Scott Ranks of Models of a Theory - math.berkeley.edumattht/slides/nd2015.pdf · We will answer...

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

Overview

The Scott rank of a countable structure is a measure of the complexity ofdescribing that structure.

Must a Πin2 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 Σ11 classes of ordinals.

Can every computable structure of high Scott rank be approximatedby structures of lower Scott rank?

Answer: No, there is a computable structure of high Scott rankwhich cannot be approximated.

What is the Scott height of Lω1ω?

Answer: δ12 .

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

Countable structure theory

All of our languages and structures will be countable.

Some of the results are about computable structures. A structure iscomputable if its domain is ω and its atomic diagram is computable.

Infinitary logic

Lω1ω is the infinitary logic which allows countable conjunctions anddisjunctions. By a “theory” we mean a sentence of Lω1ω.

A formula is Σinα if it has α-many alternations of quantifiers and begins

with a disjunction / existential quantifier.

A formula is Πinα if it has α-many alternations of quantifiers and begins

with a conjunction / universal quantifier.

Example

There is a Πin2 formula which describes the class of torsion groups. It

consists of the group axioms together with:

(∀x)⩔n∈N

nx = 0.

Infinitary logic

Lω1ω is the infinitary logic which allows countable conjunctions anddisjunctions. By a “theory” we mean a sentence of Lω1ω.

A formula is Σinα if it has α-many alternations of quantifiers and begins

with a disjunction / existential quantifier.

A formula is Πinα if it has α-many alternations of quantifiers and begins

with a conjunction / universal quantifier.

Example

There is a Πin2 formula which describes the class of torsion groups. It

consists of the group axioms together with:

(∀x)⩔n∈N

nx = 0.

Back-and-forth relations

Theorem (Scott)

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

Definition

The standard (non-symmetric) back-and-forth relations ≤α on A, forα < ω1, are defined by:

a ≤0 b if for each quantifier-free formula ψ(x) with Godel number lessthan 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 thatbd ≤β ac .

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.

Back-and-forth relations

Theorem (Scott)

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

Definition

The standard (non-symmetric) back-and-forth relations ≤α on A, forα < ω1, are defined by:

a ≤0 b if for each quantifier-free formula ψ(x) with Godel number lessthan 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 thatbd ≤β ac .

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.

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 sameautomorphism orbit of A.

ThenSR(A) = sup(SR(a) + 1 ∶ a ∈ A).

Theorem (Montalban)

Let A be a countable structure, and α a countable ordinal. The followingare equivalent:

A has a Πinα+1 Scott sentence.

Every automorphism orbit in A is Σinα -definable without parameters.

A is uniformly (boldface) ∆0α-categorical without parameters.

Every Πinα type realized in A is implied by a Σin

α formula.

No tuple in A is α-free.

SR(A) is the least ordinal α such that A has a Πinα+1 Scott sentence.

Theorem (Montalban)

Let A be a countable structure, and α a countable ordinal. The followingare equivalent:

A has a Πinα+1 Scott sentence.

Every automorphism orbit in A is Σinα -definable without parameters.

A is uniformly (boldface) ∆0α-categorical without parameters.

Every Πinα type realized in A is implied by a Σin

α formula.

No tuple in A is α-free.

SR(A) is the least ordinal α such that A has a Πinα+1 Scott sentence.

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 )?

Simple theories with no simplemodels

Simple theories with no simple models

Question (Montalban)

If T is a Πin2 sentence, must T have a model of Scott rank 1?

Theorem

Fix α < ω1. There is a Πin2 sentence T whose models all have Scott rank α.

The construction for this theorem contains many of the ideas required forour other results.

Simple theories with no simple models

Question (Montalban)

If T is a Πin2 sentence, must T have a model of Scott rank 1?

Theorem

The construction for this theorem contains many of the ideas required forour other results.

The most basic construction

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

We have relations (≡β)β<α on pairs of elements at the same level of thetree 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 relationsAi .

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 atthe 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.

Modifications

The T we have presented is not Πin2 . But we can make it Πin

2 by addingSkolem functions in a clever way.

Theorem

Computable models of high Scottrank

Computable structures of high Scott rank

Definition

ωCK1 is the least non-computable ordinal.

Theorem (Nadel)

A computable structure has Scott rank ≤ ωCK1 + 1.

Theorem (Harrison)

There is a computable linear order of order type ωCK1 ⋅ (1 + η) with Scott

rank ωCK1 + 1.

Theorem (Makkai, Knight, Millar)

There is a computable structure of Scott rank ωCK1 .

A computable structure has high Scott rank if it has Scott rank ωCK1 or

ωCK1 + 1.

Definition

Theorem (Nadel)

Theorem (Harrison)

rank ωCK1 + 1.

ωCK1 + 1.

Definition

Theorem (Nadel)

Theorem (Harrison)

rank ωCK1 + 1.

