) ∧∀ x ( ϕ ( x ) → ϕ ( S ( x )) → ∀ xϕ ( x ) for ϕ ∈ L Q . • IΣ n extends Q by induction for Σ 0 n formulae: 1. Σ 0 x ) | ϕ ∈ ∆ 0 n = {∃ x n ∀ x n − 1 ...Qx 1 ϕ ( � 0 } and 2. ϕ ∈ ∆ 0 0 iff all quantifiers in ϕ are bounded (i.e., of the forms ∀ x < t and ∃ x < t ). • PRA extends Q by 1. L PRA := L Q ∪ {F | F ∈ PrimRec } ; 2. induction for quantifier-free L PRA formulae. • ZFC extends Q ... really? in which sense? – p. 19
Interpretation An interpretation I of L in L ′ consists of: • an L ′ formula υ I ( x ) , called universe; x ) of L , an L ′ formula f I ( y,� • for function f ( � x ) ; x ) of L , an L ′ formula R I ( � • for relation R ( � x ) . – p. 20
Interpretation An interpretation I of L in L ′ consists of: • an L ′ formula υ I ( x ) , called universe; x ) of L , an L ′ formula f I ( y,� • for function f ( � x ) ; x ) of L , an L ′ formula R I ( � • for relation R ( � x ) . Extend I to all L -terms and L -formulae: • if t ( � x ) ≡ f ( t 1 ( � x ) , ..., t k ( � x )) , then x ) ≡ ∃ z 1 , ..., z k ( � t I ( y,� i ≤ k t iI ( z i ,� x ) ∧ f I ( y, z 1 , .., z n )) ; – p. 20
Interpretation An interpretation I of L in L ′ consists of: • an L ′ formula υ I ( x ) , called universe; x ) of L , an L ′ formula f I ( y,� • for function f ( � x ) ; x ) of L , an L ′ formula R I ( � • for relation R ( � x ) . Extend I to all L -terms and L -formulae: • if t ( � x ) ≡ f ( t 1 ( � x ) , ..., t k ( � x )) , then x ) ≡ ∃ z 1 , ..., z k ( � t I ( y,� i ≤ k t iI ( z i ,� x ) ∧ f I ( y, z 1 , .., z n )) ; • if ϕ ≡ R ( t 1 ( � x ) , ..., t k ( � x )) , then ϕ I ≡ ∃ z 1 , ..., z k ( � i ≤ k t iI ( z i ,� x ) ∧ R I ( z 1 , .., z n )) ; – p. 20
Interpretation An interpretation I of L in L ′ consists of: • an L ′ formula υ I ( x ) , called universe; x ) of L , an L ′ formula f I ( y,� • for function f ( � x ) ; x ) of L , an L ′ formula R I ( � • for relation R ( � x ) . Extend I to all L -terms and L -formulae: • if t ( � x ) ≡ f ( t 1 ( � x ) , ..., t k ( � x )) , then x ) ≡ ∃ z 1 , ..., z k ( � t I ( y,� i ≤ k t iI ( z i ,� x ) ∧ f I ( y, z 1 , .., z n )) ; • if ϕ ≡ R ( t 1 ( � x ) , ..., t k ( � x )) , then ϕ I ≡ ∃ z 1 , ..., z k ( � i ≤ k t iI ( z i ,� x ) ∧ R I ( z 1 , .., z n )) ; • ( ϕ ∧ ψ ) I ≡ ϕ I ∧ ψ I ; and ( ¬ ϕ ) I ≡ ¬ ϕ I ; – p. 20
