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Randomizations of Scattered Theories

• H. Jerome Keisler

January 9, 2015

• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 1 / 1

• 1. Overview

Intuitively, a randomization of a first order theory T is a continuous theory T R whose models contain random elements of models of T. In many cases, T R has properties similar to those of T.

• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 2 / 1

• 1. Overview

Intuitively, a randomization of a first order theory T is a continuous theory T R whose models contain random elements of models of T. In many cases, T R has properties similar to those of T. In a forthcoming paper, Uri Andrews and the author showed that if T has countably many non-isomorphic countable models, then T R has few separable models.

• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 2 / 1

• 1. Overview

Intuitively, a randomization of a first order theory T is a continuous theory T R whose models contain random elements of models of T. In many cases, T R has properties similar to those of T. In a forthcoming paper, Uri Andrews and the author showed that if T has countably many non-isomorphic countable models, then T R has few separable models. This means that up to isomorphism, every separable model is determined in a simple way by assigning a probability to each isomorphism type of countable models of T.

• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 2 / 1

• 1. Overview

Intuitively, a randomization of a first order theory T is a continuous theory T R whose models contain random elements of models of T. In many cases, T R has properties similar to those of T. In a forthcoming paper, Uri Andrews and the author showed that if T has countably many non-isomorphic countable models, then T R has few separable models. This means that up to isomorphism, every separable model is determined in a simple way by assigning a probability to each isomorphism type of countable models of T. In the other direction, if T R has few separable models, then T is scattered in the sense of Morley. Assuming the absolute form of Vaught’s conjecture, T has countably many non-isomorphic countable models if and

• nly if T R has few separable models.
• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 2 / 1

• 1. Overview

Intuitively, a randomization of a first order theory T is a continuous theory T R whose models contain random elements of models of T. In many cases, T R has properties similar to those of T. In a forthcoming paper, Uri Andrews and the author showed that if T has countably many non-isomorphic countable models, then T R has few separable models. This means that up to isomorphism, every separable model is determined in a simple way by assigning a probability to each isomorphism type of countable models of T. In the other direction, if T R has few separable models, then T is scattered in the sense of Morley. Assuming the absolute form of Vaught’s conjecture, T has countably many non-isomorphic countable models if and

• nly if T R has few separable models.

I will also discuss what happens without Vaught’s conjecture.

• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 2 / 1

Some References

1 I. Ben Yaacov, A. Berenstein, C. W. Henson, A. Usvyatsov. Model

Theory for Metric Structures. Cambridge U. Press (2008), 315–427.

2 I. Ben Yaacov and H. J. Keisler. Randomizations as Metric

Structures, Confluentes Mathematici 1 (2009), 197-223.

3 U. Andrews and H. J. Keisler. Separable Models of Randomizations.

To appear, JSL.

• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 3 / 1

Some References

1 I. Ben Yaacov, A. Berenstein, C. W. Henson, A. Usvyatsov. Model

Theory for Metric Structures. Cambridge U. Press (2008), 315–427.

2 I. Ben Yaacov and H. J. Keisler. Randomizations as Metric

Structures, Confluentes Mathematici 1 (2009), 197-223.

3 U. Andrews and H. J. Keisler. Separable Models of Randomizations.

To appear, JSL.

4 R. L. Vaught. Denumerable Models of Complete Theories. In

Infinitistic Methods, Pergamon Press (1961), 303-321.

5 D. Scott. Logic with Denumerably Long Formulas and Finite Strings

• f Quantifiers. In Theory of Models, North-Holland (1965), 329-341.

6 M. Morley. The Number of Countable Models. JSL 25 (1970), 14-18. 7 J. Steel. On Vaught’s Conjecture. Cabal Seminar 76–77, 193–208. 8 J. Baldwin, S. Friedman, M. Koerwien, and M. Laskowski. Three Red

Herrings around Vaught’s Conjecture. To appear, TAMS.

• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 3 / 1

• 2. Continuous Structures

Structures live on complete metric spaces.

• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 4 / 1

• 2. Continuous Structures

Structures live on complete metric spaces. Prestructures live on pseudometric spaces.

• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 4 / 1

• 2. Continuous Structures

Structures live on complete metric spaces. Prestructures live on pseudometric spaces. Completion of a prestructure: A structure formed by identifying elements at distance 0 and completing the metric.

• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 4 / 1

• 2. Continuous Structures

Structures live on complete metric spaces. Prestructures live on pseudometric spaces. Completion of a prestructure: A structure formed by identifying elements at distance 0 and completing the metric. Formulas take values in [0, 1] with 0 meaning true.

• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 4 / 1

• 2. Continuous Structures

Structures live on complete metric spaces. Prestructures live on pseudometric spaces. Completion of a prestructure: A structure formed by identifying elements at distance 0 and completing the metric. Formulas take values in [0, 1] with 0 meaning true. Connectives 0, 1, Φ/2, Φ −

· Ψ.

Quantifiers sup, inf.

• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 4 / 1

• 2. Continuous Structures

Structures live on complete metric spaces. Prestructures live on pseudometric spaces. Completion of a prestructure: A structure formed by identifying elements at distance 0 and completing the metric. Formulas take values in [0, 1] with 0 meaning true. Connectives 0, 1, Φ/2, Φ −

· Ψ.

Quantifiers sup, inf. N | = Φ means the sentence Φ has value 0 in N.

• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 4 / 1

• 2. Continuous Structures

Structures live on complete metric spaces. Prestructures live on pseudometric spaces. Completion of a prestructure: A structure formed by identifying elements at distance 0 and completing the metric. Formulas take values in [0, 1] with 0 meaning true. Connectives 0, 1, Φ/2, Φ −

· Ψ.

Quantifiers sup, inf. N | = Φ means the sentence Φ has value 0 in N. Th(N) = {Φ : N | = Φ}, N ≡ N ′ means Th(N) = Th(N ′).

• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 4 / 1

• 2. Continuous Structures

Structures live on complete metric spaces. Prestructures live on pseudometric spaces. Completion of a prestructure: A structure formed by identifying elements at distance 0 and completing the metric. Formulas take values in [0, 1] with 0 meaning true. Connectives 0, 1, Φ/2, Φ −

· Ψ.

Quantifiers sup, inf. N | = Φ means the sentence Φ has value 0 in N. Th(N) = {Φ : N | = Φ}, N ≡ N ′ means Th(N) = Th(N ′). Every prestructure is ≡ its completion.

• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 4 / 1

• 2. Continuous Structures

Structures live on complete metric spaces. Prestructures live on pseudometric spaces. Completion of a prestructure: A structure formed by identifying elements at distance 0 and completing the metric. Formulas take values in [0, 1] with 0 meaning true. Connectives 0, 1, Φ/2, Φ −

· Ψ.

Quantifiers sup, inf. N | = Φ means the sentence Φ has value 0 in N. Th(N) = {Φ : N | = Φ}, N ≡ N ′ means Th(N) = Th(N ′). Every prestructure is ≡ its completion. N is separable if its completion has a countable dense subset.

• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 4 / 1

• 2. Continuous Structures

Structures live on complete metric spaces. Prestructures live on pseudometric spaces. Completion of a prestructure: A structure formed by identifying elements at distance 0 and completing the metric. Formulas take values in [0, 1] with 0 meaning true. Connectives 0, 1, Φ/2, Φ −

· Ψ.

Quantifiers sup, inf. N | = Φ means the sentence Φ has value 0 in N. Th(N) = {Φ : N | = Φ}, N ≡ N ′ means Th(N) = Th(N ′). Every prestructure is ≡ its completion. N is separable if its completion has a countable dense subset. By a model of U we mean a prestructure N | = U.

• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 4 / 1

• 3. Randomization of a Theory

Let T be a complete first order theory with T | = ∃x∃y(x = y). Let L be the signature of T. LR is the continuous signature with:

• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 5 / 1

• 3. Randomization of a Theory

Let T be a complete first order theory with T | = ∃x∃y(x = y). Let L be the signature of T. LR is the continuous signature with: Two sorts, K for random elements and E for events.

• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 5 / 1

• 3. Randomization of a Theory

Let T be a complete first order theory with T | = ∃x∃y(x = y). Let L be the signature of T. LR is the continuous signature with: Two sorts, K for random elements and E for events. The Boolean operations ⊤, ⊥, ⊓, ⊔, ¬ of sort E.

• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 5 / 1

• 3. Randomization of a Theory

Let T be a complete first order theory with T | = ∃x∃y(x = y). Let L be the signature of T. LR is the continuous signature with: Two sorts, K for random elements and E for events. The Boolean operations ⊤, ⊥, ⊓, ⊔, ¬ of sort E. A unary relation µ of sort E for the probability of an event.

• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 5 / 1

• 3. Randomization of a Theory

Let T be a complete first order theory with T | = ∃x∃y(x = y). Let L be the signature of T. LR is the continuous signature with: Two sorts, K for random elements and E for events. The Boolean operations ⊤, ⊥, ⊓, ⊔, ¬ of sort E. A unary relation µ of sort E for the probability of an event. For each first order formula ϕ(·) with n free variables, an n-ary function [ [ϕ(·)] ] of sort Kn → E.

• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 5 / 1

• 3. Randomization of a Theory

Let T be a complete first order theory with T | = ∃x∃y(x = y). Let L be the signature of T. LR is the continuous signature with: Two sorts, K for random elements and E for events. The Boolean operations ⊤, ⊥, ⊓, ⊔, ¬ of sort E. A unary relation µ of sort E for the probability of an event. For each first order formula ϕ(·) with n free variables, an n-ary function [ [ϕ(·)] ] of sort Kn → E. Distance relations dK for sort K and dE for sort E.

• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 5 / 1

• 3. Randomization of a Theory

Let T be a complete first order theory with T | = ∃x∃y(x = y). Let L be the signature of T. LR is the continuous signature with: Two sorts, K for random elements and E for events. The Boolean operations ⊤, ⊥, ⊓, ⊔, ¬ of sort E. A unary relation µ of sort E for the probability of an event. For each first order formula ϕ(·) with n free variables, an n-ary function [ [ϕ(·)] ] of sort Kn → E. Distance relations dK for sort K and dE for sort E. The randomization of T is the continuous theory T R with signature LR and the axioms on the next page.

• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 5 / 1

• 4. Axioms for T R

For readability, use ∀, ∃, . =, = for inf, sup, dE, −

· .

Transfer: [ [ϕ] ] . = ⊤ where ϕ ∈ T. Validity: ∀ x([ [ψ( x)] ] . = ⊤) where ∀ x ψ( x) is logically valid in first order logic. Event: ∀V ∃x∃y(V . = [ [x = y] ]) Fullness: ∀ y∃x([ [ϕ(x, y)] ] . = [ [(∃xϕ)( y)] ])

• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 6 / 1

• 4. Axioms for T R

For readability, use ∀, ∃, . =, = for inf, sup, dE, −

· .

Transfer: [ [ϕ] ] . = ⊤ where ϕ ∈ T. Validity: ∀ x([ [ψ( x)] ] . = ⊤) where ∀ x ψ( x) is logically valid in first order logic. Event: ∀V ∃x∃y(V . = [ [x = y] ]) Fullness: ∀ y∃x([ [ϕ(x, y)] ] . = [ [(∃xϕ)( y)] ]) The usual Boolean algebra axioms in sort E. Measure: µ[⊤] = 1, µ[⊥] = 0, ∀V ∀W (µ(V ) + µ(W ) = µ(V ⊔ W ) + µ(V ⊓ W )) Atomless: ∀V ∃W (µ(V ⊓ W ) = µ(V )/2)

• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 6 / 1

• 4. Axioms for T R

For readability, use ∀, ∃, . =, = for inf, sup, dE, −

· .