ωCK1 + 1.

Approximations of structures

Let A be a computable structure.

SR(A) = ωCK1 if each automorphism orbit is definable by Σα formulas for

some α, but there is no computable bound on the α required.

SR(A) = ωCK1 + 1 if there is an automorphism orbit which is not defined by

a computable formula.

Let A be a computable structure.

SR(A) = ωCK1 if each automorphism orbit is definable by Σα formulas for

some α, but there is no computable bound on the α required.

SR(A) = ωCK1 + 1 if there is an automorphism orbit which is not defined by

a computable formula.

Let A be a computable structure of high Scott rank.

Definition

A is (strongly) computably approximable if every computable infinitarysentence ϕ true in A is also true in some computable B ≇ A withSR(B) < ωCK

Question (Calvert and Knight)

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

Let A be a computable structure of high Scott rank.

Definition

A is (strongly) computably approximable if every computable infinitarysentence ϕ true in A is also true in some computable B ≇ A withSR(B) < ωCK

Question (Calvert and Knight)

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

Modifications to the earlier construction

We modify the earlier construction.

Now index the back-and-forth relations ≡α by elements of ωCK1 ⋅ (1 + η).

We can get a computable model of T .

But not all elements of ωCK1 ⋅ (1 + η) are ordinals. What happens?

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 iswell-ordered.

Definition

The well-founded collapse wfc(L) of L is obtained by collapsing thenon-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 ifthey satisfy the definition of back-and-forth relations, with ordinalsreplaced by elements of L.

Definition

Computing the Scott rank

Let A be a structure and let (≤α)α∈L be non-standard back-and-forthrelations. If x ≡α y for some α ∈ L ∖wfp(L), then x and y are in the sameautomorphism orbit.

Now: x and y are in the same orbit if and only if they are at the same levelin 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 ωCK1 ⋅ (1 + η) ∖ ωCK

1 , SR(x) = ωCK1 . So if A ⊧ T ,

SR(A) = ωCK1 + 1.

1 , γ ≤ β.

1 , SR(x) = ωCK1 . So if A ⊧ T ,

SR(A) = ωCK1 + 1.

1 , γ ≤ β.

1 , SR(x) = ωCK1 . So if A ⊧ T ,

SR(A) = ωCK1 + 1.

Theorem

There is a computable model A of Scott rank ωCK1 + 1 and a Πc

2 sentenceψ such that:

A ⊧ ψ

B ⊧ ψÔ⇒ SR(B) = ωCK1 + 1.

Corollary

There is a computable model A of Scott rank ωCK1 + 1 which is not

computably approximable.

The general construction for wfp

What about SR(A) = ωCK1 ?

For each n, have a set Rn ⊆ ωCK1 ⋅ (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 inRn ∩ ω

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

1 ∩Rn, γ ≤ β.

Then SR(x) ∈ ωCK1 for all x . We get SR(A) = ωCK

1 ∩Rn, γ ≤ β.

Theorem

There is a computable model A of Scott rank ωCK1 and a Πc

2 sentence ψsuch that:

A ⊧ ψ

B ⊧ ψÔ⇒ SR(B) = ωCK1 .

Corollary

There is a computable model A of Scott rank ωCK1 which is not

Theorem

There is a computable model A of Scott rank ωCK1 and a Πc

2 sentence ψsuch that:

A ⊧ ψ

B ⊧ ψÔ⇒ SR(B) = ωCK1 .

Corollary

There is a computable model A of Scott rank ωCK1 which is not

Classifying the Scott spectra

Classifying the Scott spectra, Version 1

Question

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 Σ11 class.

Question

Theorem (ZFC + PD)

1 wfp(C),

2 wfc(C), or

3 wfp(C) ∪wfc(C)

Let C be a pseudo-elementary class of linear orders.

Add a new sort to the structures. The new sort is a linear order in C,together with the witnesses.

Replace the back-and-forth relations ≡α by a single ternary relation x ≡a y .Now, the back-and-forth relations are indexed by the new sort.

Name each element of the new sort by a constant so that it does notaffect the Scott rank.

If M ⊧ T , and the order sort is well-founded with order type α, thenSR(M) = α.

What if the order sort is not well-founded?

Scott rank for non-well-founded linear orders

Let A be a model of T , with (L,≤) the ordered sort.

We get SR(x) ≤ wfp(L), and this is achieved by some x . SoSR(A) = wfc(L).

Making the same modification as before to get Scott rank ωCK1 , we can

get SR(A) = wfp(L).

Question

Theorem (ZFC + PD)

1 wfp(C),

2 wfc(C), or

3 wfp(C) ∪wfc(C)

The role of projective determinacy

What is projective determinacy used for?

Definition

A set is projective if it is Σ1n for some n.

