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

1

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

2

Ramsey Theorem for PA: Let B be a model

of 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.

3

Theorem (McAloon 82) Let M be a model

of 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

4

Ketonen and Solovay (’81) related the follow-

ing three notions:

1. α-largeness

2. Ramsey Theory

3. Wainer functions

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α-LARGENESSε0 is the least ordinal α such that ωα = α

Cantor normal form:

α = ωα1x1 + ωα2x2 + . . . + ωαkxk

where αi are ordinals, α1 > α2 > . . . > αk andk, x1, x2, . . . , xk ∈ ω − {0}

Fundamental sequence:For each ordinal 0 < α < ε0, we define the x-thordinal in the fundamental sequence {α}(x) asfollowsα = 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, sayα = γ + ωβ · a, {α}(x) = γ + {ωβ · a}(x)

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Sequence of ordinals cofinal in ε0ω0 = 1 and ωn+1 = ωωn

Sommer in 95 formalizes the whole theory ofordinals below ε0 in I∆0, including the notionof fundamental sequence, Cantor normal form,ωn’s

Defn: A sequence J = x1 < x2 < . . . is α-large if there is (a code for) a computationsequence C =< c0, c1, . . . , c2r+2 > where c2i is adecreasing sequence of ordinals, c2i+1 = xi ∈ Jand 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

7

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

then J = Jnˆ· · · J1, 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 thereis 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 ordinalωα · 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 allx ≤ j0 there is i s.t. {α}(x) = α2i

x-unwinding of α

6. If J is cofinal definable then J is α-large forall α < ε0

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Connections with Ramsey theory by Ketonen

and Solovay

Thm (Inductive lemma) Let n ≥ 1 and let ω ≤α < ε0. 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|| = ∑kj=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 < ε0, 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 limitordinal

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 < ε0

(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

of 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

of B0,

4) there exists b s.t. for all α < ε0, 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 sufficientconditions for an initial segment B of A modelof I∆0 to be a model of PA and n-elementarysubstructure of A

2) When does a ∈ A belong to an initial seg-ment B of A model of PA and n-elementarysubstructure of A?

Lemma: Let A be a model of I∆0, and letn > 0. Suppose I ⊆ A is a set of order typeω that is diagonal indiscernible for all elementsof B0 and bounds witnesses for all elements ofBn−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 A15

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 Bsuch 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

of 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 < ε0.

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

of BTn−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 < ε0 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 BTn−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 < ε0,

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 Γ22

Cor. Let A be a nonstandard model of I∆0.Then the following are equivalent:

1) there is a nonstandard n-elementary initialsegment B model of PA,

2) there exists a set I of order type ω suchthat I is diagonal indiscernible for all elementsof B0 and bounds witnesses for all elements ofBn−1,

3) there exist b and c coding satisfaction ofΣn formulas by tuples u ≤ b, and for all r,there exists Ir of size r such that Ir is diagonalindiscernible for the first r elements of B0 andbounds witnesses for the first r elements ofBn−1,

4) there exist nonstandard b and c coding sat-isfaction of Σn formulas by tuples u ≤ b, andfor all standard ordinals α < ε0 and all finiteΓ ⊆ 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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