Interpretation An interpretation I of L in L ′ consists of: • an L ′ formula υ I ( x ) , called universe; x ) of L , an L ′ formula f I ( y,� • for function f ( � x ) ; x ) of L , an L ′ formula R I ( � • for relation R ( � x ) . Extend I to all L -terms and L -formulae: • if t ( � x ) ≡ f ( t 1 ( � x ) , ..., t k ( � x )) , then x ) ≡ ∃ z 1 , ..., z k ( � t I ( y,� i ≤ k t iI ( z i ,� x ) ∧ f I ( y, z 1 , .., z n )) ; • if ϕ ≡ R ( t 1 ( � x ) , ..., t k ( � x )) , then ϕ I ≡ ∃ z 1 , ..., z k ( � i ≤ k t iI ( z i ,� x ) ∧ R I ( z 1 , .., z n )) ; • ( ϕ ∧ ψ ) I ≡ ϕ I ∧ ψ I ; and ( ¬ ϕ ) I ≡ ¬ ϕ I ; • ( ∀ yϕ ( y )) I ≡ ∀ y ( υ I ( y ) → ϕ ( y ) I ) . – p. 20
Interpretation 2 Given an interpretation I of L in L ′ . • I is an interpretation in an L ′ theory T ′ iff 1. T ′ ⊢ ∃ xυ I ( x ) ; 2. T ′ ⊢ ∀ � x ) → ∃ ! y ( υ I ( y ) ∧ f I ( y,� x ( υ I ( � x ))) . – p. 21
Interpretation 2 Given an interpretation I of L in L ′ . • I is an interpretation in an L ′ theory T ′ iff 1. T ′ ⊢ ∃ xυ I ( x ) ; 2. T ′ ⊢ ∀ � x ) → ∃ ! y ( υ I ( y ) ∧ f I ( y,� x ( υ I ( � x ))) . • I is an interpretation of an L theory T in T ′ iff 1. (as above); 2. (as above); 3. if T ⊢ ϕ then T ′ ⊢ ϕ I for any ϕ ∈ L . – p. 21
Theories ess. containing Q “ T ′ ess. contains T ” = “ ∃ interpretation of T in T ′ ”. – p. 22
Theories ess. containing Q “ T ′ ess. contains T ” = “ ∃ interpretation of T in T ′ ”. • ZFC essentially contains Q by von Neumann interpretation v : 1. υ v ( x ) ≡ “ x is a finite von Neumann ordinal”; 2. S v ( y, x ) ≡ y = x ∪ { x } , etc.; – p. 22
Theories ess. containing Q “ T ′ ess. contains T ” = “ ∃ interpretation of T in T ′ ”. • ZFC essentially contains Q by von Neumann interpretation v : 1. υ v ( x ) ≡ “ x is a finite von Neumann ordinal”; 2. S v ( y, x ) ≡ y = x ∪ { x } , etc.; • modal extensions of PA (directly) contains Q ; – p. 22
Theories ess. containing Q “ T ′ ess. contains T ” = “ ∃ interpretation of T in T ′ ”. • ZFC essentially contains Q by von Neumann interpretation v : 1. υ v ( x ) ≡ “ x is a finite von Neumann ordinal”; 2. S v ( y, x ) ≡ y = x ∪ { x } , etc.; • modal extensions of PA (directly) contains Q ; • Heyting Arithmetic HA (ess.) contains Q by • HA literally extends PA in ∧ , ¬ , ∀ , with extra-operators ∨ , ∃ (like modality); – p. 22
