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A Miniaturized “Predicativity” – “slow growing” analogues.
Stanley S. Wainer1 (Leeds UK) HAPPY BIRTHDAY GERHARD December 2013.
1Earlier parts of this work were done jointly with Elliott Spoors and were
A Miniaturized Predicativity slow growing analogues. Stanley S. - - PowerPoint PPT Presentation
A Miniaturized Predicativity slow growing analogues. Stanley S. - - PowerPoint PPT Presentation
A Miniaturized Predicativity slow growing analogues. Stanley S. Wainer 1 (Leeds UK) HAPPY BIRTHDAY GERHARD December 2013. 1 Earlier parts of this work were done jointly with Elliott Spoors and were partially supported by the 2012
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§1. EA(I; O) – Leivant (1995), Ostrin-Wainer (2005),
◮ Quantified numerical “output” variables a, b, c, . . . . ◮ Unquantified “input” variables x, y, z, . . . (constants). ◮ Terms 0, Succ, +, ×, π, π0, π1, . . . with usual axioms. ◮ “Predicative/bounded/pointwise Induction” up to x:
A(0) ∧ ∀a(A(a) → A(a + 1)) → ∀a ≤ xA(a).
◮ Define f (x)↓ ≡ ∃aCf (x, a) for some Σ1 formula Cf . ◮ Then EA(I; O) ⊢ f (x)↓ if and only if f is elementary.
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EA(I; O) ⊂ EA(I; O)+ – Spoors-Wainer (2012)
EA(I; O) is not “user-friendly” since composition of functions f : I → O cannot be proved straightforwardly – however Wirz (2005) developed a variety of derived rules showing this. To remedy this, add a Σ1-“Reflection Rule” as in Cantini (2002): Σ( x), ∃aA(a, x) Σ( x), ∃yA(y, x) where the only free parameters are inputs
- x. And add I-quantifiers:
Γ, A(x) Γ, ∀yA(y) Γ, A(t( x)) Γ, ∃yA(y) . Note: the inductions are still restricted to EA(I; O) formulas only. Then if ⊢ f (x)↓ and ⊢ g(x)↓ we can directly prove ∀yf (y)↓ and (by reflection) ∃y(g(x) = y). Therefore EA(I; O)+ ⊢ f (g(x))↓.
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§2. EA(I1, I2; O) = EA(I1; O)+(I2)+.
Add to EA(I1; O)+ a new layer of I2–inputs u, v, . . . and a new level of inductions: A(0) ∧ ∀a(A(a) → A(a + 1)) → A(u) where A is now any EA(I1; O)+ formula. Then:
◮ EA(I1; O) ⊢ 2x↓ ◮ EA(I1; O)+ ⊢ ∀x∃y(2x = y) ◮ EA(I1; O)+ ⊢ ∃y(2x a = y) → ∃y(2x a+1 = y) ◮ EA(I1; O)+(I2) ⊢ ∀x∃y(2x u = y)
Now add I2–quantifier rules and a Σ1–reflection rule for I2. This allows compositions of the superexponential etc., so EA(I1; O)+(I2)+ ⊢ E4(u) ↓ .
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§3. Pointwise Transfinite Induction.
Usual transfinite induction TI(A, α) may be written: A(0) ∧ ∀γ(A(γ) → A(γ + 1)) ∧ ∀λ(∀iA(λi) → A(λ)) → A(α)
Definition
“Weak, pointwise transfinite induction” PTI(Ax,α): A(0) ∧ ∀γ(A(γ) → A(γ + 1)) ∧ ∀λ(∀i ≤ xA(λi) → A(λ)) → A(α)
- r
A(0) ∧ ∀γ(A(γ) → A(γ + 1)) ∧ ∀λ(A(λx) → A(λ)) → A(α). The idea goes back to U. Schmerl (1982). This is enough to define the Slow Growing function Gα(x) and is equivalent to Bounded Induction “up to” Gα(x): A(0) ∧ ∀a(A(a) → A(a + 1)) → ∀a ≤ Gα(x)A(a).
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Tree Ordinals α ≺ Γ0 and their Gα’s.
Definition
◮ α ∈ Ω if α = 0 or ∃β ∈ Ω(α = β + 1) or α : N → Ω. ◮ G : N × Ω → N is given by
Gn(0) = 0, Gn(β + 1) = Gn(β) + 1. Gn(λ) = Gn(λn) .
◮ ψ : Ω × Ω → Ω is given by
ψ0(β) = β+2β, ψα+1(β) = ψ2β
α (β),
ψλ(β) = sup ψλn(β) .
◮ F0(m) = m + 2m, Fn+1(m) = F 2m n (m) . Then Fω(n) = Fn(n).
Theorem
(i) |ψα(ω)| = Veblen φα(0) for α ≻ 0. (ii) Gn(ψα(β)) = FGn(α)(Gn(β)). So Gn(ψα(α)) = Fω(Gn(α)).
