∆-Definability of Sequence-Coding Operations
th Let p1 = 2, p2 = 3,..., pi = i prime number.
N If a = ha1, . . . , aki, then |a| = k and( a)i = ai. All of the sequence-coding operations are ∆-definable (hence, representable). We have∆-formulas: Divides(y, x): ≡ (∃z ≤ x)[x = y · z], Prime(x): ≡ S0 < x ∧ (∀y ≤ x)[Divides(y, x) → (y = 1 ∨ y = x)] We have∆-formulas: Divides(y, x): ≡ (∃z ≤ x)[x = y · z], Prime(x): ≡ S0 < x ∧ (∀y ≤ x)[Divides(y, x) → (y = 1 ∨ y = x)] | {z } ¬Divides(y, x) ∨ (y = 1 ∨ y = x) Remark: Technically, ¬Divides(y, x) is not a∆-formula. However, it is equivalent to a∆-formula. The set 2 PrimePair := {(pi, pi+1): i ≥ 1} ⊆ N is defined by the∆-formula PrimePair(x, y):≡ Prime(x) ∧ Prime(y) ∧ x < y ∧ (∀z < y)[Prime(z) → z ≤ x]. The set 2 PrimePair := {(pi, pi+1): i ≥ 1} ⊆ N is defined by the∆-formula PrimePair(x, y):≡ Prime(x) ∧ Prime(y) ∧ x < y ∧ (∀z < y)[Prime(z) → z ≤ x]. The set This formula expresses: “c is divisible by2 and for every prime pair( pi, pi+1), if pi+1 divides c, so pi divides c.” Continuing, we define the set Yardstick := {h0, 1, 2, . . . , k − 1i : k ∈ N} | {z } 1 2 3 k 2 3 5 ...(pk) by the∆-formula Yardstick(x):≡ Divides(2, x) ∧ ¬Divides(4, x) ∧ (∀y ≤ x)(∀z ≤ x)(∀i < x) PrimePair(y, z) ∧ Divides(z, x) . → Divides(y E i, x) ↔ Divides(z E Si, x) |{z} | {zi+1 } yi z We next define the set IthPrime := {(i, pi): i ∈ N≥1} by the∆-formula Yardstick(x) ∧ Divides(yi, x) ∧ IthPrime(i, y):≡ Prime(y) ∧ (∃x ≤ some term) . ¬Divides(yi+1, x) Question: What term suffices for this bounded quantifier? We next define the set IthPrime := {(i, pi): i ∈ N≥1} by the∆-formula Yardstick(x) ∧ 2 i Divides(yi, x) ∧ IthPrime(i, y):≡ Prime(y) ∧ (∃x ≤ y ) . ¬Divides(yi+1, x) Question: What term suffices for this bounded quantifier? 1 2 i Answer: We want x to be the i-th yardstick number( p1) (p2) ... (pi) . This is at most 2 3 i (i+1) i2 (pi)(pi) (pi) ··· (pi) = (pi) 2 ≤ (pi) . i2 Therefore, if y = pi, it suffices to take x ≤ y . We next define the set IthPrime := {(i, pi): i ∈ N≥1} by the∆-formula Yardstick(x) ∧ 2 i Divides(yi, x) ∧ IthPrime(i, y):≡ Prime(y) ∧ (∃x ≤ y ) . ¬Divides(yi+1, x) Question: What term suffices for this bounded quantifier? 1 2 i Answer: We want x to be the i-th yardstick number( p1) (p2) ... (pi) . This is at most 2 3 i (i+1) i2 (pi)(pi) (pi) ··· (pi) = (pi) 2 ≤ (pi) . i2 Therefore, if y = pi, it suffices to take x ≤ y . i i2 Alternatively, since pi ≤ (i + 1) (easy fact), we could instead use x ≤ (i + 1) . We next define the set IthPrime := {(i, pi): i ∈ N≥1} by the∆-formula Yardstick(x) ∧ i·i Divides(yi, x) ∧ IthPrime(i, y):≡ Prime(y) ∧ (∃x ≤ y ) . ¬Divides(yi+1, x) Exercise. Convince yourself that IthPrime(i, y) indeed defines IthPrime. That is, show that • N |= IthPrime(k, pk) for every k ≥ 1, • N |= ¬IthPrime(a, b) whenever a = 0 or b =6 pa. We next define the