Theories ess. containing Q “ T ′ ess. contains T ” = “ ∃ interpretation of T in T ′ ”. • ZFC essentially contains Q by von Neumann interpretation v : 1. υ v ( x ) ≡ “ x is a finite von Neumann ordinal”; 2. S v ( y, x ) ≡ y = x ∪ { x } , etc.; • modal extensions of PA (directly) contains Q ; • Heyting Arithmetic HA (ess.) contains Q by • HA literally extends PA in ∧ , ¬ , ∀ , with extra-operators ∨ , ∃ (like modality); • relaxing the notion of interpretation so that double negation translation N is included: ( ϕ ∨ ψ ) N ≡ ¬ ( ¬ ϕ N ∧¬ ψ N ) ; ( ∃ xϕ ( x )) N ≡ ¬∀ x ¬ ϕ ( x ) N ; etc. – p. 22
Presburger Arithmetic PresA Language L PresA = { 0 , S, + } ; Axioms 1. ¬ ( S ( x ) = 0) ; 2. S ( x ) = S ( y ) → x = y ; 3. x = 0 ∨ ∃ y ( x = S ( y )) ; 4. x +0 = x ; and x + S ( y ) = S ( x + y ) ; 5. induction for all L PresA formulae. – p. 23
Presburger Arithmetic PresA Language L PresA = { 0 , S, + } ; Axioms 1. ¬ ( S ( x ) = 0) ; 2. S ( x ) = S ( y ) → x = y ; 3. x = 0 ∨ ∃ y ( x = S ( y )) ; 4. x +0 = x ; and x + S ( y ) = S ( x + y ) ; 5. induction for all L PresA formulae. Remarks • essentially, the · -free fragment of PA ; – p. 23
Presburger Arithmetic PresA Language L PresA = { 0 , S, + } ; Axioms 1. ¬ ( S ( x ) = 0) ; 2. S ( x ) = S ( y ) → x = y ; 3. x = 0 ∨ ∃ y ( x = S ( y )) ; 4. x +0 = x ; and x + S ( y ) = S ( x + y ) ; 5. induction for all L PresA formulae. Remarks • essentially, the · -free fragment of PA ; • introduced by M. Presburger in 1929; – p. 23
Presburger Arithmetic PresA Language L PresA = { 0 , S, + } ; Axioms 1. ¬ ( S ( x ) = 0) ; 2. S ( x ) = S ( y ) → x = y ; 3. x = 0 ∨ ∃ y ( x = S ( y )) ; 4. x +0 = x ; and x + S ( y ) = S ( x + y ) ; 5. induction for all L PresA formulae. Remarks • essentially, the · -free fragment of PA ; • introduced by M. Presburger in 1929; • proven by him to be complete, i.e., PreA ⊢ ϕ ⇐ ⇒ N | = ϕ for any ϕ ∈ L PresA ; – p. 23
Presburger Arithmetic PresA Language L PresA = { 0 , S, + } ; Axioms 1. ¬ ( S ( x ) = 0) ; 2. S ( x ) = S ( y ) → x = y ; 3. x = 0 ∨ ∃ y ( x = S ( y )) ; 4. x +0 = x ; and x + S ( y ) = S ( x + y ) ; 5. induction for all L PresA formulae. Remarks • essentially, the · -free fragment of PA ; • introduced by M. Presburger in 1929; • proven by him to be complete, i.e., PreA ⊢ ϕ ⇐ ⇒ N | = ϕ for any ϕ ∈ L PresA ; • hence not essentially contains Q . – p. 23
Theory of real closed fields RCF Language L RCF := { 0 , 1 , − , + , · , < } ; Axioms 1. x +0 = x ; x +( − x ) = 0 ; x + y = y + x ; 2. x · 0 = 0 ; x · ( y + z ) = x · y + x · z ; x · y = y · x ; 3. x < y → x + z < y + z ; x > 0 ∧ y > 0 → x · y > 0 ; 4. x > 0 → ∃ y ( x = y · y ) ; 5. ∀ x 2 n +1 ...x 0 ( x 2 n +1 � = 0 → ∃ y ( � i ≤ 2 n +1 x i · y i = 0)) . – p. 24
