Seconda Universita di Napoli Clermont-Ferrand 2006 *joint work with - - PDF document

Strong initial segments of models of I∆0 Paola D’Aquino∗ Seconda Universita’ di Napoli

Clermont-Ferrand 2006 *joint work with Julia Knight

Language L contains +, ·, 0, 1, < PA: L-theory axiomatized by basic axioms for + and ·, and induction ∀¯ y((θ(0, ¯ y) ∧ ∀x(θ(x, ¯ y) → θ(x + 1, ¯ y)) → ∀xθ(x + 1, ¯ y)) for any formula θ I∆0: subsystem of PA where induction is only ∆0-induction (θ ∈ ∆0). I∆0 ⊢ expotentiation total (Parikh,71) exp = ∀x > 1∀y∃z(xy = z) I∆0 ⊂ I∆0 + exp

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Defn [Bn formulas] The B0 formulas are the ∆0 formulas. The Σn+1 formulas have the form (∃u) ϕ, where ϕ is a Bn formula. The Bn+1 formulas are obtained from the Σn+1 formulas by taking Boolean combinations and adding bounded quantifiers. Combinatorial principles are ubiquitous in arith- metical theories, e.g. Ramsey theory, pigeon- hole principle

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Ramsey Theorem for PA: Let B be a model

• f PA.

Let I be a cofinal definable set, and let F : I[n] → c = {0, . . . , c − 1} be a definable partition of I[n], where n is standard and c ∈ B. Then there is a cofinal definable set J ⊆ I that is homogeneous for F. Defn Let I be a subset of an L-structure A. We say that I is diagonal indiscernible for ϕ(u, x) if for all i < j, k in I, A | = (∀u ≤ i) [ϕ(u, j) ↔ ϕ(u, k)] . Proposition Let A be a model of PA, and let I be a cofinal definable set. For any finite r and any finite set Γ of formulas (with free variables split), there is a set J ⊆ I of size at least r that is diagonal indiscernible for all ϕ(u, x) ∈ Γ.

• Cor. In the same hypothesis get J ⊆ I cofinal

definable set of diagonal indiscernible for all ϕ(u, x) ∈ Γ.

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Theorem (McAloon 82) Let M be a model

• f I∆0.

Then M has a nonstandard initial segment I which is a model of PA The proof uses diagonal indiscernibles

• Thm. Let A be a model of I∆0. Let I be a

subset of A of order type ω such that

• 1. for i, j ∈ I, A |

= i < j → i2 < j,

• 2. I is diagonal indiscernible for all ∆0-formulas.

Then B = {x ∈ A : x < i for some i ∈ I} is a model of PA. We want refinements of McAloon’s result

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Ketonen and Solovay (’81) related the follow- ing three notions:

• 1. α-largeness
• 2. Ramsey Theory
• 3. Wainer functions

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α-LARGENESS ǫ0 is the least ordinal α such that ωα = α Cantor normal form: α = ωα1x1 + ωα2x2 + . . . + ωαkxk where αi are ordinals, α1 > α2 > . . . > αk and k, x1, x2, . . . , xk ∈ ω − {0} Fundamental sequence: For each ordinal 0 < α < ǫ0, we define the x-th

• rdinal in the fundamental sequence {α}(x) as

follows α = β + 1, {α}(x) = β, for all x α = ωβ+1, {α}(x) = ωβ · x α = ωβ, where β is a limit ordinal, {α}(x) = ω{β}(x) α = ωβ · (a + 1), where a = 1, {α}(x) = ωβ · a + {ωβ}(x) α with Cantor normal form ending in ωβ ·a, say α = γ + ωβ · a, {α}(x) = γ + {ωβ · a}(x)

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Sequence of ordinals cofinal in ǫ0 ω0 = 1 and ωn+1 = ωωn Sommer in 95 formalizes the whole theory of

• rdinals below ǫ0 in I∆0, including the notion
• f fundamental sequence, Cantor normal form,