Definition

The axiom of projective determinacy says that for any Gale-Stewart game,if the victory set is projective, then one of the players has a winningstrategy (is determined).

Projective determinacy follows from some large cardinal axioms and is notknown to be inconsistent with ZFC.

Definition

Martin’s theorem

Definition

A cone is a set of the form {X ∶X ≥ Y } for some Y .

Theorem (Martin’s Theorem)

If A is Turing invariant and determined, then it either contains or isdisjoint from a cone.

It is an old result that if T is a theory with models of unbounded Scottrank, then for every α a T -admissible ordinal, T has a model A withSR(A) ≥ ωA1 = α.

But we do not know how to decide whether such A has Scott rank ωA1 orωA1 + 1 (or perhaps there are A with each). Projective determinacy saysthat one of these possibilities happens on a cone.

Martin’s theorem

Definition

Martin’s theorem

Definition

Martin’s theorem

Definition

Classification of the Scott spectra, Version 2

Definition

A club is a set of countable ordinals which

unbounded below ω1,

closed in the order topology.

Definition

A stationary set is one which intersect all clubs.

A stationary set contains {ωX1 ∶X ≥T Y } for some Y .

Definition

A set of countable ordinals A is a Σ11 class of ordinals if there is a Σ1

1 classC such that

A = C ∩On.

Theorem (ZFC + PD)

The Scott spectra of Lω1ω-sentences are the Σ11 classes C of ordinals with

the property that if C is unbounded below ω1, then either

C is stationary, or

{α∶α + 1 ∈ C} is stationary.

Definition

A set of countable ordinals A is a Σ11 class of ordinals if there is a Σ1

1 classC such that

A = C ∩On.

Theorem (ZFC + PD)

The Scott spectra of Lω1ω-sentences are the Σ11 classes C of ordinals with

the property that if C is unbounded below ω1, then either

C is stationary, or

{α∶α + 1 ∈ C} is stationary.

Pseudo-elementary classes

Let C be a pseudo-elementary class of linear orders, as used in theconstruction earlier.

If C is not Πin2 , we can make it Πin

2 using Morleyizations.

This does not affect the Scott rank, because the elements of the order sortare named by constants.

Theorem (ZFC + PD)

Every Scott spectrum is the Scott spectrum of a Πin2 theory.

Theorem (ZFC + PD)

Definition

A class C of structures is an Lω1ω-pseudo-elementary class (PCLω1ω-class)

if there is an Lω1ω-sentence T in an expanded language such that thestructures in C are the reducts of models of T .

Theorem (ZFC + PD)

Every Scott spectrum of a PCLω1ω-class is the Scott spectrum of an

Lω1ω-sentence.

Scott heights of computabletheories

Scott height of Lω1ω

Definition (Scott heights)

sh(C) = sup SS(C).

sh(Lω1,ω) is the supremum, over the computable Lω1ω-sentences Twith sh(T ) < ω1, of the Scott height of the models of T .

sh(PCLω1ω) is the supremum, over the computable PCLω1ω

-classes Cwith sh(C) < ω1, of the Scott height of C.

By a counting argument, sh(Lω1,ω) and sh(PCLω1ω) are countable

ordinals.

Question (Sacks)

What is sh(Lω1,ω)?

Definition (Scott heights)

sh(C) = sup SS(C).

sh(Lω1,ω) is the supremum, over the computable Lω1ω-sentences Twith sh(T ) < ω1, of the Scott height of the models of T .

sh(PCLω1ω) is the supremum, over the computable PCLω1ω

-classes Cwith sh(C) < ω1, of the Scott height of C.

By a counting argument, sh(Lω1,ω) and sh(PCLω1ω) are countable

ordinals.

Question (Sacks)

What is sh(Lω1,ω)?

Definition

δ12 is the least ordinal which has no ∆12 presentation.

Theorem (Sacks)

sh(Lω1,ω) ≤ δ12 .

Theorem (Marker)

sh(PCLω1ω) = δ12

Theorem

sh(Lω1,ω) = δ12 .

Definition

Theorem (Sacks)

sh(Lω1,ω) ≤ δ12 .

Theorem (Marker)

sh(PCLω1ω) = δ12

Theorem

sh(Lω1,ω) = δ12 .

Definition

Theorem (Sacks)

sh(Lω1,ω) ≤ δ12 .

Theorem (Marker)

sh(PCLω1ω) = δ12

Theorem

sh(Lω1,ω) = δ12 .

Definition

Theorem (Sacks)

sh(Lω1,ω) ≤ δ12 .

Theorem (Marker)

sh(PCLω1ω) = δ12

Theorem

sh(Lω1,ω) = δ12 .

Open questions

Question

Classify the Scott spectra of Lω1ω-sentences in ZFC.

Question

Classify the Scott spectra of computable Lω1ω-sentences.

Question

Classify the Scott spectra of first-order theories.

Thanks!

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