Theory of real closed fields RCF Language L RCF := { 0 , 1 , − , + , · , < } ; Axioms 1. x +0 = x ; x +( − x ) = 0 ; x + y = y + x ; 2. x · 0 = 0 ; x · ( y + z ) = x · y + x · z ; x · y = y · x ; 3. x < y → x + z < y + z ; x > 0 ∧ y > 0 → x · y > 0 ; 4. x > 0 → ∃ y ( x = y · y ) ; 5. ∀ x 2 n +1 ...x 0 ( x 2 n +1 � = 0 → ∃ y ( � i ≤ 2 n +1 x i · y i = 0)) . Remarks • proven by Tarski (1951) to admit quantifier-elimination; and so • RFC ⊢ ϕ ⇐ ⇒ ( R , 0 , 1 , − , + , · , < ) | = ϕ ; – p. 24
Theory of real closed fields RCF Language L RCF := { 0 , 1 , − , + , · , < } ; Axioms 1. x +0 = x ; x +( − x ) = 0 ; x + y = y + x ; 2. x · 0 = 0 ; x · ( y + z ) = x · y + x · z ; x · y = y · x ; 3. x < y → x + z < y + z ; x > 0 ∧ y > 0 → x · y > 0 ; 4. x > 0 → ∃ y ( x = y · y ) ; 5. ∀ x 2 n +1 ...x 0 ( x 2 n +1 � = 0 → ∃ y ( � i ≤ 2 n +1 x i · y i = 0)) . Remarks • proven by Tarski (1951) to admit quantifier-elimination; and so • RFC ⊢ ϕ ⇐ ⇒ ( R , 0 , 1 , − , + , · , < ) | = ϕ ; • hence not essentially contains Q . – p. 24
Quiz 2 — Which is correct? • Hilbert’s Programme looks for: a complete and decidable axiomatization of real numbers . Gödel Incompleteness Theorem answers: “ impossible ”. • Tarski’s Theorem (1951): quantifier elimination of real closed field . As a consequence, it yields: a complete and decidable axiomatization of ( R , 0 , 1 , − , + , · , < ) . – p. 25
The Statement (5) If a first order theory T satisfies the following: • T is consistent; • T is recursively axiomatizable; • T essentially contains Robinson Arithmetic Q , then the following hold: 1st incompleteness: T is not complete; – p. 26
The Statement (5) If a first order theory T satisfies the following: • T is consistent; • T is recursively axiomatizable; • T essentially contains Robinson Arithmetic Q , then the following hold: 1st incompleteness: T is not complete; 2nd incompleteness: T cannot prove a sentence which represents the consistency of T . – p. 26
Nelson’s trick There is an interpretation of IΣ 0 + Ω 1 in Q . • Main idea: define an L Q formula W ( x ) which intuitively means “ < is well-founded below x ”; – p. 27
Nelson’s trick There is an interpretation of IΣ 0 + Ω 1 in Q . • Main idea: define an L Q formula W ( x ) which intuitively means “ < is well-founded below x ”; • Nelson’s interpretation n should be 1. υ n ( x ) ≡ W ( x ) ; 2. S n ( y, x ) ≡ y = S ( x ) ; + n ( z, x, y ) ≡ z = x + y ; · n ( z, x, y ) ≡ z = x · y . 3. = n ( x, y ) ≡ x = y ; < n ( x, y ) ≡ x < y . – p. 27
