Reminder of Notation Language is always L NT = (0 , S, + , , E, < - PowerPoint PPT Presentation

Reminder of Notation Language is always L NT = (0 , S, + , · , E, < ). N is the natural numbers as L NT -structure N = ( N , 0 , S, + , · , E, < ). N = { N 1 , . . . , N 11 } is the set of axioms of Robinson Arithmetic. For a ∈ N , we let a stand for the variable-free term SS . . . S 0. � �� � a times For a variable-free term t , we let t N ∈ N stand for the interpretation of t in N . (For example, ( SSS 0 · SS 0) N equals 6.)

The Power of Robinson Arithmetic Robinson Arithmetic. The eleven axioms of N are: (N1) ( ∀ x ) ¬ [ Sx = 0] (N2) ( ∀ x )( ∀ y )[ Sx = Sy → x = y ] (N3) ( ∀ x )[ x + 0 = x ] (N4) ( ∀ x )( ∀ y )[ x + Sy = S ( x + y )] (N5) ( ∀ x )[ x · 0 = 0] (N6) ( ∀ x )( ∀ y )[( x · Sy ) = ( x · y ) + x ] (N7) ( ∀ x )[ xE 0 = S 0] (N8) ( ∀ x )( ∀ y )[ xE ( Sy ) = ( xEy ) · x ] (N9) ( ∀ x ) ¬ [ x < 0] (N10) ( ∀ x )( ∀ y )[ x < Sy ↔ ( x < y ∨ x = y )] (N11) ( ∀ x )( ∀ y )[ x < y ∨ x = y ∨ y < x ].

The Power of Robinson Arithmetic Lemma 2.8.4. For all natural numbers a and b : 1. If a = b , then N ⊢ a = b . 2. If a � = b , then N ⊢ a � = b . 3. If a < b , then N ⊢ a < b . 4. If a � < b , then N ⊢ ¬ ( a < b ). 5. N ⊢ a + b = a + b . 6. N ⊢ a · b = a · b . N ⊢ aEb = a b . 7. Lemma 5.3.10. N ⊢ ( t = t N ) for every variable-free term t . (Proof by induction on t , on blackboard.) For example, if t : ≡ ( SS 0 + S 0) · SS 0, then this lemma tells us N ⊢ (( SS 0 + S 0) · SS 0 = SSSSSS 0) .

The Power of Robinson Arithmetic Lemma 5.3.11 (Rosser’s Lemma). For every a ∈ N , � � N ⊢ ( ∀ x < a ) x = 0 ∨ x = 1 ∨ · · · ∨ x = a − 1 . Proof by induction on a (on blackboard)

The Power of Robinson Arithmetic Lemma 5.3.11 (Rosser’s Lemma). For every a ∈ N , � � N ⊢ ( ∀ x < a ) x = 0 ∨ x = 1 ∨ · · · ∨ x = a − 1 . Proof by induction on a (on blackboard) Corollary 5.3.12. For every a ∈ N and formula ϕ ( x ), � � � � N ⊢ ( ∀ x < a ) ϕ ( x ) ↔ ϕ (0) ∧ ϕ (1) ∧ · · · ∧ ϕ ( a − 1) � �� � � � � � ϕ x 0 ∧ ϕ x 1 ∧ · · · ∧ ϕ x that is, ( ∀ x < a ) ϕ ↔ a − 1 (Proof given as Exercise 11 in Section 5.3; solution on page 319. Good exercise to try on your own! )

The Power of Robinson Arithmetic Proposition 5.3.13. If ϕ is a Σ -sentence such that N | = ϕ , then N ⊢ ϕ . In other words, N proves every Σ-sentence which is true in N .

The Power of Robinson Arithmetic Proposition 5.3.13. If ϕ is a Σ -sentence such that N | = ϕ , then N ⊢ ϕ . In other words, N proves every Σ-sentence which is true in N . RECALL: As we have discussed before, N does not prove every sentence which is true in N . In particular, N �⊢ ( ∀ x ) ¬ [ x < x ] and N �⊢ ( ∀ x )( ∀ y )[ x + y = y + x ].

