SLIDE 1 ∆-Definability of Sequence-Coding Operations Let p1 = 2, p2 = 3, . . . , pi = ith prime number. N<N is the set of finite sequences of natural numbers. Sequence-coding function : N<N → N is defined by a1, . . . , ak :=
k
pai+1
i
. If a = a1, . . . , ak, then |a| = k and (a)i = ai. All of the sequence-coding operations are ∆-definable (hence, representable).
SLIDE 2
We have ∆-formulas: Divides(y, x) :≡ (∃z ≤ x)[x = y · z], Prime(x) :≡ S0 < x ∧ (∀y ≤ x)[Divides(y, x) → (y = 1 ∨ y = x)]
SLIDE 3 We have ∆-formulas: Divides(y, x) :≡ (∃z ≤ x)[x = y · z], Prime(x) :≡ S0 < x ∧ (∀y ≤ x)[Divides(y, x) → (y = 1 ∨ y = x)
- ¬Divides(y, x) ∨ (y = 1 ∨ y = x)
] Remark: Technically, ¬Divides(y, x) is not a ∆-formula. However, it is equivalent to a ∆-formula.
SLIDE 4
The set PrimePair := {(pi, pi+1) : i ≥ 1} ⊆ N2 is defined by the ∆-formula PrimePair(x, y) :≡ Prime(x) ∧ Prime(y) ∧ x < y ∧ (∀z < y)[Prime(z) → z ≤ x].
SLIDE 5 The set PrimePair := {(pi, pi+1) : i ≥ 1} ⊆ N2 is defined by the ∆-formula PrimePair(x, y) :≡ Prime(x) ∧ Prime(y) ∧ x < y ∧ (∀z < y)[Prime(z) → z ≤ x]. The set Codenumber = { a1, . . . , ak
1
pa2+1
2
···pak+1
k
: (a1, . . . , ak) ∈ N<N} ⊆ N is defined by the ∆-formula Codenumber(c) :≡ Divides(SS0, c) ∧ (∀z < c)(∀y < z)
- PrimePair(y, z) ∧ Divides(z, c)
- → Divides(y, c)
- .
This formula expresses: “c is divisible by 2 and for every prime pair (pi, pi+1), if pi+1 divides c, so pi divides c.”
SLIDE 6 Continuing, we define the set Yardstick := {0, 1, 2, . . . , k − 1
: k ∈ N} 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
- yi
, x) ↔ Divides(z E Si
zi+1
, x)
SLIDE 7
We next define the set IthPrime := {(i, pi) : i ∈ N≥1} by the ∆-formula IthPrime(i, y) :≡ Prime(y) ∧ (∃x ≤ some term) Yardstick(x) ∧ Divides(yi, x) ∧ ¬Divides(yi+1, x) . Question: What term suffices for this bounded quantifier?
SLIDE 8
We next define the set IthPrime := {(i, pi) : i ∈ N≥1} by the ∆-formula IthPrime(i, y) :≡ Prime(y) ∧ (∃x ≤ yi2) Yardstick(x) ∧ Divides(yi, x) ∧ ¬Divides(yi+1, x) . Question: What term suffices for this bounded quantifier? Answer: We want x to be the i-th yardstick number (p1)1(p2)2 . . . (pi)i. This is at most (pi)(pi)2(pi)3 · · · (pi)i = (pi)(i+1
2 ) ≤ (pi)i2.
Therefore, if y = pi, it suffices to take x ≤ yi2.
SLIDE 9
We next define the set IthPrime := {(i, pi) : i ∈ N≥1} by the ∆-formula IthPrime(i, y) :≡ Prime(y) ∧ (∃x ≤ yi2) Yardstick(x) ∧ Divides(yi, x) ∧ ¬Divides(yi+1, x) . Question: What term suffices for this bounded quantifier? Answer: We want x to be the i-th yardstick number (p1)1(p2)2 . . . (pi)i. This is at most (pi)(pi)2(pi)3 · · · (pi)i = (pi)(i+1
2 ) ≤ (pi)i2.
Therefore, if y = pi, it suffices to take x ≤ yi2. Alternatively, since pi ≤ (i + 1)i (easy fact), we could instead use x ≤ (i + 1)i2.
SLIDE 10 We next define the set IthPrime := {(i, pi) : i ∈ N≥1} by the ∆-formula IthPrime(i, y) :≡ Prime(y) ∧ (∃x ≤ yi·i) Yardstick(x) ∧ Divides(yi, x) ∧ ¬Divides(yi+1, x) .