Nelson’s trick There is an interpretation of IΣ 0 + Ω 1 in Q . • Main idea: define an L Q formula W ( x ) which intuitively means “ < is well-founded below x ”; • Nelson’s interpretation n should be 1. υ n ( x ) ≡ W ( x ) ; 2. S n ( y, x ) ≡ y = S ( x ) ; + n ( z, x, y ) ≡ z = x + y ; · n ( z, x, y ) ≡ z = x · y . 3. = n ( x, y ) ≡ x = y ; < n ( x, y ) ≡ x < y . • Compare to ϕ �→ ϕ WF in set theory. – p. 27
Nelson’s trick There is an interpretation of IΣ 0 + Ω 1 in Q . • Main idea: define an L Q formula W ( x ) which intuitively means “ < is well-founded below x ”; • Nelson’s interpretation n should be 1. υ n ( x ) ≡ W ( x ) ; 2. S n ( y, x ) ≡ y = S ( x ) ; + n ( z, x, y ) ≡ z = x + y ; · n ( z, x, y ) ≡ z = x · y . 3. = n ( x, y ) ≡ x = y ; < n ( x, y ) ≡ x < y . • Compare to ϕ �→ ϕ WF in set theory. As a consequence, “ T ess. contains Q ” ⇐ ⇒ “ T ess. contains IΣ 0 + Ω 1 ” – p. 27
Numeralwise representation R ⊆ ω n is numeralwise represented by ϕ ( � x ) iff • Q ⊢ ϕ ( k 1 , ..., k n ) ⇐ ⇒ R ( k 1 , ..., k n ) and • Q ⊢ ¬ ϕ ( k 1 , ..., k n ) ⇐ ⇒ ¬ R ( k 1 , ..., k n ) , where k := S ( ... ( S (0) ... ) . � �� � k – p. 28
Numeralwise representation R ⊆ ω n is numeralwise represented by ϕ ( � x ) iff • Q ⊢ ϕ ( k 1 , ..., k n ) ⇐ ⇒ R ( k 1 , ..., k n ) and • Q ⊢ ¬ ϕ ( k 1 , ..., k n ) ⇐ ⇒ ¬ R ( k 1 , ..., k n ) , where k := S ( ... ( S (0) ... ) . � �� � k We have the following L Q formulae ( Σ 0 1 completeness): • Q ⊢ neg( � � , k ) ⇐ ⇒ k = � ¬ ϕ ϕ � and Q ⊢ ¬ neg( � � , k ) ⇐ ⇒ k � = � ¬ ϕ ϕ � ; – p. 28
Numeralwise representation R ⊆ ω n is numeralwise represented by ϕ ( � x ) iff • Q ⊢ ϕ ( k 1 , ..., k n ) ⇐ ⇒ R ( k 1 , ..., k n ) and • Q ⊢ ¬ ϕ ( k 1 , ..., k n ) ⇐ ⇒ ¬ R ( k 1 , ..., k n ) , where k := S ( ... ( S (0) ... ) . � �� � k We have the following L Q formulae ( Σ 0 1 completeness): • Q ⊢ neg( � � , k ) ⇐ ⇒ k = � ¬ ϕ ϕ � and Q ⊢ ¬ neg( � � , k ) ⇐ ⇒ k � = � ¬ ϕ ϕ � ; • Q ⊢ Prf T ( � Λ � ) ⇐ ⇒ Λ is a T -proof of ϕ � , � ϕ Q ⊢ ¬ Prf T ( � Λ � ) ⇐ ⇒ Λ is not a T -proof of ϕ � , � ϕ (if T is recursively axiomatizable). – p. 28
