Quantifier Rank Spectrum of

L∞,ω (PhD Thesis Defense)by Nathanael Leedom Ackerman

April 19, 2006

Definition 1. If L is a relational language then

Lω1,ω(L) is the smallest collection of formulassuch that if φ(x) ∈ Lω1,ω(L) then

• L ⊆ Lω1,ω(L)

• ¬φ(x) ∈ Lω1,ω(L)

• (∀y)φ(x) ∈ Lω1,ω(L)

• (∃y)φ(x) ∈ Lω1,ω(L)

and if {ψi(x) : i ∈ ω} ⊆ Lω1,ω(L) where⋃

i∈ωFreeVariables(φi) is finite then

• ∧i∈ω ψi(x) ∈ Lω1,ω(L)

• ∨i∈ω ψi(x) ∈ Lω1,ω(L)

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Definition 2. If L is a relational language and

φ(x) ∈ Lω1,ω(L) we define the quantifier rankof φ(x) (qr(φ(x))) by induction:

• qr(R(x)) = 0 if R is a relation in L.

• qr(¬φ(x)) = qr(φ(x)).

• qr(∧i∈ω ψi(x)) = sup{qr(ψi) : i ∈ ω}.

• qr((∀y)ψ(x)) = qr(ψ(x)) + 1.

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Definition 3. If M , N are models of the lan-

guage L then we say M is equivalent to N up

to α (M ≡α N) if and only if for all φ ∈ Lω1,ω(L)with qr(φ) ≤ α

M |= φ ⇔ N |= φ

Definition 4. We say that the quantifier rank

of M (qr(M)) is α if α is the least ordinal such

that for all models N of L

M ≡α N ⇒ (∀β < ω1)M ≡β N

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We know, by the following theorem of

Dana Scott, that in the case of countable mod-

els this is notion is well defined and further the

quantifier rank of any countable model is itself

countable.

Theorem 5 (Scott). If M is a countable model

of the language L then there is a formula φMof Lω1,ω such that

• M |= φM

• For all models N of LN |= φM → N ∼= M

Further, as we will only be interested in

countable models for this talk we will assume

all models are countable and all models have

countable quantifier rank.

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Definition 6. Let φ ∈ Lω1,ω(L). We define thequantifier rank spectrum of φ (qr(φ)) to be

{qr(M) : M |= φ ∧ |M | = ω}

In this talk we will primarily be interested

in particularly well behaved formulas.

Definition 7. Let φ ∈ Lω1,ω. We say that φ isScattered if

(∀α ∈ qr(φ))|{M : M |= φ ∧ |M | = ω∧ qr(M) = α}| = ω

Theorem 8 (Morley). Let φ ∈ Lω1,ω. Then φis scattered if and only if |{M : M |= φ and M iscountable}| = ω or ω1 in all forcing extensionsof the universe.

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The main result of Part I of my thesis and

the main result of this talk is

Theorem 9. Let ω ∗α be a limit ordinal. Thenthere is a scattered sentence φω∗α such that

• Quantifier rank of φω∗α ≤ ω

• Quantifier rank spectrum of φω∗α is un-bounded in ω ∗ α

• φω∗α is scattered.

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Definition 10. Let LP = {Pn : Pn is an n-arypredicate}.

Definition 11. Let TP be universal closure of

the following LP sentences:

(∀i1, · · · in ∈ n)Pn(x1, · · · , xn)→ Pn(xi1, · · · , xin)

Pn+1(x0, · · · , xn) → Pn(x1, · · · , xn)

This theory puts a tree structure under

subseteq (⊆) on the finite subsets of our model.

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Definition 12. Define the color of a ∈ M (‖a‖)as follows:

• ¬P (a) ↔ ‖a‖ = −∞

• P (a) ↔ ‖a‖ ≥ 0

• If the tree extending a is wellfounded then‖a‖ = sup{‖ab‖ : b ∈ M}

• ‖a‖ = ∞ otherwise.

Definition 13. Let M |= TP . Then the Spec-trum of M (Spec(M)) = {‖a‖ : a ⊆ M}

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Definition 14. Let f : m → ORD such thatf(m +1)+1 = f(m). Then we say that f is a

slow slant line.

