Dorottya Sziraki

Model Theoretic Spectrum Functions

and Algebraic Logic

M. Sc. Thesis

Lorand Eotvos University

Budapest, 2012

Lorand Eotvos University

Faculty of Science

M. Sc. Thesis

Model Theoretic Spectrum Functions

and Algebraic Logic

Author:

Dorottya Sziraki

M. Sc. Pure Mathematics

Supervisor:

Gabor Sagi

MTA Renyi Institute

Examination Committee:

Committee Chairman: Prof. Dr. Laszlo Simon . . . . . . . . . . . . . . . . . . . . . . . .

Committee Members: Dr. Balazs Csikos . . . . . . . . . . . . . . . . . . . . . . . .

Dr. Alice Fialowski . . . . . . . . . . . . . . . . . . . . . . . .

Prof. Dr. Tibor Jordan . . . . . . . . . . . . . . . . . . . . . . . .

Dr. Tamas Mori . . . . . . . . . . . . . . . . . . . . . . . .

Dr. Peter Sziklai . . . . . . . . . . . . . . . . . . . . . . . .

Dr. Janos Toth . . . . . . . . . . . . . . . . . . . . . . . .

Acknowledgements

I wish to express my heartfelt gratitude to my supervisor, Gabor Sagi, for all his

help, guidance and encouragement. I am deeply indebted to my parents for their

loving support and patience. Especially, I would like to give my loving thanks to

Gergo Nemes whose love, support and encouragement enabled me to write this

thesis.

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Table of Contents

Acknowledgements i

Table of Contents ii

1 Introduction 1

1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Preliminaries from Algebraic Logic 4

2.1 Algebraizing First Order Logic With Cylindric Algebras . . . . . . . 4

2.2 Dimension Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 Generalized Cylindrifications and Substitutions . . . . . . . . . . . 8

2.4 Algebraizing Logic Without Equality with Quasi-Polyadic Algebras 10

3 Representations of Certain Cylindric and

Quasi-Polyadic Algebras 15

3.1 Constructing Representations . . . . . . . . . . . . . . . . . . . . . 15

3.1.1 The Equality-Free Case . . . . . . . . . . . . . . . . . . . . . 15

3.1.2 The Case with Equality . . . . . . . . . . . . . . . . . . . . 18

3.2 On Non-Isomorphic Representations . . . . . . . . . . . . . . . . . . 19

3.2.1 The Equality-Free Case . . . . . . . . . . . . . . . . . . . . . 20

3.2.2 The Case with Equality . . . . . . . . . . . . . . . . . . . . 22

4 A New Proof for Morley’s Original Theorem 24

4.1 A Variant about Elementary Bi-Embeddability . . . . . . . . . . . . 25

5 Some Examples of Models 27

5.1 Elementary Bi-Embeddability and Isomorphism . . . . . . . . . . . 27

5.2 Reversibility of Elementary Embeddings . . . . . . . . . . . . . . . 30

5.2.1 Basic Definitions and Lemmas . . . . . . . . . . . . . . . . . 30

5.2.2 Our Counterexample . . . . . . . . . . . . . . . . . . . . . . 32

5.2.3 Our Counterexample in the Equality-Free Case . . . . . . . 36

References 38

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1 Introduction

Let Σ be a complete first order theory in a countable language (with equality). For

any cardinality κ, I (Σ, κ) denotes the number of pairwise non-isomorphic models

of Σ. It is evident that for a theory Σ in a countable language, I (Σ,ℵ0) ≤ 2ℵ0 ,

and some well known examples show that I (Σ,ℵ0) can be 2ℵ0 or any countable

cardinality other than 2, (which, according to a theorem of Vaught’s, is not a

possible value for I (Σ,ℵ0)). Vaught also conjectured that we cannot have ℵ0 <

I (Σ,ℵ0) < 2ℵ0 , i.e., if Σ is a complete theory in a countable language, and, counting

up to isomorphism, Σ has more than ℵ0 models, then Σ has 2ℵ0 many models.

This conjecture was first published in 1961, in [21]. Since then, it has become an

important open problem, and has been mentioned in practically all monographs

on model theory. See for example Hodges [12], on page 339, (and Shelah [19],

Chang-Keisler [6], Buechler [4], and Marker [14]).

Vaught’s conjecture has been researched intensively. For completeness, we

recall some special cases of the conjecture which have been answered affirmatively.

• Bouscaren and Lascar proved in [2] that the conjecture is true for ω-stable

Σ of finite Morley rank.

• Shelah proved in [18] that the conjecture is true for ω-stable Σ.

• Buechler proved in [3] that the conjecture is true for superstable Σ of finite

U -rank.

The best general result is Morley’s theorem, which states that if Σ is a complete

theory in a countable language (with equality), then

I (Σ,ℵ0) > ℵ1, implies I (Σ,ℵ0) = 2ℵ0 .

This theorem was published in [15]; (see also Theorem 4.4.16 of [14]). The original

proof uses infinitary logic and some descriptive set theoretical results.