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PTI(α) in EA(I, ..; O) theories.
◮ Recall: For α ≺ ε0, G(α) ∈ E3, but G(ε0) ∈ E3. ◮ Hence: EA(I; O) ⊢ PTI(α ≺ ε0), but EA(I; O) ⊢ PTI(ε0). ◮ Thus: EA(I; O)+W = ε0 = φ1(0). ◮ Similarly: EA(I1, I2; O)+W = φ2(0) etcetera. ◮ And: EA(I1, . . . , Iω; O)∞ ≺Γ0W = Γ0. ◮ Wainer-Williams (2005): ID1(I; O)W = φεΩ+1(0) but
ID1(I; O) ≡ PA. Note: J¨ ager-Probst (2013) and Ranzi-Strahm (2013), SIDν have full (unstratified) numerical induction in the base theory.
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§4. EA(I1, I2, . . . , Iω; O)∞
≺Γ0.
Sequents are: n : Ik ; . . . , m : Ii ⊢α Γ with ω ≥ k > . . . > i. Logic Rules are as follows where β ≺n α ≺ Γ0 : (∃Ii)n : Ik ; . . . , m : Ii ⊢β
C ℓ
n : Ik ; . . . , m : Ii ⊢β Γ, A(ℓ) n : Ik ; . . . , m : Ii ⊢α Γ, ∃x(Ii(x) ∧ A(x)) (∀Ii) { n : Ik ; . . . , max(m, j) : Ii ⊢β Γ, A(j) }j n : Ik ; . . . , m : Ii ⊢α Γ, ∀x(Ii(x) → A(x)) level(A) < k and (∨), (∧) and (Cut) as usual, together with Computation Rules: (Ax) n; . . . , m ⊢α
C ℓ if ℓ ≤ q(m)
(C)n; . . . m ⊢β
C m′
n; . . . m′ ⊢β
C ℓ
n; . . . m ⊢α
C ℓ
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Reading Off Bounding Functions.
Ordinal assignment is “slow growing”: |{β : β ≺n α}| = Gn(α).
Lemma
If n; m ⊢α
C k then k ≤ qGn(2α)(m).
Theorem (Basic bounding principle)
If EA(I;O)+ ⊢ f (x)↓ then, by embedding and cut-reduction, there is an α ≺ ε0 such that for every x := n, n; − ⊢α
0 ∃aCf (n, a). Then
∃a ≤ kCf (n, a) where k = qGn(2α)(0). So f ∈ E3.
Lemma (E4 bounding)
Let B1(α, n) = qGn(2α)(0) be the bounding function at level 1. Then B2(α, n) = B1(α)Gn(2α)(n) is the bound at level 2: n2; n1, − ⊢α
C k
⇒ k ≤ B2(α, max(n2, n1)) . This bound is E4-definable. Etcetera.
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§5. Level ω – Ackermann.
Suppressing ordinal bounds, EA(I1, I2, . . . , Iω; O)+
∞ proves:
∀xr∃y r(F 2a
r (x) = y) → ∀xr∃y r(F 2a+1 r
(x) = y) Hence by induction on a, using repeated cuts: k : Ir+1; ⊢ ∀xr∃y r(Fr(x) = y) → ∀xr∃y r(F 2k
r (x) = y)
By Cut on ∀xr∃y r(Fr(x) = y) using k : Ir+1 ⊢C k : Ir, k : Ir+1 ⊢ ∃y r(Fr+1(k) = y) Then k : Ir+1 ⊢ ∃y r+1(Fr+1(k) = y) so ∀xr+1∃y r+1(Fr+1(x) = y). So r : Iω ⊢ ∀xr∃y r(Fr(x) = y). But note: r : Iω ⊢C r : Ir. Therefore r : Iω ⊢ ∃y ω(Fω(r) = y).
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§6. “Predicativity” in EA(I1, . . . , Iω; O)∞
≺Γ0.
◮ Sequents are: n : Ik; ..m : Ii ⊢α Γ where ordinal bounds
α ≺ Γ0 are autonomously generated according to the rule: PTI(β) ⇒ PTI(ψβ(β)).
◮ This holds because if Gn(β) is computable in the system, so is
Gn(ψβ(β)) = FGn(β)(Gn(β)) = Fω(Gn(β)) . Only finite iterations of Fω are possible, so can’t reach Γ0.
◮ Collapsing Principle: n : Iω; m : Ii ⊢α C k
⇒ m : Ii; ⊢a
C k
where a = Gn(ψα(α)) = Fω(Gn(α). Computational bounds are finite F-terms, elementary in Fω.
◮
EA(I1, . . . Iω; O)∞
≺Γ0 ⊢ Eω(x : Iω) ↓
EA(I1, . . . Iω; O)∞
≺Γ0W = Γ0 .
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