Definition 15. Let f be a slant line. We say

that two tuples 〈ai : i ∈ n〉 and 〈bi : i ∈ n〉 arethe same up to f if for all S ⊆ n

• ‖〈ai : i ∈ S〉‖ = ‖〈bi : i ∈ S‖〉

• ‖〈ai : i ∈ S〉‖ > f(|S|) and ‖〈bi : i ∈ S‖〉 >f(|S|)

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Tuples

Color

ω ∗ α + 6

ω ∗ α + 5

ω ∗ α + 4

ω ∗ α + 3

ω ∗ α + 2

ω ∗ α + 1

ω ∗ α

a1 a2 a3 a1a2 a2a3 a1a3 a1a2a3

fg

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Definition 16. Let LR = LP ∪ {Ri,j≤ : Ri,j≤ is an

i + j-ary predicate}.

We will abuse notation and consider Ri,j≤ as apredicate of two arguments, one of arity i and

one of arity j.

Definition 17. Let TR be universal closure of

the following LR sentences:

TP

R≤(x,y) ↔ [[¬P (x)] ∨ [P (x) → P (y)∧(∀a)(∃b)R≤(xa,yb)]]

This expanded theory TR will be useful

because we have the following theorem

Theorem 18. If M |= TR and has no tuples ofcolor ∞ then M |= (∀a, b)R≤(a, b) ↔ ‖a‖ ≤ ‖b‖

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The “nice” scattered sentences will have what

we call a Collection of Archetypes. The collec-

tion of archetypes for a sentence T will consist

of four pieces of information

• A set AT(T ) of archetypes

• A partial order 〈2−AT(T ),≤〉 of on certainpairs of archetypes (called consistent pairs

of archetypes)

• A collection BP(T ) of base predicates (alongwith consistent pairs of base predicates

〈2−BP (T ),≤〉)

• An “Extra Information” function EIT : AT(T )∪{M : M |= T} → X ×ORD

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Definition 19. Let T be our theory with nice

properties in a language L. Further let M |=TP . Then define

L(M) = L1 ∪ L2 ∪ {Q, R2≤} ∪ {ci : i ∈M}

Definition 20. Let T (M) be universal closureof the following L(M) sentences:Q:

• Q(x) ↔ ∨a∈M x = ca

• Q |= φ(ca1, · · · can) in L2 iffM |= φ(a1, · · · an)

• Q(x) ∧ ¬Q(y) → ¬U(x,y) where U is anypredicate other than R2≤ and |x| > 0

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L2 :

• (∀x)(∃c)Q(c)R2≤(x, c)

• (∀c)(∃x)¬Q(x) ∧R2≤(x, c)

Other Axioms:

• ¬Q |= T2

• ¬Q |= T1

• ¬Q |= P1(x) → P2(x)

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• Homogeneity:For all (A, A∗), (B, B∗) consistent pairs ofbase predicates such that (A, A∗) ≤ (B, B∗),and all m ∈ ω

¬Q |=[(∀x)[A1(x) ∧A2∗(x)] →(∃y1, · · ·ym)(B1(x,y) ∧B2∗ (x,y))]

• Completeness:

(∀x)(∃y)∨

(A,A′)∈2−BP (T )(A, A′)(xy)

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We are going to want our theory T to have

properties which allow us to prove the following

Theorem 21. If

• M, N |= T (M)

• M |L1 ≡ω∗α N |L1

• M |L2 ∼= N |L2

then M ≡ω∗α NTheorem 22. If Spec(M) ⊆ Spec(M) thenthere is a model M ′ |= T (M) such that M ′|L1 ∼=M .

Theorem 23. If ω ∗ α < Spec(M) which isa limit ordinal then there are M, N such that

M ≡ω∗α N and Spec(M)∪Spec(N) ⊆ Spec(M).

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We are now ready to give our definition of a

collection of archetypes

(Truth on Atomic Formulas for Archetypes)

If M |= φ(x) and N |= φ(y) where φ is anarchetype then for every atomic formula ψ,

M |= ψ(x) iff N |= ψ(y).

(Truth on Color)

If φ ∈ AT(T ) and φ(x1, · · · , xn), φ(y1, · · · , yn)then ‖{xi : i ∈ S‖ = ‖{yi : i ∈ S‖ for all S ⊆ n.

(Truth on Atomic Formulas for Base Predicates)

If M |= A(x) and N |= B(y) where B is abase predicatethen for every atomic formula

ψ, M |= ψ(x) iff N |= ψ(y).

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(Restriction of Arity for Archetypes)

If φ is an archetype on a tuple x and y is a

subset of x then we can restrict φ to y and get

an archetype.