Numeralwise representation R ⊆ ω n is numeralwise represented by ϕ ( � x ) iff • Q ⊢ ϕ ( k 1 , ..., k n ) ⇐ ⇒ R ( k 1 , ..., k n ) and • Q ⊢ ¬ ϕ ( k 1 , ..., k n ) ⇐ ⇒ ¬ R ( k 1 , ..., k n ) , where k := S ( ... ( S (0) ... ) . � �� � k We have the following L Q formulae ( Σ 0 1 completeness): • Q ⊢ neg( � � , k ) ⇐ ⇒ k = � ¬ ϕ ϕ � and Q ⊢ ¬ neg( � � , k ) ⇐ ⇒ k � = � ¬ ϕ ϕ � ; • Q ⊢ Prf T ( � Λ � ) ⇐ ⇒ Λ is a T -proof of ϕ � , � ϕ Q ⊢ ¬ Prf T ( � Λ � ) ⇐ ⇒ Λ is not a T -proof of ϕ � , � ϕ (if T is recursively axiomatizable). Then it is natural to define Con( T ) : ≡ ¬∃ x Prf T ( x, � ⊥ � ) . – p. 28
Ambiguity Even if the following hold for all Λ and ϕ : • Q ⊢ Prf T ( � Λ � ) ⇐ ⇒ Λ is a T -proof of ϕ and � , � ϕ Q ⊢ ¬ Prf T ( � Λ � ) ⇐ ⇒ Λ is not a T -proof of ϕ ; � , � ϕ • Q ⊢ Prf ∗ T ( � Λ � ) ⇐ ⇒ Λ is a T -proof of ϕ and � , � ϕ Q ⊢ ¬ Prf ∗ T ( � Λ � ) ⇐ ⇒ Λ is not a T -proof of ϕ ; � , � ϕ – p. 29
Ambiguity Even if the following hold for all Λ and ϕ : • Q ⊢ Prf T ( � Λ � ) ⇐ ⇒ Λ is a T -proof of ϕ and � , � ϕ Q ⊢ ¬ Prf T ( � Λ � ) ⇐ ⇒ Λ is not a T -proof of ϕ ; � , � ϕ • Q ⊢ Prf ∗ T ( � Λ � ) ⇐ ⇒ Λ is a T -proof of ϕ and � , � ϕ Q ⊢ ¬ Prf ∗ T ( � Λ � ) ⇐ ⇒ Λ is not a T -proof of ϕ ; � , � ϕ we do not have • Q ⊢ ∀ x, y (Prf T ( x, y ) ↔ Prf ∗ T ( x, y )) , nor – p. 29
Ambiguity Even if the following hold for all Λ and ϕ : • Q ⊢ Prf T ( � Λ � ) ⇐ ⇒ Λ is a T -proof of ϕ and � , � ϕ Q ⊢ ¬ Prf T ( � Λ � ) ⇐ ⇒ Λ is not a T -proof of ϕ ; � , � ϕ • Q ⊢ Prf ∗ T ( � Λ � ) ⇐ ⇒ Λ is a T -proof of ϕ and � , � ϕ Q ⊢ ¬ Prf ∗ T ( � Λ � ) ⇐ ⇒ Λ is not a T -proof of ϕ ; � , � ϕ we do not have • Q ⊢ ∀ x, y (Prf T ( x, y ) ↔ Prf ∗ T ( x, y )) , nor • Q ⊢ Con( T ) ↔ Con ∗ ( T ) , where Con ∗ ( T ) : ≡ ¬∃ x Prf ∗ T ( x, � ⊥ � ) . – p. 29
Ambiguity Even if the following hold for all Λ and ϕ : • Q ⊢ Prf T ( � Λ � ) ⇐ ⇒ Λ is a T -proof of ϕ and � , � ϕ Q ⊢ ¬ Prf T ( � Λ � ) ⇐ ⇒ Λ is not a T -proof of ϕ ; � , � ϕ • Q ⊢ Prf ∗ T ( � Λ � ) ⇐ ⇒ Λ is a T -proof of ϕ and � , � ϕ Q ⊢ ¬ Prf ∗ T ( � Λ � ) ⇐ ⇒ Λ is not a T -proof of ϕ ; � , � ϕ we do not have • Q ⊢ ∀ x, y (Prf T ( x, y ) ↔ Prf ∗ T ( x, y )) , nor • Q ⊢ Con( T ) ↔ Con ∗ ( T ) , where Con ∗ ( T ) : ≡ ¬∃ x Prf ∗ T ( x, � ⊥ � ) . The point here: T ⊢ ϕ ( k ) for all k ∈ ω �⇒ T ⊢ ∀ xϕ ( x ) . – p. 29
Quiz 3 — Which is correct? • Gödel 2nd Incompleteness (1931): PA cannot prove a sentence which represents the consistency of PA . • Kreisel’s Remark (1960): PA does prove a sentence which represents the consistency of PA . – p. 30
2. A Brief Look at the Proofs – p. 31
