[PPT] - Scott Ranks of Models of a Theory Matthew Harrison-Trainor PowerPoint Presentation

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Scott Ranks of Models of a Theory

Matthew Harrison-Trainor

University of California, Berkeley

Notre Dame, September 2015

Matthew Harrison-Trainor Scott Ranks of Models of a Theory Notre Dame, 2015 1 / 38

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Overview

The Scott rank of a countable structure is a measure of the complexity of describing that structure. Must a Πin

2 theory have a model of Scott rank ≤ α?

Answer: No, it may have only models of high Scott rank. What are the possible Scott spectra of theories? Answer: Certain Σ1

1 classes of ordinals.

Can every computable structure of high Scott rank be approximated by structures of lower Scott rank? Answer: No, there is a computable structure of high Scott rank which cannot be approximated. What is the Scott height of Lω1ω? Answer: δ1

2.

We will answer these questions as applications of a general construction.

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Countable structure theory

All of our languages and structures will be countable. Some of the results are about computable structures. A structure is computable if its domain is ω and its atomic diagram is computable.

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Infinitary logic

Lω1ω is the infinitary logic which allows countable conjunctions and

disjunctions. By a “theory” we mean a sentence of Lω1ω.

A formula is Σin

α if it has α-many alternations of quantifiers and begins

with a disjunction / existential quantifier. A formula is Πin

α if it has α-many alternations of quantifiers and begins

with a conjunction / universal quantifier.

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Infinitary logic

Lω1ω is the infinitary logic which allows countable conjunctions and

disjunctions. By a “theory” we mean a sentence of Lω1ω.

A formula is Σin

α if it has α-many alternations of quantifiers and begins

with a disjunction / existential quantifier. A formula is Πin

α if it has α-many alternations of quantifiers and begins

with a conjunction / universal quantifier.

Example

There is a Πin

2 formula which describes the class of torsion groups. It

consists of the group axioms together with: (∀x) ⩔

n∈N

nx = 0.

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Back-and-forth relations

Theorem (Scott)

Let A be a countable structure. There is an Lω1ω-sentence ϕ, the Scott sentence of A, such that B ⊧ ϕ if and only if B ≅ A.

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Back-and-forth relations

Theorem (Scott)

Let A be a countable structure. There is an Lω1ω-sentence ϕ, the Scott sentence of A, such that B ⊧ ϕ if and only if B ≅ A.

Definition

The standard (non-symmetric) back-and-forth relations ≤α on A, for α < ω1, are defined by: ¯ a ≤0 ¯ b if for each quantifier-free formula ψ(¯ x) with G¨

del number less

than the length of ¯ a, if A ⊧ ψ(¯ a) then A ⊧ ψ(¯ b). For α > 0, ¯ a ≤α ¯ b if for each β < α and ¯ d there is ¯ c such that ¯ b ¯ d ≤β ¯ a¯ c. Let ¯ a ≡α ¯ b if ¯ a ≤α ¯ b and ¯ b ≤α ¯ a, ¯ a ≡α ¯ b if and only if ¯ a and ¯ b satisfy the same Σα formulas.

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Scott rank, version 1

Let A be a structure.

Definition (Scott rank, version 1)

SR(¯ a) is the least α such that: if ¯ a ≡α ¯ b, then ¯ a and ¯ b are in the same automorphism orbit of A. Then SR(A) = sup(SR(¯ a) + 1 ∶ ¯ a ∈ A).

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Scott rank, version 2

Theorem (Montalb´ an)

Let A be a countable structure, and α a countable ordinal. The following are equivalent: A has a Πin

α+1 Scott sentence.

Every automorphism orbit in A is Σin

α -definable without parameters.

A is uniformly (boldface) ∆0

α-categorical without parameters.

Every Πin

α type realized in A is implied by a Σin α formula.

No tuple in A is α-free.

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Scott rank, version 2

Theorem (Montalb´ an)

Let A be a countable structure, and α a countable ordinal. The following are equivalent: A has a Πin

α+1 Scott sentence.

Every automorphism orbit in A is Σin

α -definable without parameters.

A is uniformly (boldface) ∆0

α-categorical without parameters.

Every Πin

α type realized in A is implied by a Σin α formula.