The Power of Robinson Arithmetic Proposition 5.3.13. If ϕ is a Σ -sentence such that N | = ϕ , then N ⊢ ϕ . PROOF. Let ϕ be a Σ-sentence such that N | = ϕ . We argue by induction on the complexity of ϕ .

The Power of Robinson Arithmetic Proposition 5.3.13. If ϕ is a Σ -sentence such that N | = ϕ , then N ⊢ ϕ . PROOF. Let ϕ be a Σ-sentence such that N | = ϕ . We argue by induction on the complexity of ϕ . Base case ϕ is atomic or ¬ (atomic). Suppose (for example) ϕ is t < u . = ϕ means that t N < u N . Then N | So by Lemma 2.8.4, N ⊢ t N < u N . = t = t N and N | By Lemma 5.3.10 (which we just proved), N | = u = u N . Therefore, N ⊢ t < u (using the (E3) axiom).

The Power of Robinson Arithmetic Proposition 5.3.13. If ϕ is a Σ -sentence such that N | = ϕ , then N ⊢ ϕ . PROOF. Let ϕ be a Σ-sentence such that N | = ϕ . We argue by induction on the complexity of ϕ . Suppose ϕ : ≡ ( α ∨ β ). Without loss of generality, assume N | = α . By induction hypothesis, N ⊢ α . Therefore, N ⊢ ϕ by (PC) rule.

The Power of Robinson Arithmetic Proposition 5.3.13. If ϕ is a Σ -sentence such that N | = ϕ , then N ⊢ ϕ . PROOF. Let ϕ be a Σ-sentence such that N | = ϕ . We argue by induction on the complexity of ϕ . NOTE: We do not need to consider the case ϕ : ≡ ¬ α , since Σ-sentences are not closed under negation.

The Power of Robinson Arithmetic Proposition 5.3.13. If ϕ is a Σ -sentence such that N | = ϕ , then N ⊢ ϕ . PROOF. Let ϕ be a Σ-sentence such that N | = ϕ . We argue by induction on the complexity of ϕ . Suppose ϕ : ≡ ( ∃ y ) α . = α y Since N | = ϕ , there exists a ∈ N such that N | a . Note that α y a is a Σ-sentence with lower complexity than ϕ (that is, fewer ∨ and ∀ symbols). (NOTE: α y a possibly has greater length as a string.) By induction hypothesis, N ⊢ α y a . By (Q2) axiom: ⊢ α y a → ( ∃ y ) α . (Since a is variable-free, it is substitutable for y in α .) Therefore, N ⊢ ϕ by (PC) rule.

The Power of Robinson Arithmetic Proposition 5.3.13. If ϕ is a Σ -sentence such that N | = ϕ , then N ⊢ ϕ . PROOF. Let ϕ be a Σ-sentence such that N | = ϕ . We argue by induction on the complexity of ϕ . Suppose ϕ : ≡ ( ∀ y < u ) α where u is a variable-free term. = α y a for every a < u N . Since N | = ϕ , it follows that N | By the induction hypothesis, N ⊢ α y a for every a < u N . By Corollary 4.3.8 (the corollary of Rosser’s Lemma), we have � � � � α y 0 ∧ α y 1 ∧ · · · ∧ α y N ⊢ ( ∀ y < u N ) α ↔ . u N − 1 By (PC) rule, N ⊢ ( ∀ y < u N ) α. By Lemma 4.3.6, N ⊢ u = u N . This lets us derive N ⊢ ( ∀ y < u ) α ] as required. Q.E.D.