- Exercise. Convince yourself that IthPrime(i, y) indeed defines IthPrime.
That is, show that
= IthPrime(k, pk) for every k ≥ 1,
= ¬IthPrime(a, b) whenever a = 0 or b = pa.
SLIDE 11 We next define the set IthPrime := {(i, pi) : i ∈ N≥1} by the ∆-formula IthPrime(i, y) :≡ Prime(y) ∧ (∃x ≤ yi·i) Yardstick(x) ∧ Divides(yi, x) ∧ ¬Divides(yi+1, x) .
- Remark. The set IthPrime ⊆ N2 corresponds to the function N → N
defined by i → pi. Therefore, we say that the formula IthPrime(i, y) defines the function i → pi.
SLIDE 12 Continuing, the set Length :=
- (a1, . . . , ak, k) : k ≥ 1 and (a1, . . . , ak) ∈ Nk
is defined by the ∆-formula Length(c, ℓ) :≡ Codenumber(c) ∧ (∃y ≤ c) IthPrime(ℓ, y) ∧ Divides(y, c) ∧ (∀z ≤ c)[PrimePair(y, z) → ¬Divides(z, c)]
SLIDE 13 Continuing, the set Length :=
- (a1, . . . , ak, k) : k ≥ 1 and (a1, . . . , ak) ∈ Nk
is defined by the ∆-formula Length(c, ℓ) :≡ Codenumber(c) ∧ (∃y ≤ c) IthPrime(ℓ, y) ∧ Divides(y, c) ∧ (∀z ≤ c)[PrimePair(y, z) → ¬Divides(z, c)]
The set IthElement :=
- (aj, j, a1, . . . , ak) : 1 ≤ j ≤ k and (a1, . . . , ak) ∈ Nk
is defined by the ∆-formula IthElement(e, i, c) :≡ Codenumber(c) ∧ (∃y ≤ c)
- IthPrime(i, y) ∧ Divides(ySe, c) ∧ ¬Divides(ySSe, c)
- .
SLIDE 14
∆-Definability of Sequence-Coding Operations For practice, try writing a ∆-formula that defines the set Concatenation ⊆ N3 of triples of the form (a1, . . . , ak, b1, . . . , bℓ, a1, . . . , ak, b1, . . . , bℓ).
SLIDE 15 G¨
- del Numbers of Terms and Formulas
We assign a unique number to each symbol in LNT as follows: ¬ 1 ∨ 3 ∀ 5 = 7 9 S 11 + 13 · 15 E 17 < 19 ( 21 ) 23 vi 2i 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 #(s1), . . . , #(sn) 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.
SLIDE 16 Def 5.7.1. For each term t and formula ϕ, the G¨
are defined as follows: ¬α = 1, α +t1t2 = 13, t1, t2 (α ∨ β) = 3, α, β · t1t2 = 15, t1, t2 (∀vi)(α) = 5, vi, α Et1t2 = 17, t1, t2 =t1t2 = 7, t1, t2 <t1t2 = 19, t1, t2 0 = 9 vi = 2i. St = 11, t
SLIDE 17 Def 5.7.1. For each term t and formula ϕ, the G¨
are defined as follows: ¬α = 1, α +t1t2 = 13, t1, t2 (α ∨ β) = 3, α, β · t1t2 = 15, t1, t2 (∀vi)(α) = 5, vi, α Et1t2 = 17, t1, t2 =t1t2 = 7, t1, t2 <t1t2 = 19, t1, t2 0 = 9 vi = 2i. St = 11, t
- Obs. t and ϕ are never divisible by 7. (Why?)
SLIDE 18 Def 5.7.1. For each term t and formula ϕ, the G¨
are defined as follows: ¬α = 1, α +t1t2 = 13, t1, t2 (α ∨ β) = 3, α, β · t1t2 = 15, t1, t2 (∀vi)(α) = 5, vi, α Et1t2 = 17, t1, t2 =t1t2 = 7, t1, t2 <t1t2 = 19, t1, t2 0 = 9 vi = 2i. St = 11, t
= 7, 9, 11, 9 = 7, 210, 11, 210 = 28310255(21231025+1). Notice how fast SSSS0 grows: SSSS0 = 11, 11, 11, 11, 9 = 2123212321232123210
SLIDE 19
Next Steps (Section 5.8) ∆-definability of sets Terms := {t : terms t} = {a ∈ N : a = t for some term t}, Formulas := {ϕ : formulas ϕ} = {a ∈ N : a = ϕ for some formula ϕ}.