(Completeness for Archetypes)

If φ is an archetype which describes a tuple x

and x∧y extends x then there is some archetypewhich describes x∧y.

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(Amalgamation for Archetypes)

If φ and ψ are archetypes which agree on the

what they force to be true on their common

domain then there is a consistent extension of

A and B which forces all “new” colors to be

−∞.

(Amalgamation for Base Predicates)

If A and B are base predicates which agree on

the what they force to be true on their com-

mon domain then there is a consistent exten-

sion of A and B which forces all “new” colors

to be −∞.

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(Homogeneity for Base Predicates)

If B is a base predicate which forces another

base predicate A to hold and M |= A(a) thenthere are infinitely many extensions {bi : i ∈ ω}of a such that M |= A(a∧b)

(Uniqueness of Base Predicate)

This says that each tuple realizes at most one

base predicate

(Completeness of Extra Information)

This says that the extra information predicate

for a model is just the union of the extra infor-

mation from each archetype which is realized.

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Now we come to two of the most important

properties of a collection of archetypes.

(Prediction)

If σ, τ are archetypes such that τ(x,y) forces

σ(x) then there is an archetype ητ(a) and a

base predicate Aτ such that

• M |= (∃x,y)τ(x,y) if and only if M |= (∃a)ητ(a)

• (∀M |= T ) M |= [ητ(a)∧σ(x)∧Aτ(x,y, z, a)] →τ(x,y).

If η(a)

σ(x)

A(x,y, z, a)

then η(a)

σ(x)

A(x,y, z, a)

τ(x,y)

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(Prediction up to a Slant Line)

If σ, σ′, τ are archetypes such that

• τ(x,y) forces σ(x)

• σ and σ′ force the colors on their domainsto be the same up to a slant line sl

• sl(1) = ω ∗ λ + |xy|+ n

then there is an archetype ητ |sl(a) and a basepredicate Aτ |sl such that

• If M |= (∃a)σ′(a) then M |= (∃b)(ητ |sl(b)

• For all M |= T if M |= [ητ |sl(a) ∧ σ′(x) ∧Aτ |sl(x,y,z, a)]∧τ ′(x,y) then τ and τ ′ forcethe colors of the tuples they describe to be

the same up to slant line sl

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(Consistency of Color)

If (φ, φ′) is a consistent pair archetypes thenany color φ′ forces must be at least as large asthe color φ forces on the same tuple.

(Consistency of ≤)≤ on consistent archetype pairs is transitiveand if (φ0, φ1) ≤ (ψ0, ψ1) then ψi is the restric-tion of φi to its domain.

(Restriction of Arity for 2-Seq. of Archetypes)

If (φ0, φ1)(x,y) ≤ (ψ0, ψ1)(x) and (ζ0, ζ1) is arestriction of (φ0, φ1)(x,y) to x,z with z ⊆ ythen (ζ0, ζ1) ≤ (ψ0, ψ1)

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(Amalgamation for 2-Sequences of Archetypes)

If (φ0, φ1) and (ψ0, ψ1) are consistent pairs of

archetypes which each force the same informa-

tion on their common domain then the amal-

gamations which give all “new” tuples color

−∞ is also a consistent archetype pair and ≤(φ0, φ1) and (ψ0, ψ1).

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(Homogeneity of 2-Sequences of Archetypes)

Suppose

• (σ, σ′), (τ, τ ′), (η, η′) are consistent pairs ofarchetypes

• (η, η′)(x,y) ≤ (σ, σ′)(x)

• (η, η′)(x,y) forces (B, B′)(x,y)

• (τ, τ ′)(x), (σ, σ′)(x) both force (A, A′)(x)

(where A, A′, B, B′ are base predicates). Thenthere is a consistent pair of archetypes (ζ, ζ′)such that

• (ζ, ζ′)(x,y) ≤ (τ, τ ′)(x)

• (ζ, ζ′)(x,y) forces (B, B′)(x,y)

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(Completeness of 2-Sequences of Base Predicate)

If (τ, τ ′) is a consistent sequence of archetypessuch that (τ, τ ′) forces (A, A′) and σ, σ′ arearchetypes such that

• σ(x) forces A(x)

• σ′(x) forces A′(x)

• Every color which σ′ forces is at least asgreat as the color σ forces on the same

tuple

Then (σ, σ′) is a consistent pair of archetypes.

