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 formulas
such 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∈ωFree
Variables(φi) is finite then
• ∧i∈ω ψi(x) ∈ Lω1,ω(L)
• ∨i∈ω ψi(x) ∈ Lω1,ω(L)
1
Definition 2. If L is a relational language and
φ(x) ∈ Lω1,ω(L) we define the quantifier rank
of 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.
2
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
3
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 φM
of Lω1,ω such that
• M |= φM
• For all models N of L
N |= 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.
4
Definition 6. Let φ ∈ Lω1,ω(L). We define the
quantifier 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 is
Scattered if
(∀α ∈ qr(φ))|{M : M |= φ ∧ |M | = ω
∧ qr(M) = α}| = ω
Theorem 8 (Morley). Let φ ∈ Lω1,ω. Then φ
is scattered if and only if |{M : M |= φ and M is
countable}| = ω or ω1 in all forcing extensions
of the universe.
5
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. Then
there 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.
6
Definition 10. Let LP = {Pn : Pn is an n-ary
predicate}.
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.
7
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}
8
Definition 14. Let f : m → ORD such that
f(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〉 are
the 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|)
9
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
10
Definition 16. Let LR = LP ∪ {Ri,j≤ : R
i,j≤ is an
i + j-ary predicate}.
We will abuse notation and consider Ri,j≤ as a
predicate 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 of
color ∞ then M |= (∀a, b)R≤(a, b) ↔ ‖a‖ ≤ ‖b‖
11
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 certain
pairs of archetypes (called consistent pairs
of archetypes)
• A collection BP(T ) of base predicates (along
with consistent pairs of base predicates
〈2−BP (T ),≤〉)
• An “Extra Information” function EIT : AT(T )∪{M : M |= T} → X ×ORD
12
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 closure
of 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 any
predicate 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)
14
• Homogeneity:
For all (A, A∗), (B, B∗) consistent pairs of
base 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)
15
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 N
Theorem 22. If Spec(M) ⊆ Spec(M) then
there is a model M ′ |= T (M) such that M ′|L1 ∼=M .
Theorem 23. If ω ∗ α < Spec(M) which is
a limit ordinal then there are M, N such that
M ≡ω∗α N and Spec(M)∪Spec(N) ⊆ Spec(M).
16
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 an
archetype 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 a
base 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 archetype
which 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 −∞.
19
(Homogeneity for Base Predicates)
If B is a base predicate which forces another
base predicate A to hold and M |= A(a) then
there 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.
20
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)
21
(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 domains
to 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 base
predicate 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 then
any color φ′ forces must be at least as large as
the color φ forces on the same tuple.
(Consistency of ≤)
≤ on consistent archetype pairs is transitive
and 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 a
restriction of (φ0, φ1)(x,y) to x,z with z ⊆ y
then (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 of
archetypes
• (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). Then
there 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 archetypes
such that (τ, τ ′) forces (A, A′) and σ, σ′ are
archetypes such that
• σ(x) forces A(x)
• σ′(x) forces A′(x)
• Every color which σ′ forces is at least as
great 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 consistent
pair 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 model
which realizes both τ and σ′. Then there is
an archetype τ ′ such that (τ, τ ′) is a consistent
pair of archetypes and (τ, τ ′) ≤ (σ, σ′)
Tuples
Color
τ σ
τ ′σ′
Tuples
Color
τ σ
τ ′σ′
η
η′
27
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).
28
We start with
σ(x)
σ(x)′
then we have
σ(x)
σ(x)′
B(xy)
B′(xy)′
29
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
32
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 partial
isomorphisms which witness that M ≡ω∗α N .
33
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 such
that 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}
34
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 back
and forth property and hence witnesses that
M ≡ω∗α N
35
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 of
color∞ 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
36
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
37