ωn’s Defn: A sequence J = x1 < x2 < . . . is α- large if there is (a code for) a computation sequence C =< c0, c1, . . . , c2r+2 > where c2i is a decreasing sequence of ordinals, c2i+1 = xi ∈ J and c0 = α, c1 = x0, and c2(i+1) = {c2i}(c2i+1) and c2r+2 = 0. Notation: (J, C)α Example: The set X = {3, 4, 5, 6} is ω-large, ω(3, 4, 5, 6) = 3(4, 5, 6) = 2(5, 6) = 1(6) = 0(∅) = 0 giving the computation sequence C =< ω, 3, 3, 4, 2, 5, 1, 6, 0 >. Sommer has α-largeness in I∆0 + exp

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Properties:

• 1. If J is α-large, where α = ωβ1x1+. . .+ωβnxn

then J = Jnˆ· · ·ˆJ1, where Ji is ωβixi-large.

• 2. If (J, C)ωα then there exists J′ ⊆ J and C′

such that (J′, C′)α If C′ = (α, x0, β1, x1, . . . , βr−1, xr, 0) then there is a subsequence C′′ of C C′′ = (ωα, x0, ωβ1, x1, . . . , ωβr−1xr, ω0),

• 3. If (J, C)ωα·x then for all y < x, the ordinal

ωα · y appears in C. 4. If J is ωn+2-large, with first element ≥ c, then there exists J′ ⊆ J that is (ωn+1 + ω3 + c + 3)-large.

• 5. (J, C)α and j0 1st element of J then for all

x ≤ j0 there is i s.t. {α}(x) = α2i x-unwinding of α

• 6. If J is cofinal definable then J is α-large for

all α < ǫ0

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Connections with Ramsey theory by Ketonen and Solovay Thm (Inductive lemma) Let n ≥ 1 and let ω ≤ α < ǫ0. Suppose F : J[n+1] → c. If J is θ-large, where θ = ωα+ω3+max{c, ||α||}+3, then there is an α-large set I ⊆ J such that for increasing tuples x, y and x, z in Jn+1, F(x, y) = F(x, z). where ||0|| = 0 and if α = ωα1m1+. . .+ωαkmk, then ||α|| = k

j=1 mj · (||αj|| + 1)

Cor (pigeon-hole principle) Let F : J → c. If J is θ-large, where θ = ωα+1+ω3+max{c, ||α||}+ 3, then there is an α-large set I ⊆ J on which F is constant.

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The following simple but wasteful version of Inductive lemma e pigeonhole principle hold Thm (Inductive lemma) Suppose F : J[n+1] → c, where J is ωk+2-large and min(J) ≥ c. Then there is an ωk-large I ⊆ J such that for in- creasing tuples x, y and x, z in Jn+1, F(x, y) = F(x, z), or even (ωk + 1)-large. Cor (pigeon-hole principle) Suppose F : J → c, where J is ωk+2-large and min(J) ≥ c. Then there is an ωk-large I ⊆ J on which F is con-

• stant. There is also a set that is (ωk+1)-large.

Ramsey Theorem for α-largeness Suppose n ≥ 1. Let F : J[n] → c, where J is ωk+2n-large, consisting of elements ≥ c. Then there is an ωk-large, or even (ωk + 1)-large homogeneous set I ⊆ J.

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• Thm. Let A be a model of PA. Let Γ be a

finite subset of formulas with the free variables

• split. Let r ∈ N. If J is (ω1+2n(r−1) + 1)-large,

and for x, y ∈ J, A | = x < y → gΓ(x) < y, then there is a subset of J of size r that is diagonal indiscernible for all elements of Γ (n=max length of tuples, gΓ primitive recursive function bounding the number of equivalence classes).

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Wainer hierarchy: For α < ǫ0, Fα(x) is defined as follows F0(x) = x + 1, Fα+1(x) = F (x+1)

α

(x), Fα(x) = max{F{α}(j)(x) : j ≤ x} for α a limit

• rdinal

Ketonen and Solovay related the notion of α- largeness to the functions of the Wainer Hier-

• archy. They introduce the function

Gα(x) = µy([x, y] is α − large),

• Thm. For any α < ǫ0

(i) Fα(n) ≤ Gωα(n + 1); (ii) Gωα(n) ≤ Fα(n + 1) Sommer proves Theorem in I∆0 + exp.