No tuple in A is α-free.

Definition (Scott rank, version 2)

SR(A) is the least ordinal α such that A has a Πin

α+1 Scott sentence.

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Scott spectra

Let T be an Lω1ω-sentence.

Definition

The Scott spectrum of T is the set SS(T) = {α ∈ ω1∶α is the Scott rank of a countable model of T}.

Main Question

What do we know about SS(T)?

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Simple theories with no simple models

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Simple theories with no simple models

Question (Montalb´ an)

If T is a Πin

2 sentence, must T have a model of Scott rank 1?

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Simple theories with no simple models

Question (Montalb´ an)

If T is a Πin

2 sentence, must T have a model of Scott rank 1?

Theorem

Fix α < ω1. There is a Πin

2 sentence T whose models all have Scott rank α.

The construction for this theorem contains many of the ideas required for

ur other results.

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The most basic construction

Fix α an ordinal. The models of T will be ranked trees. The root node has rank α. If a node has rank β, then it has infinitely many children of rank γ for each γ < β.

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The most basic construction

Fix α an ordinal. The models of T will be ranked trees. The root node has rank α. If a node has rank β, then it has infinitely many children of rank γ for each γ < β. We have relations (≡β)β<α on pairs of elements at the same level of the tree and with the same rank.

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The most basic construction

Fix α an ordinal. The models of T will be ranked trees. The root node has rank α. If a node has rank β, then it has infinitely many children of rank γ for each γ < β. We have relations (≡β)β<α on pairs of elements at the same level of the tree and with the same rank. T says that x ≡β y are back-and-forth relations, i.e., x ≡0 y if and only if x and y satisfy the same unary atomic relations Ai. for β > 0, x ≡β y if and only if

▸ for all children x′ of x and γ < β, there is a child y ′ of y with x′ ≡γ y ′. ▸ for all children y ′ of y and γ < β, there is a child x′ of x with x′ ≡γ y ′. Matthew Harrison-Trainor Scott Ranks of Models of a Theory Notre Dame, 2015 11 / 38

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The most basic construction

Fix α an ordinal. The models of T will be ranked trees. The root node has rank α. If a node has rank β, then it has infinitely many children of rank γ for each γ < β. We have relations (≡β)β<α on pairs of elements at the same level of the tree and with the same rank. T says that x ≡β y are back-and-forth relations, i.e., x ≡0 y if and only if x and y satisfy the same unary atomic relations Ai. for β > 0, x ≡β y if and only if

▸ for all children x′ of x and γ < β, there is a child y ′ of y with x′ ≡γ y ′. ▸ for all children y ′ of y and γ < β, there is a child x′ of x with x′ ≡γ y ′.

We make sure that x and y are in the same orbit if and only if they are at the same level in the tree, have the same tree rank β < α, and x ≡β y.

Matthew Harrison-Trainor Scott Ranks of Models of a Theory Notre Dame, 2015 11 / 38

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The most basic construction

Fix α an ordinal. The models of T will be ranked trees. The root node has rank α. If a node has rank β, then it has infinitely many children of rank γ for each γ < β. We have relations (≡β)β<α on pairs of elements at the same level of the tree and with the same rank. T says that x ≡β y are back-and-forth relations, i.e., x ≡0 y if and only if x and y satisfy the same unary atomic relations Ai. for β > 0, x ≡β y if and only if

▸ for all children x′ of x and γ < β, there is a child y ′ of y with x′ ≡γ y ′. ▸ for all children y ′ of y and γ < β, there is a child x′ of x with x′ ≡γ y ′.

We make sure that x and y are in the same orbit if and only if they are at the same level in the tree, have the same tree rank β < α, and x ≡β y. For each x, SR(x) is the tree rank of x. So if A ⊧ T, SR(A) = α.

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Modifications

The T we have presented is not Πin

2 . But we can make it Πin 2 by adding

Skolem functions in a clever way.

Theorem

Fix α < ω1. There is a Πin

2 sentence T whose models all have Scott rank α.

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Computable models of high Scott rank

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Computable structures of high Scott rank

Definition

ωCK

1

is the least non-computable ordinal.

Theorem (Nadel)

A computable structure has Scott rank ≤ ωCK

1

+ 1.