Definable and Representable Sets A set A ⊆ N k is Σ / Π / ∆ -definable if there exists a Σ / Π / ∆-formula ϕ ( x 1 , . . . , x k ) such that • N | = ϕ ( a 1 , . . . , a k ) for every ( a 1 , . . . , a k ) ∈ A = ¬ ϕ ( b 1 , . . . , b k ) for every ( b 1 , . . . , b k ) ∈ N k \ A . • N |

Definable and Representable Sets A set A ⊆ N k is Σ / Π / ∆ -definable if there exists a Σ / Π / ∆-formula ϕ ( x 1 , . . . , x k ) such that • N | = ϕ ( a 1 , . . . , a k ) for every ( a 1 , . . . , a k ) ∈ A = ¬ ϕ ( b 1 , . . . , b k ) for every ( b 1 , . . . , b k ) ∈ N k \ A . • N | A set A ⊆ N k is representable if there exists a formula ϕ ( x 1 , . . . , x k ) such that • N ⊢ ϕ ( a 1 , . . . , a k ) for every ( a 1 , . . . , a k ) ∈ A • N ⊢ ¬ ϕ ( b 1 , . . . , b k ) for every ( b 1 , . . . , b k ) ∈ N k \ A .

Definable and Representable Sets A set A ⊆ N k is Σ / Π / ∆ -definable if there exists a Σ / Π / ∆-formula ϕ ( x 1 , . . . , x k ) such that • N | = ϕ ( a 1 , . . . , a k ) for every ( a 1 , . . . , a k ) ∈ A = ¬ ϕ ( b 1 , . . . , b k ) for every ( b 1 , . . . , b k ) ∈ N k \ A . • N | A set A ⊆ N k is representable if there exists a formula ϕ ( x 1 , . . . , x k ) such that • N ⊢ ϕ ( a 1 , . . . , a k ) for every ( a 1 , . . . , a k ) ∈ A • N ⊢ ¬ ϕ ( b 1 , . . . , b k ) for every ( b 1 , . . . , b k ) ∈ N k \ A . A set A ⊆ N k is weakly representable if there exists a formula ϕ ( x 1 , . . . , x k ) such that • N ⊢ ϕ ( a 1 , . . . , a k ) for every ( a 1 , . . . , a k ) ∈ A • N �⊢ ϕ ( b 1 , . . . , b k ) for every ( b 1 , . . . , b k ) ∈ N k \ A .

Definable and Representable Sets A function f : A → N where A ⊆ N k is definable or representable according to the corresponding set { ( a 1 , . . . , a k , b ) : f ( a 1 , . . . , a k ) = b } ⊆ N k +1 . Example. The function a �→ a 2 is ∆-definable, since it is defined by the ∆-formula ϕ ( x, y ) : ≡ ( y = x · x ) (or ( y = x E SS 0)).

The Power of Robinson Arithmetic Proposition 5.3.13. If ϕ is a Σ -sentence such that N | = ϕ , then N ⊢ ϕ . Corollary 5.3.15. Every ∆ -definable set is representable. This fact is extremely useful: it lets us show that various sets and functions are representable!

The Power of Robinson Arithmetic Proposition 5.3.13. If ϕ is a Σ -sentence such that N | = ϕ , then N ⊢ ϕ . Corollary 5.3.15. Every ∆ -definable set is representable. This fact is extremely useful: it lets us show that various sets and functions are representable! Proof of Proposition ⇒ Corollary: Suppose A ⊆ N k is defined by the ∆-formula ϕ ( x 1 , . . . , x n ). Both ϕ ( x 1 , . . . , x n ) and ¬ ϕ ( x 1 , . . . , x n ) are logically equivalent to Σ-formulas. Therefore, Proposi- tion 5.3.13 implies • N ⊢ ϕ ( a 1 , . . . , a k ) (since N | = ϕ ( a 1 , . . . , a k )) for every ( a 1 , . . . , a k ) ∈ A = ¬ ϕ ( b 1 , . . . , b k )) for every ( b 1 , . . . , b k ) ∈ N k \ A . • N ⊢ ¬ ϕ ( b 1 , . . . , b k ) (since N |

Representable Functions and Computer Programs (Section 5.4) Various mathematical model of “computable” sets and functions were proposed in the 1930s: • Turing machines (most intuitive model) • Church’s λ -calculus • G¨ odel’s recursive functions • representable functions Remarkably, all these models capture the same notion: a set A ⊆ N k (or function f : N k → N ) is representable iff it is λ -computable iff it is Turing computable iff it is recursive. The equivalence of these various notions of “computable” is a mathematical theorem.