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(Extension of 0-Colors)

Suppose (σ, σ′) is a consistent pair of archetypes.Further assume that τ ′(x,y) forces σ′(x). Then,if τ(x,y) forces σ(x) and forces all “new” tu-

ples to have color −∞, (τ, τ ′) is a consistentpair of archetypes and (τ, τ ′) ≤ (σ, σ′)

(Extension of 1-Colors)

Suppose (σ, σ′) is a consistent pair of archetypes,τ ′(x,y) forces σ′(x) and there is some modelwhich realizes both τ and σ′. Then there isan archetype τ ′ such that (τ, τ ′) is a consistentpair of archetypes and (τ, τ ′) ≤ (σ, σ′)

Tuples

Color

τ σ

τ ′σ′

Tuples

Color

τ σ

τ ′σ′

η

η′

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Theorem 24. Let N |= T (M). If

• (σ0, σ1), (τ0, τ1) ∈ 2−AT(T )

• (τ0, τ1)(x,y) ≤ (σ0, σ1)(x)

• τi is realized in N |Li

then N |= (∀x)(σ0, σ1)(x) → (∃y)(τ0, τ1)(x,y).

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We start with

σ(x)

σ(x)′

then we have

σ(x)

σ(x)′

B(xy)

B′(xy)′

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We know there exists

σ(x)

σ(x)′

B(xy)

B′(xy)′

ητ(a)

ητ ′(b)

and we have by Prediction

σ(x)

σ(x)′

B(xy)

B′(xy)′

ητ(a)

ητ ′(b)

A(axyz)

A′(bxyz′)

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So we have

σ(x)

σ(x)′

B(xy)

B′(xy)′

ητ(a)

ητ ′(b)

A(axyz)

A′(bxyz′)

E(abxyzz′)

E′(abxyzz′)

and this implies

σ(x)

σ(x)′

τ(xy)

τ ′(xy)′

ητ(a)

ητ ′(b)

A(axyz)

A′(bxyz′)

E(abxyzz′)

E′(abxyzz′)

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Theorem 25. If

• (σ0, σ1),(σ′0, σ′1), (τ0, τ1) ∈ 2−AT(T )

• (τ0, τ1)(x,y) ≤ (σ0, σ1)(x)

• σ0|sl = σ′0|sl

then there is a τ ′0 such that (τ ′0, τ1) ≤ (σ′0, σ′1)and τ ′0|sl = τ0|sl

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Lemma 26. If

• M, N |= T

• M ≡ω∗α N

• Iη∗ω+n = {f : M → N s.t. f is a bijection,|dom(f)| < ω, there exists a slant line sl <(η + 1) ∗ ω such that if M |= σf(dom(f))and N |= τf(range(f)) then σf |sl = τf |sland where sl(|dom(f)|+ n) ≥ η ∗ ω}

Then 〈Iη : η < ω ∗ α〉 is a sequence of partialisomorphisms which witness that M ≡ω∗α N .

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Theorem 27. If

• M, N |= T (M)

• M |L1 ≡ω∗α N |L1

• M |L2 ∼= N |L2

then M ≡ω∗α N

Proof. Let Iω∗η+n = {f :

• |dom(f)| < ω,

• There exists a slant line sl < (η+1)∗ω suchthat if M |= (σ0, σ1)(dom(f)) and N |=(τ0, τ1)(range(f)) then σ0|sl = τ0|sl, τ1 =σ1 and where sl(|dom(f)|+ n) ≥ η ∗ ω}

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We know that Iη is non-empty for all η <

ω ∗ α by the previous lemma, and by the pre-vious theorems we know (with out to much

work) that in fact 〈Iη : η ∈ ω ∗ α〉 has the backand forth property and hence witnesses that

M ≡ω∗α N

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Theorem 28. If Θ is as in “The Vaught’s Con-

jecture: A Counter Example” then Θ has a

collection of archetypes and Θ is scattered.

Theorem 29. If M, N |= Θ, have no tuples ofcolor∞ and Spec(M)∩ORD,Spec(N)∩ORD ≥ω ∗ α then M ≡ω∗α N

Theorem 30. If N,Mmodels Θ and if Spec(N) ⊆Spec(M) then there is a model N ′ |= Θ(M)such that N ′|L1 ∼= N and N ′|L2 ∼= M

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Theorem 31. IfM |= Θ and Spec(M) = {−∞}∪ω ∗ α then

• Θ(M) has quantifier rank ω

• Quantifier Rank Spectrum(Θ(M)) is un-bounded in ω ∗ α

• Θ(M) is Scattered

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