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Computable structures of high Scott rank

Definition

ωCK

1

is the least non-computable ordinal.

Theorem (Nadel)

A computable structure has Scott rank ≤ ωCK

1

+ 1.

Theorem (Harrison)

There is a computable linear order of order type ωCK

1

⋅ (1 + η) with Scott rank ωCK

1

+ 1.

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Computable structures of high Scott rank

Definition

ωCK

1

is the least non-computable ordinal.

Theorem (Nadel)

A computable structure has Scott rank ≤ ωCK

1

+ 1.

Theorem (Harrison)

There is a computable linear order of order type ωCK

1

⋅ (1 + η) with Scott rank ωCK

1

+ 1.

Theorem (Makkai, Knight, Millar)

There is a computable structure of Scott rank ωCK

1

. A computable structure has high Scott rank if it has Scott rank ωCK

1

r

ωCK

1

+ 1.

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Approximations of structures

Let A be a computable structure. SR(A) = ωCK

1

if each automorphism orbit is definable by Σα formulas for some α, but there is no computable bound on the α required.

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Approximations of structures

Let A be a computable structure. SR(A) = ωCK

1

if each automorphism orbit is definable by Σα formulas for some α, but there is no computable bound on the α required. SR(A) = ωCK

1

+ 1 if there is an automorphism orbit which is not defined by a computable formula.

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Approximations of structures

Let A be a computable structure of high Scott rank.

Definition

A is (strongly) computably approximable if every computable infinitary sentence ϕ true in A is also true in some computable B ≇ A with SR(B) < ωCK

1

.

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Approximations of structures

Let A be a computable structure of high Scott rank.

Definition

A is (strongly) computably approximable if every computable infinitary sentence ϕ true in A is also true in some computable B ≇ A with SR(B) < ωCK

1

.

Question (Calvert and Knight)

Is every computable model of high Scott rank is computably approximable?

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Modifications to the earlier construction

We modify the earlier construction. Now index the back-and-forth relations ≡α by elements of ωCK

1

⋅ (1 + η).

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Modifications to the earlier construction

We modify the earlier construction. Now index the back-and-forth relations ≡α by elements of ωCK

1

⋅ (1 + η). We can get a computable model of T.

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Modifications to the earlier construction

We modify the earlier construction. Now index the back-and-forth relations ≡α by elements of ωCK

1

⋅ (1 + η). We can get a computable model of T. But not all elements of ωCK

1

⋅ (1 + η) are ordinals. What happens?

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Non-standard ordinals

Let (L,≤) be a linear order. We consider L to be a non-standard ordinal.

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Non-standard ordinals

Let (L,≤) be a linear order. We consider L to be a non-standard ordinal.

Definition

The well-founded part wfp(L) of L is the initial segment which is well-ordered.

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Non-standard ordinals

Let (L,≤) be a linear order. We consider L to be a non-standard ordinal.

Definition

The well-founded part wfp(L) of L is the initial segment which is well-ordered.

Definition

The well-founded collapse wfc(L) of L is obtained by collapsing the non-well-founded part to a single element. If A is well-ordered, wfc(L) = wfp(L). Otherwise, wfc(L) = wfp(L) + 1.

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Non-standard ordinals

Let (L,≤) be a linear order. We consider L to be a non-standard ordinal.

Definition

The well-founded part wfp(L) of L is the initial segment which is well-ordered.

Definition

The well-founded collapse wfc(L) of L is obtained by collapsing the non-well-founded part to a single element. If A is well-ordered, wfc(L) = wfp(L). Otherwise, wfc(L) = wfp(L) + 1.

Definition

If A is a structure, (≤α)α∈L are non-standard back-and-forth relations if they satisfy the definition of back-and-forth relations, with ordinals replaced by elements of L.

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Computing the Scott rank

Lemma

Let A be a structure and let (≤α)α∈L be non-standard back-and-forth

relations. If ¯

x ≡α ¯ y for some α ∈ L ∖ wfp(L), then ¯ x and ¯ y are in the same automorphism orbit.