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Refinements of McAloon Thm (McAloon, Som- mer, D’A., Paris, Dimitracopoulos) Let A be a model of I∆0, and let a be a non- standard element. TFAE: 1) there is an initial segment B of A such that a ∈ B and B is a model of PA; 2) there is an infinite set I of order type ω, con- sisting of elements greater than a, such that if i < j in I, A | = i2 < j, and I is diagonal indiscernible for all ϕ(u, x) in B0; 3) there exist b and c s.t. c codes satisfaction

• f bounded formulas by tuples < b, and for all

finite r, there is a sequence Ir of size r, with a < Ir < b, s.t. if i < j in Ir, A | = i2 < j, and Ir is diagonal indiscernible for the first r elements

• f B0,

4) there exists b s.t. for all α < ǫ0, Fα(a) ↓< b.

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Defn. Let A and B be L-structures. We say that B is an n-elementary substructure of A if for all Bn- formulas ϕ(x) and all b in B, B | = ϕ(b) iff A | = ϕ(b). Notation: B ≤n A, Tarski Criterion: Let B ≤0 A, and let n > 0. Suppose that for all Bn−1 formulas ϕ(x, u), and for all b in B if there exists d ∈ A such that A | = ϕ(b, d), then there exists d′ in B such that A | = ϕ(b, d′). Then B ≤n A.

• Defn. Let A be a L-structure and let ϕ(u, x) be

a formula with the free variables splitted into u and x. We say that I bounds witnesses for ϕ(u, x) if for all i, j ∈ I such that A | = i < j, and all a ≤ i in A, A | = (∃x) ϕ(a, x) → (∃x ≤ j) ϕ(a, x) .

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QUESTIONS: 1) Give necessary and sufficient conditions for an initial segment B of A model

• f I∆0 to be a model of PA and n-elementary

substructure of A 2) When does a ∈ A belong to an initial seg- ment B of A model of PA and n-elementary substructure of A? Lemma: Let A be a model of I∆0, and let n > 0. Suppose I ⊆ A is a set of order type ω that is diagonal indiscernible for all elements

• f B0 and bounds witnesses for all elements of

Bn−1. Let B be the downward closure of I. Then is a model of PA and B ≤n A. In order to guarantee the existence of such el- ements we distinguish two cases: Case 1: N ≤n A Case 2: N ≤n A

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N ≤n A

Lemma: Suppose N ≤n A. If I is an infinite subset of N, and β(x, u) is Bn−1, then there is an infinite set J ⊆ I that bounds witnesses for β(x, u). Thm: Suppose A is a nonstandard model of I∆0 such that N ≤n A. TFAE: 1) There is a nonstandard initial segment B such that B ≤n A and B is a model of PA. 2) There exist b and c such that b is nonstan- dard and c codes satisfaction of Σn formulas in A by tuples x ≤ b. In 2)⇒1) we get finite approximations to a set I which bounds witnesses for Bn−1 and is a set

• f diagonal indiscernibles for B0

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Case 2: N ≤n A Lemma Let B be a model of PA. If I is a cofinal definable set, and β(u, x) is in Bn−1, then there is a cofinal definable set J ⊆ I that bounds witnesses for β(u, x).

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Functions Fα,Γ We define partial functions Fα,Γ, for finite Γ ⊆ BT

n−1 and α < ǫ0.

Fα,Γ(a) = the code of a sequence C witness- ing the existence of an α-large sequence J = {j0, j1, j2, . . .}, with a < J, and J bounds wit- nesses for all elements of Γ I.e. C = (c0, c1, c2, . . . , 0) where c0 = α. If α0 = 0, then C has length 1. If c0 = 0, then c1 is the first z > a such that for all ϕ(u, x) ∈ Γ and all u ≤ a, if there exists x satisfying ϕ(u, x), then there is such an x ≤ z. Given c2x = β = 0, and c2x+1, we have c2x+2 = {β}(c2x+1), call it β′. If β′ = 0, then the sequence C has length 2x + 2, while if β′ = 0, then J continues with Fβ′,Γ(c2x+1).

