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

Reminder of Notation Language is always LNT = (0, S, +, ·, E, <). N is the natural numbers as LNT-structure N = (N, 0, S, +, ·, E, <). N = {N1, . . . , N11} is the set of axioms of Robinson Arithmetic. For a ∈ N, we let a stand for the variable-free term SS . . . S

• a times

0. For a variable-free term t, we let tN ∈ N stand for the interpretation of t in N. (For example, (SSS0·SS0)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)[xE0 = S0] (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. 7. N ⊢ aEb = ab. Lemma 5.3.10. N ⊢ (t = tN) for every variable-free term t. (Proof by induction on t, on blackboard.) For example, if t :≡ (SS0 + S0) · SS0, then this lemma tells us N ⊢ ((SS0 + S0) · SS0 = SSSSSS0).

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)
• that is,
• (∀x < a)ϕ
• ↔
• ϕx

0 ∧ ϕx 1 ∧ · · · ∧ ϕx 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. Then N | = ϕ means that tN < uN. So by Lemma 2.8.4, N ⊢ tN < uN. By Lemma 5.3.10 (which we just proved), N | = t = tN and N | = u = uN. 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. Since N | = ϕ, there exists a ∈ N such that N | = αy

a.

Note that αy

a is a Σ-sentence with lower complexity than ϕ (that is, fewer ∨

and ∀ symbols). (NOTE: αy

a possibly has greater length as a string.)

By induction hypothesis, N ⊢ αy

a.

By (Q2) axiom: ⊢ αy

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. Since N | = ϕ, it follows that N | = αy

a for every a < uN.

By the induction hypothesis, N ⊢ αy

a for every a < uN.

By Corollary 4.3.8 (the corollary of Rosser’s Lemma), we have N ⊢

• (∀y < uN)α
• ↔
• αy

0 ∧ αy 1 ∧ · · · ∧ αy uN−1

• .

By (PC) rule, N ⊢ (∀y < uN)α. By Lemma 4.3.6, N ⊢ u = uN. This lets us derive N ⊢ (∀y < u)α] as required. Q.E.D.

Definable and Representable Sets A set A ⊆ Nk is Σ/Π/∆-definable if there exists a Σ/Π/∆-formula ϕ(x1, . . . , xk) such that

• N |

= ϕ(a1, . . . , ak) for every (a1, . . . , ak) ∈ A

• N |

= ¬ϕ(b1, . . . , bk) for every (b1, . . . , bk) ∈ Nk \ A.

Definable and Representable Sets A set A ⊆ Nk is Σ/Π/∆-definable if there exists a Σ/Π/∆-formula ϕ(x1, . . . , xk) such that

• N |

= ϕ(a1, . . . , ak) for every (a1, . . . , ak) ∈ A

• N |

= ¬ϕ(b1, . . . , bk) for every (b1, . . . , bk) ∈ Nk \ A. A set A ⊆ Nk is representable if there exists a formula ϕ(x1, . . . , xk) such that

• N ⊢ ϕ(a1, . . . , ak) for every (a1, . . . , ak) ∈ A
• N ⊢ ¬ϕ(b1, . . . , bk) for every (b1, . . . , bk) ∈ Nk \ A.

Definable and Representable Sets A set A ⊆ Nk is Σ/Π/∆-definable if there exists a Σ/Π/∆-formula ϕ(x1, . . . , xk) such that

• N |

= ϕ(a1, . . . , ak) for every (a1, . . . , ak) ∈ A

• N |

= ¬ϕ(b1, . . . , bk) for every (b1, . . . , bk) ∈ Nk \ A. A set A ⊆ Nk is representable if there exists a formula ϕ(x1, . . . , xk) such that

• N ⊢ ϕ(a1, . . . , ak) for every (a1, . . . , ak) ∈ A
• N ⊢ ¬ϕ(b1, . . . , bk) for every (b1, . . . , bk) ∈ Nk \ A.

A set A ⊆ Nk is weakly representable if there exists a formula ϕ(x1, . . . , xk) such that

• N ⊢ ϕ(a1, . . . , ak) for every (a1, . . . , ak) ∈ A
• N ⊢ ϕ(b1, . . . , bk) for every (b1, . . . , bk) ∈ Nk \ A.

Definable and Representable Sets A function f : A → N where A ⊆ Nk is definable or representable according to the corresponding set {(a1, . . . , ak, b) : f(a1, . . . , ak) = b} ⊆ Nk+1.

• Example. The function a → a2 is ∆-definable, since it is defined by the

∆-formula ϕ(x, y) :≡ (y = x · x) (or (y = x E SS0)).

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 ⊆ Nk is defined by the ∆-formula ϕ(x1, . . . , xn). Both ϕ(x1, . . . , xn) and ¬ϕ(x1, . . . , xn) are logically equivalent to Σ-formulas. Therefore, Proposi- tion 5.3.13 implies

• N ⊢ ϕ(a1, . . . , ak) (since N |

= ϕ(a1, . . . , ak)) for every (a1, . . . , ak) ∈ A

• N ⊢ ¬ϕ(b1, . . . , bk) (since N |

= ¬ϕ(b1, . . . , bk)) for every (b1, . . . , bk) ∈ Nk \ A.

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¨
• del’s recursive functions
• representable functions

Remarkably, all these models capture the same notion: a set A ⊆ Nk (or function f : Nk → 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.

Representable Functions and Computer Programs (Section 5.4) Theorem.

• If A ⊆ Nk is representable, then there is an algorithm (computer program)

which, given (a1, . . . , ak) ∈ Nk as input, determines in finite time whether

• r not (a1, . . . , ak) ∈ A.
• Conversely, if there exists a computer program which determines member-

ship in a set A ⊆ Nk, then A is representable.

Representable Functions and Computer Programs (Section 5.4) Theorem.

• If A ⊆ Nk is representable, then there is an algorithm (computer program)

which, given (a1, . . . , ak) ∈ Nk as input, determines in finite time whether

• r not (a1, . . . , ak) ∈ A.
• Conversely, if there exists a computer program which determines member-

ship in a set A ⊆ Nk, then A is representable. The Church-Turing Thesis. A function on the natural numbers is com- putable by a human being following an algorithm, ignoring resource limitations, if and only if it is computable by a Turing machine or any other equivalent no- tion (e.g., representability).