Transfer: [ [ϕ] ] . = ⊤ where ϕ ∈ T. Validity: ∀ x([ [ψ( x)] ] . = ⊤) where ∀ x ψ( x) is logically valid in first order logic. Event: ∀V ∃x∃y(V . = [ [x = y] ]) Fullness: ∀ y∃x([ [ϕ(x, y)] ] . = [ [(∃xϕ)( y)] ]) The usual Boolean algebra axioms in sort E. Measure: µ[⊤] = 1, µ[⊥] = 0, ∀V ∀W (µ(V ) + µ(W ) = µ(V ⊔ W ) + µ(V ⊓ W )) Atomless: ∀V ∃W (µ(V ⊓ W ) = µ(V )/2) Distance: ∀x∀y dK(x, y) = µ[ [x = y] ], ∀V ∀W dE(V , W ) = µ(V ∆W )

• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 6 / 1

• 4. Axioms for T R

For readability, use ∀, ∃, . =, = for inf, sup, dE, −

· .

Transfer: [ [ϕ] ] . = ⊤ where ϕ ∈ T. Validity: ∀ x([ [ψ( x)] ] . = ⊤) where ∀ x ψ( x) is logically valid in first order logic. Event: ∀V ∃x∃y(V . = [ [x = y] ]) Fullness: ∀ y∃x([ [ϕ(x, y)] ] . = [ [(∃xϕ)( y)] ]) The usual Boolean algebra axioms in sort E. Measure: µ[⊤] = 1, µ[⊥] = 0, ∀V ∀W (µ(V ) + µ(W ) = µ(V ⊔ W ) + µ(V ⊓ W )) Atomless: ∀V ∃W (µ(V ⊓ W ) = µ(V )/2) Distance: ∀x∀y dK(x, y) = µ[ [x = y] ], ∀V ∀W dE(V , W ) = µ(V ∆W ) Fact: T R is complete.

• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 6 / 1

• 5. Borel Randomization of a Model

By countable we will mean of power at most ℵ0. The Borel randomization of a model M of T is the continuous prestructure MR = (M[0,1), E) with signature LR where:

• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 7 / 1

• 5. Borel Randomization of a Model

By countable we will mean of power at most ℵ0. The Borel randomization of a model M of T is the continuous prestructure MR = (M[0,1), E) with signature LR where: ([0, 1), E, µ) is the Lebesgue probability space on [0, 1) .

• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 7 / 1

• 5. Borel Randomization of a Model

By countable we will mean of power at most ℵ0. The Borel randomization of a model M of T is the continuous prestructure MR = (M[0,1), E) with signature LR where: ([0, 1), E, µ) is the Lebesgue probability space on [0, 1) . M[0,1) is the set of f : [0, 1) → M with countable range such that f −1(a) ∈ E for each a ∈ M.

• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 7 / 1

• 5. Borel Randomization of a Model

By countable we will mean of power at most ℵ0. The Borel randomization of a model M of T is the continuous prestructure MR = (M[0,1), E) with signature LR where: ([0, 1), E, µ) is the Lebesgue probability space on [0, 1) . M[0,1) is the set of f : [0, 1) → M with countable range such that f −1(a) ∈ E for each a ∈ M. For each formula ψ( x) of L and tuple f in M[0,1), [ [ψ( f )] ] := {t ∈ [0, 1) : M | = ψ( f (t))} (which is in E).

• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 7 / 1

• 5. Borel Randomization of a Model

By countable we will mean of power at most ℵ0. The Borel randomization of a model M of T is the continuous prestructure MR = (M[0,1), E) with signature LR where: ([0, 1), E, µ) is the Lebesgue probability space on [0, 1) . M[0,1) is the set of f : [0, 1) → M with countable range such that f −1(a) ∈ E for each a ∈ M. For each formula ψ( x) of L and tuple f in M[0,1), [ [ψ( f )] ] := {t ∈ [0, 1) : M | = ψ( f (t))} (which is in E). dK(f , g) := µ([ [f = g] ]) and dE(A, B) := µ(A△B).

• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 7 / 1

• 5. Borel Randomization of a Model

By countable we will mean of power at most ℵ0. The Borel randomization of a model M of T is the continuous prestructure MR = (M[0,1), E) with signature LR where: ([0, 1), E, µ) is the Lebesgue probability space on [0, 1) . M[0,1) is the set of f : [0, 1) → M with countable range such that f −1(a) ∈ E for each a ∈ M. For each formula ψ( x) of L and tuple f in M[0,1), [ [ψ( f )] ] := {t ∈ [0, 1) : M | = ψ( f (t))} (which is in E). dK(f , g) := µ([ [f = g] ]) and dE(A, B) := µ(A△B). Facts: M is countable iff MR is separable. M is a model of T iff MR is a model of T R.

• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 7 / 1

• 6. Density Functions

Let M be a big model of T and I(T) be the set of isomorphism types of countable elementary submodels of M.

• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 8 / 1

• 6. Density Functions

Let M be a big model of T and I(T) be the set of isomorphism types of countable elementary submodels of M. p : I(T) → [0, 1] is a density function for T if

i∈I(T) p(i) = 1.

• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 8 / 1

• 6. Density Functions

Let M be a big model of T and I(T) be the set of isomorphism types of countable elementary submodels of M. p : I(T) → [0, 1] is a density function for T if

i∈I(T) p(i) = 1.

Lemma

(Andrews and HJK) Suppose that For each i ∈ I(T), Mi ∈ i. Bi, i ∈ I is a partition of [0, 1) into Borel sets. The function p(i) = µ(Bi) is a density function for T. K is the set of all f ∈ M[0,1) such that (∀i ∈ I(T))f (Bi) ⊆ Mi. Then (K, E) is a separable model of T R.

• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 8 / 1

• 6. Density Functions

Let M be a big model of T and I(T) be the set of isomorphism types of countable elementary submodels of M. p : I(T) → [0, 1] is a density function for T if

i∈I(T) p(i) = 1.

Lemma

(Andrews and HJK) Suppose that For each i ∈ I(T), Mi ∈ i. Bi, i ∈ I is a partition of [0, 1) into Borel sets. The function p(i) = µ(Bi) is a density function for T. K is the set of all f ∈ M[0,1) such that (∀i ∈ I(T))f (Bi) ⊆ Mi. Then (K, E) is a separable model of T R. If N is isomorphic to (K, E) above, say p is a density function for N.

• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 8 / 1

• 7. Few Separable Models

Theorem

(Andrews and HJK) (i) A separable model of T R has at most one density function. (ii) Any two separable models of T R with the same density function are isomorphic.

• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 9 / 1

• 7. Few Separable Models

Theorem

(Andrews and HJK) (i) A separable model of T R has at most one density function. (ii) Any two separable models of T R with the same density function are isomorphic. We say that T R has few separable models if every separable model of T R has a density function.

Theorem

(Andrews and HJK) If I(T) is countable, then T R has few separable

• models. So the isomorphism types of separable models of T R are fully

characterized by their density functions.

• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 9 / 1

• 7. Few Separable Models

Theorem

(Andrews and HJK) (i) A separable model of T R has at most one density function. (ii) Any two separable models of T R with the same density function are isomorphic. We say that T R has few separable models if every separable model of T R has a density function.

Theorem

(Andrews and HJK) If I(T) is countable, then T R has few separable

• models. So the isomorphism types of separable models of T R are fully

characterized by their density functions. Question: If T R has few separable models, what can one say about T?

• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 9 / 1

• 8. Scattered Theories and Vaught’s Conjecture

Theorem

(Scott 1965) Each isomorphism type i ∈ I(T) is definable by a sentence in the infinitary logic Lω1ω.

• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 10 / 1

• 8. Scattered Theories and Vaught’s Conjecture

Theorem

(Scott 1965) Each isomorphism type i ∈ I(T) is definable by a sentence in the infinitary logic Lω1ω.

Definition

T is scattered if for each countable ordinal α, only countably many i ∈ I(T) are definable by sentences in Lω1ω of quantifier rank < α.

• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 10 / 1

• 8. Scattered Theories and Vaught’s Conjecture

Theorem

(Scott 1965) Each isomorphism type i ∈ I(T) is definable by a sentence in the infinitary logic Lω1ω.

Definition

T is scattered if for each countable ordinal α, only countably many i ∈ I(T) are definable by sentences in Lω1ω of quantifier rank < α.

Theorem

(Morley 1970) If T is scattered then |I(T)| ≤ ℵ1. If T is not scattered then |I(T)| = 2ℵ0.

• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 10 / 1

• 8. Scattered Theories and Vaught’s Conjecture

Theorem

(Scott 1965) Each isomorphism type i ∈ I(T) is definable by a sentence in the infinitary logic Lω1ω.

Definition

T is scattered if for each countable ordinal α, only countably many i ∈ I(T) are definable by sentences in Lω1ω of quantifier rank < α.

Theorem

(Morley 1970) If T is scattered then |I(T)| ≤ ℵ1. If T is not scattered then |I(T)| = 2ℵ0. Vaught’s conjecture (1961): |I(T)| < 2ℵ0 implies |I(T)| ≤ ℵ0. Absolute VC (Steel 1977): T is scattered implies |I(T)| ≤ ℵ0.

• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 10 / 1

• 9. Scattered versus Few Separable Models

Theorem

If T R has few separable models, then T is scattered.

• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 11 / 1

• 9. Scattered versus Few Separable Models

Theorem

If T R has few separable models, then T is scattered.

Corollary

Assume the absolute Vaught conjecture. For every T, the following are equivalent: T is scattered; T R has few separable models; I(T) is countable.

• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 11 / 1

• 9. Scattered versus Few Separable Models

Theorem

If T R has few separable models, then T is scattered.

Corollary

Assume the absolute Vaught conjecture. For every T, the following are equivalent: T is scattered; T R has few separable models; I(T) is countable.

Theorem

Assume Martin’s axiom for ℵ1. For every T, the following are equivalent: T is scattered; T R has few separable models.

• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 11 / 1

• 9. Scattered versus Few Separable Models

Theorem

If T R has few separable models, then T is scattered.

Corollary

Assume the absolute Vaught conjecture. For every T, the following are equivalent: T is scattered; T R has few separable models; I(T) is countable.

Theorem

Assume Martin’s axiom for ℵ1. For every T, the following are equivalent: T is scattered; T R has few separable models. Question: Can the above equivalence be proved in ZFC?

• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 11 / 1

Some References

1 I. Ben Yaacov, A. Berenstein, C. W. Henson, A. Usvyatsov. Model

Theory for Metric Structures. Cambridge U. Press (2008), 315–427.

2 I. Ben Yaacov and H. J. Keisler. Randomizations as Metric

Structures, Confluentes Mathematici 1 (2009), 197-223.

3 U. Andrews and H. J. Keisler. Separable Models of Randomizations.

To appear, JSL.

4 R. L. Vaught. Denumerable Models of Complete Theories. In

Infinitistic Methods, Pergamon Press (1961), 303-321.

5 D. Scott. Logic with Denumerably Long Formulas and Finite Strings

• f Quantifiers. In Theory of Models, North-Holland (1965), 329-341.

6 M. Morley. The Number of Countable Models. JSL 25 (1970), 14-18. 7 J. Steel. On Vaught’s Conjecture. Cabal Seminar 76–77, 193–208. 8 J. Baldwin, S. Friedman, M. Koerwien, and M. Laskowski. Three Red

Herrings around Vaught’s Conjecture. To appear, TAMS.

• H. Jerome Keisler

Randomizations of Scattered Theories January 9, 2015 12 / 1