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Computing the Scott rank

Lemma

Let A be a structure and let (≤α)α∈L be non-standard back-and-forth

relations. If ¯

x ≡α ¯ y for some α ∈ L ∖ wfp(L), then ¯ x and ¯ y are in the same automorphism orbit. Now: x and y are in the same orbit if and only if they are at the same level in the tree, have the same tree rank β, and x ≡γ y for all γ ∈ ωCK

1

, γ ≤ β.

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Computing the Scott rank

Lemma

Let A be a structure and let (≤α)α∈L be non-standard back-and-forth

relations. If ¯

x ≡α ¯ y for some α ∈ L ∖ wfp(L), then ¯ x and ¯ y are in the same automorphism orbit. Now: x and y are in the same orbit if and only if they are at the same level in the tree, have the same tree rank β, and x ≡γ y for all γ ∈ ωCK

1

, γ ≤ β. If the tree rank of x is in ωCK

1

⋅ (1 + η) ∖ ωCK

1

, SR(x) = ωCK

1

. So if A ⊧ T, SR(A) = ωCK

1

+ 1.

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Computable structures of high Scott rank

Theorem

There is a computable model A of Scott rank ωCK

1

+ 1 and a Πc

2 sentence

ψ such that: A ⊧ ψ B ⊧ ψ ⇒ SR(B) = ωCK

1

+ 1.

Corollary

There is a computable model A of Scott rank ωCK

1

+ 1 which is not computably approximable.

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The general construction for wfp

What about SR(A) = ωCK

1

?

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The general construction for wfp

What about SR(A) = ωCK

1

? For each n, have a set Rn ⊆ ωCK

1

⋅ (1 + η). At level n, have all of the tree be in Rn. Also, at level n, if x ≡α y and x ≢α+1 y, then α ∈ Rn.

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The general construction for wfp

What about SR(A) = ωCK

1

? For each n, have a set Rn ⊆ ωCK

1

⋅ (1 + η). At level n, have all of the tree be in Rn. Also, at level n, if x ≡α y and x ≢α+1 y, then α ∈ Rn. If we are clever, we can make each Rn have a maximal element in Rn ∩ ωCK

1

.

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The general construction for wfp

What about SR(A) = ωCK

1

? For each n, have a set Rn ⊆ ωCK

1

⋅ (1 + η). At level n, have all of the tree be in Rn. Also, at level n, if x ≡α y and x ≢α+1 y, then α ∈ Rn. If we are clever, we can make each Rn have a maximal element in Rn ∩ ωCK

1

. x and y are in the same orbit if and only if they are at the same level n in the tree, have the same tree rank β, and x ≡γ y for all γ ∈ ωCK

1

∩ Rn, γ ≤ β.

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The general construction for wfp

What about SR(A) = ωCK

1

? For each n, have a set Rn ⊆ ωCK

1

⋅ (1 + η). At level n, have all of the tree be in Rn. Also, at level n, if x ≡α y and x ≢α+1 y, then α ∈ Rn. If we are clever, we can make each Rn have a maximal element in Rn ∩ ωCK

1

. x and y are in the same orbit if and only if they are at the same level n in the tree, have the same tree rank β, and x ≡γ y for all γ ∈ ωCK

1

∩ Rn, γ ≤ β. Then SR(¯ x) ∈ ωCK

1

for all ¯

x. We get SR(A) = ωCK

1

.

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Computable structures of high Scott rank

Theorem

There is a computable model A of Scott rank ωCK

1

and a Πc

2 sentence ψ

such that: A ⊧ ψ B ⊧ ψ ⇒ SR(B) = ωCK

1

.

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Computable structures of high Scott rank

Theorem

There is a computable model A of Scott rank ωCK

1

and a Πc

2 sentence ψ

such that: A ⊧ ψ B ⊧ ψ ⇒ SR(B) = ωCK

1

.

Corollary

There is a computable model A of Scott rank ωCK

1

which is not computably approximable.

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Classifying the Scott spectra

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Classifying the Scott spectra, Version 1

Question

What are the possible Scott spectra of theories?

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Classifying the Scott spectra, Version 1

Question

What are the possible Scott spectra of theories?

Theorem (ZFC + PD)

The Scott spectra of Lω1ω-sentences are exactly the sets of the form:

1 wfp(C), 2 wfc(C), or 3 wfp(C) ∪ wfc(C)

where C is a Σ1

1 class.

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