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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 unbounded in ω ∗ α
• φω∗α is scattered.
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Definition 10. Let LP = {Pn : Pn is an narypredicate}.
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 Spectrum 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 + jary 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 restriction of φi to its domain.
(Restriction of Arity for 2Seq. 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 2Sequences 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 2Sequences 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 2Sequences 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 0Colors)
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 1Colors)
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)
• σ0sl = σ′0sl
then there is a τ ′0 such that (τ ′0, τ1) ≤ (σ′0, σ′1)and τ ′0sl = τ0sl
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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 σ0sl = τ0sl, τ1 =σ1 and where sl(dom(f)+ n) ≥ η ∗ ω}
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We know that Iη is nonempty for all η <
ω ∗ α by the previous lemma, and by the previous 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 unbounded in ω ∗ α
• Θ(M) is Scattered
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