set IthPrime := {(i, pi): i ∈ N≥1} by the∆-formula Yardstick(x) ∧ i·i Divides(yi, x) ∧ IthPrime(i, y):≡ Prime(y) ∧ (∃x ≤ y ) . ¬Divides(yi+1, x) Remark. The set IthPrime ⊆ N2 corresponds to the function N → N defined by i 7→ pi. Therefore, we say that the formula IthPrime(i, y) defines the function i 7→ pi. Continuing, the set k Length := (ha1, . . . , aki, k): k ≥ 1 and (a1, . . . , ak) ∈ N is defined by the∆-formula Length(c, `):≡ Codenumber(c) IthPrime(`, y) ∧ Divides(y, c) ∧ (∃y ≤ c) . ∧ (∀z ≤ c)[PrimePair(y, z) → ¬Divides(z, c)] Continuing, the set k Length := (ha1, . . . , aki, k): k ≥ 1 and (a1, . . . , ak) ∈ N is defined by the∆-formula Length(c, `):≡ Codenumber(c) IthPrime(`, y) ∧ Divides(y, c) ∧ (∃y ≤ c) . ∧ (∀z ≤ c)[PrimePair(y, z) → ¬Divides(z, c)] The set k IthElement := (aj, j, ha1, . . . , aki) : 1 ≤ j ≤ k and (a1, . . . , ak) ∈ N is defined by the∆-formula IthElement(e, i, c):≡ Codenumber(c) ∧ (∃y ≤ c) IthPrime(i, y) ∧ Divides(ySe, c) ∧ ¬Divides(ySSe, c). ∆-Definability of Sequence-Coding Operations For practice, try writing a∆-formula that defines the set Concatenation ⊆ 3 N of triples of the form( ha1, . . . , aki, hb1, . . . , b`i, ha1, . . . , ak, b1, . . . , b`i). Godel¨ Numbers of Terms and Formulas We assign a unique number to each symbol in LNT as follows: ¬ 1 · 15 ∨ 3 E 17 ∀ 5 < 19 =7 ( 21 09 ) 23 S 11 vi 2i + 13 Suppose s :≡ s1 . . . sn is a string of symbols, which constituting a well-formed term or formula of LNT . Naively, we could encode s by the number h#(s1),..., #(sn)i where#( si) is the number corresponding to the symbol si. However, it much better to encode s according to the inductive type of terms and formulas. Def 5.7.1. For each term t and formula ϕ, the G¨odelnumbers ptq and pϕq are defined as follows: p¬αq = h1, pαqi p+t1t2q = h13, pt1q, pt2qi p(α ∨ β)q = h3, pαq, pβqi p· t1t2q = h15, pt1q, pt2qi p(∀vi)(α)q = h5, pviq, pαqi pEt1t2q = h17, pt1q, pt2qi p=t1t2q = h7, pt1q, pt2qi p p¬αq = h1, pαqi p+t1t2q = h13, pt1q, pt2qi p(α ∨ β)q = h3, pαq, pβqi p· t1t2q = h15, pt1q, pt2qi p(∀vi)(α)q = h5, pviq, pαqi pEt1t2q = h17, pt1q, pt2qi p=t1t2q = h7, pt1q, pt2qi p Obs. ptq and pϕq are never divisible by7. (Why?) Def 5.7.1. For each term t and formula ϕ, the G¨odelnumbers ptq and pϕq are defined as follows: p¬αq = h1, pαqi p+t1t2q = h13, pt1q, pt2qi p(α ∨ β)q = h3, pαq, pβqi p· t1t2q = h15, pt1q, pt2qi p(∀vi)(α)q = h5, pviq, pαqi pEt1t2q = h17, pt1q, pt2qi p=t1t2q = h7, pt1q, pt2qi p Example. p=0S0q = h7, p0q, pS0qi = h7, h9i, h11, h9iii 12 1025 = h7, 210, h11, 210ii = 28310255(2 3 +1). Notice how fast SSSS0 grows: p q 10 12 21232 12 21232 3 pSSSS0q = h11, h11, h11, h11, h9iiiii =2 3 Next Steps (Section 5.8) ∆-definability of sets Terms := {ptq : terms t} = {a ∈ N : a = ptq for some term t}, Formulas := {pϕq : formulas ϕ} = {a ∈ N : a = pϕq for some formula ϕ}.