Rosser’s trick Given Prf T such that, for all Λ and ϕ , • Q ⊢ Prf T ( � Λ � ) ⇐ ⇒ Λ is a T -proof of ϕ , � , � ϕ • Q ⊢ ¬ Prf T ( � Λ � ) ⇐ ⇒ Λ is not a T -proof of ϕ , � , � ϕ – p. 32
Rosser’s trick Given Prf T such that, for all Λ and ϕ , • Q ⊢ Prf T ( � Λ � ) ⇐ ⇒ Λ is a T -proof of ϕ , � , � ϕ • Q ⊢ ¬ Prf T ( � Λ � ) ⇐ ⇒ Λ is not a T -proof of ϕ , � , � ϕ we can define Prf ∗ T by Prf ∗ T ( x, u ) : ≡ Prf( x, u ) ∧ ( ∀ z < x ) ∀ v ¬ (neg( u, v ) ∧ Prf( z, v )) . – p. 32
Rosser’s trick Given Prf T such that, for all Λ and ϕ , • Q ⊢ Prf T ( � Λ � ) ⇐ ⇒ Λ is a T -proof of ϕ , � , � ϕ • Q ⊢ ¬ Prf T ( � Λ � ) ⇐ ⇒ Λ is not a T -proof of ϕ , � , � ϕ we can define Prf ∗ T by Prf ∗ T ( x, u ) : ≡ Prf( x, u ) ∧ ( ∀ z < x ) ∀ v ¬ (neg( u, v ) ∧ Prf( z, v )) . Then Q ⊢ Prf ∗ T ( � Λ � ) ⇐ ⇒ Λ is a T -proof of ϕ ∧ � , � ϕ there is no T -proof ∆ of ¬ ϕ ∆ Λ � < � with � � = ⇒ Λ is a T -proof of ϕ. – p. 32
Rosser’s trick Given Prf T such that, for all Λ and ϕ , • Q ⊢ Prf T ( � Λ � ) ⇐ ⇒ Λ is a T -proof of ϕ , � , � ϕ • Q ⊢ ¬ Prf T ( � Λ � ) ⇐ ⇒ Λ is not a T -proof of ϕ , � , � ϕ we can define Prf ∗ T by Prf ∗ T ( x, u ) : ≡ Prf( x, u ) ∧ ( ∀ z < x ) ∀ v ¬ (neg( u, v ) ∧ Prf( z, v )) . Then Q ⊢ Prf ∗ T ( � Λ � ) ⇐ ⇒ Λ is a T -proof of ϕ ∧ � , � ϕ there is no T -proof ∆ of ¬ ϕ ∆ Λ � < � with � � = ⇒ Λ is a T -proof of ϕ. If T is consistent, ⇐ = also holds. – p. 32
Rosser’s trick Given Prf T such that, for all Λ and ϕ , • Q ⊢ Prf T ( � Λ � ) ⇐ ⇒ Λ is a T -proof of ϕ , � , � ϕ • Q ⊢ ¬ Prf T ( � Λ � ) ⇐ ⇒ Λ is not a T -proof of ϕ , � , � ϕ we can define Prf ∗ T by Prf ∗ T ( x, u ) : ≡ Prf( x, u ) ∧ ( ∀ z < x ) ∀ v ¬ (neg( u, v ) ∧ Prf( z, v )) . Then Q ⊢ ¬ Prf ∗ T ( � Λ � ) ⇐ ⇒ Λ is not a T -proof of ϕ ∨ � , � ϕ there is a T -proof ∆ of ¬ ϕ ∆ Λ � < � with � � ⇐ = Λ is not a T -proof of ϕ. – p. 32
Rosser’s trick Given Prf T such that, for all Λ and ϕ , • Q ⊢ Prf T ( � Λ � ) ⇐ ⇒ Λ is a T -proof of ϕ , � , � ϕ • Q ⊢ ¬ Prf T ( � Λ � ) ⇐ ⇒ Λ is not a T -proof of ϕ , � , � ϕ we can define Prf ∗ T by Prf ∗ T ( x, u ) : ≡ Prf( x, u ) ∧ ( ∀ z < x ) ∀ v ¬ (neg( u, v ) ∧ Prf( z, v )) . Then Q ⊢ ¬ Prf ∗ T ( � Λ � ) ⇐ ⇒ Λ is not a T -proof of ϕ ∨ � , � ϕ there is a T -proof ∆ of ¬ ϕ ∆ Λ � < � with � � ⇐ = Λ is not a T -proof of ϕ. If T is consistent, = ⇒ also holds. – p. 32
Kreisel’s remark (1960) Since there is a proof ∆ of ¬⊥ , if T is consistent, Q ⊢ ( ∀ x < � ∆ � ) ¬ Prf( x, � ⊥ � ) . – p. 33
Recommend
More recommend