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Thm: Suppose A is a model of I∆0, and let n > 0. For an element a ∈ A, TFAE: 1) a is contained in a nonstandard n-elementary initial segment B that is a model of PA, 2) there is a set I, of order type ω, such that a < I, and I is diagonal indiscernible for all el- ements of B0 and I bounds witnesses for all elements of Bn−1, 3) there exist b > a and c such that c codes satisfaction in A of Σn formulas by tuples < b, and for each finite r, there is a sequence Ir of length r, such that a < Ir < b, and Ir is diago- nal indiscernible for the first r elements of B0, and bounds witnesses for the first r elements

• f BT

n−1,

4) there exist b and c such that c codes satisfac- tion in A of Σn formulas by tuples ≤ b, and for all α < ǫ0 and all finite Γ ⊆ Bn−1, Fα,Γ(a) ↓< b.

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Proof: 3 ⇒ 2 ⇒ 1 ⇒ 4 (3 ⇒ 2) Write a bounded formula ψ(u, a, b, c) which expresses condition 2, the bounds are in terms of b and c. By 3 it is satisfied by all standard u, then use ∆0-overspill. (2 ⇒ 1) By previous lemma (1 ⇒ 4) It follows since we are in a model of PA

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(4 ⇒ 3):

• 1. write a bounded formula ϕ(u, a, b, c) saying

that there exists b′ < b such that for each α ≤ u, there exist J1, J2, C1, C2, such that a < J1 < b′ < J2 < b, J1, J2 bound witnesses for all ϕ ≤ u in Bn−1, and Ci < b witnesses that Ji is α-large.

• 2. ϕ(u, a, b, c) is satisfied by all standard u: if Γ

is the finite set of elements of Bn−1 with codes ≤ u, and α1, . . . , αk the ordinals with codes ≤ u consider the ordinal α = ωα1+. . .+ωαk+ωm+ωm(α1)+. . .+ωm(αk). By 4, Fα,Γ(a) ↓< b, using properties of α-large sets we partition J into intervals J1,1, . . . , J1,k, J∗, J2,1, . . . , J2,k such that J1,i is ωαi-large, J∗ is ωm-large, J2,i is ωm(αi)- large. We get b′ ∈ J∗.

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• 3. By ∆0-overspill get a non standard u sat-

isfying ϕ(u, a, b, c). So there is b′ < b s.t. for all standard α there are J1, J2, C1, C2 < b, J1, J2 bound witnesses for all ϕ ≤ u in BT

n−1, and

Ci < b witnesses that Ji is α-large. 4. From b′ < J2 < b, C2 < b, J2 is α-large witnessed by C2 and J2 bounds witnesses for all standard elements of B0, it follows C2 contains the b′-unwinding of α We show that Fαi(b′) ↓< b for all αi appearing in C2. The proof proceeds by distinguishing αi being a successor ordinal or a limit ordinal. 5. Since Fα(b′) ↓< b, for all standard α < ǫ0, there is an initial segment B of A such that b′ ∈ B and B is a model of PA. Having a sufficiently large set J1 above a bounding witnesses for Γ we get a set Ir ⊆ J of size r that is diagonal indiscernible for Γ

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• Cor. Let A be a nonstandard model of I∆0.

Then the following are equivalent: 1) there is a nonstandard n-elementary initial segment B model of PA, 2) there exists a set I of order type ω such that I is diagonal indiscernible for all elements

• f B0 and bounds witnesses for all elements of

Bn−1, 3) there exist b and c coding satisfaction of Σn formulas by tuples u ≤ b, and for all r, there exists Ir of size r such that Ir is diagonal indiscernible for the first r elements of B0 and bounds witnesses for the first r elements of Bn−1, 4) there exist nonstandard b and c coding sat- isfaction of Σn formulas by tuples u ≤ b, and for all standard ordinals α < ǫ0 and all finite Γ ⊆ Bn−1, FΓ,α(0) ↓< b.

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Open problem: Give conditions under which a nonstandard model of I∆0 has a nonstandard m-elementary initial segment that is a model of IΣn, and say which elements can be included in such an initial segment.

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