In this thesis, we give a new proof of Morley’s theorem, which is based on

algebraic logical representation theory and descriptive set theory. An advantage

of our method is that it can be generalized to give two interesting variants of Mor-

ley’s theorem. Firstly, we will be able to obtain the corresponding statement for

equality-free languages. Secondly, following the terminology of [8], let us call two

models M and N elementarily bi-embeddable iff M is elementarily embeddable

into N and N is elementarily embeddable into M. We will prove that the number

of pairwise non elementarily bi-embeddable models of a theory cannot be greater

than ℵ1 but less than 2ℵ0 . Notice that elementary bi-embeddability is an equiv-

alence relation on the models of Σ of cardinality κ. Denoting the number of its

1

equivalence classes by I ′(Σ, κ), our main results are the following.

Theorems 4.4 and 4.5.

1. (Morley) If Σ is a theory in a language with equality, then I (Σ,ℵ0) > ℵ1

implies I (Σ,ℵ0) = 2ℵ0.

2. If Σ is a theory in a language without equality, then I (Σ,ℵ0) > ℵ1 implies

I (Σ,ℵ0) = 2ℵ0.

3. If Σ is a theory in a language with or without equality, then I ′(Σ,ℵ0) > ℵ1

implies I ′(Σ,ℵ0) = 2ℵ0.

Note that we do not assume the completeness of the theory Σ, because we will

not need it in our proofs.

The rest of this thesis is organized as follows. At the end of the Introduction,

we summarize our system of notation and give the definitions of some basic log-

ical concepts, to avoid ambiguity. In Section 2, we recall some basic properties

of cylindric and quasi-polyadic algebras, and study their roles in algebraizing first

order logic (with and without equality, respectively). In Section 3, we deal with

representations of cylindric and quasi-polyadic algebras, and characterize elemen-

tary embeddability and isomorphism in terms of algebraic logic, in Theorems 3.9

and 3.11. With the help of these characterizations, we prove our main results in

Section 4. In Section 5, we give a few examples of models which are related to our

results.

1.1 Notation

Our system of notation is mostly standard, but the following list may be useful.

Throughout, ω denotes the set of natural numbers, and for every n ∈ ω we have

n = {0, 1, . . . , n− 1}. Let A and B be sets. Then AB denotes the set of functions

whose domain is A and whose range is a subset of B. In addition, |A| denotes the

cardinality of A. If κ is a cardinal, then [A]κ denotes the set of subsets of A which

are of cardinality κ, and P(A) denotes the power set of A, that is, the set which

consists of all the subsets of A. For any distinct elements i, j from a given set U ,

[i/j] ∈ UU is the function on U which maps i to j and leaves every other element

fixed. In addition, IdU denotes the identity function on U . Throughout, we use

function composition in such a way that the rightmost factor acts first. That is,

for functions f, g we define f ◦ g(x) = f(g(x)). If f : A −→ B is a function and

X ⊆ A, then f ∗(X) = {f(x) : x ∈ X}. Moreover, f−1 : P(B) −→ P(A) acts

between the powersets.

Throughout Sections 2 to 4, we assume that a first order language L (with

or without equality) contains only relation symbols. We denote the sequence of

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variables of L by 〈vi : i ∈ ω〉 and the set of formulas of L by Form(L). By a theory

in L, or an L-theory, we mean an arbitrary subset of Form(L). B 4 A denotes

the fact that B is an elementary substructure of A, and f : B 4 A denotes that f

is an elementary embedding of B into A. Given a theory Σ in a language L and

a formula ϕ of L, we take Σ |= ϕ to mean that for all models M of Σ we have

A |= ϕ. We define some additional notation at the beginning of Section 5, which

will be used only in that section.

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2 Preliminaries from Algebraic Logic

This section deals with the basic properties of cylindric and quasi-polyadic alge-

bras, and their roles in algebraizing first order logic. The first four subsections

are devoted to cylindric algebras. The definitions and propositions of this section,

along with proofs, can be found in either [10] or [11]. In the last subsection, we

discuss quasi-polyadic algebras. We note that the material of this section and the

next closely follows that of Sections 1 and 2 of [20].

2.1 Algebraizing First Order Logic With Cylindric Alge-

bras

Cylindric algebras emerged in the first half of the twentieth century, due to the

work of Alfred Tarski and his students. Their original intention with this theory

was to provide an algebraic treatment of first order logic (with equality), just as

Boolean algebras do for sentential calculus. In fact, a cylindric algebra is a Boolean

algebra equipped with additional operations. When algebraizing first order logic,

these operations correspond to the quantifiers ∃vi and the formulas vi = vj.

Definition 2.1. Let α be any ordinal. A cylindric algebra of dimension α is an

algebraic structure

A = 〈A,∨,∧,−, 0, 1, ci, di,j〉i,j<α

such that 0, 1 and dij are distinguished elements of A (for all i, j < α), − and ci

are unary operations on A (for all i < α), ∧ and ∨ are binary operations on A,

and for all x, y ∈ A and i, j < α, the following axioms are satisfied:

(C0) 〈A,∨,∧,−, 0, 1〉 is a Boolean algebra,

(C1) ci0 = 0,

(C2) x ≤ cix,

(C3) ci(x ∧ ciy) = cix ∧ ciy,

(C4) cicjx = cjcix,

(C5) dii = 1,

(C6) if i 6= j, k, then djk = ci(dji ∧ dik),

(C7) if i 6= j, then ci(dij ∧ x) ∧ ci(dij ∧ −x) = 0.

CA and CAα denote the class of all cylindric algebras and the class of cylindric alge-

bras of dimension α respectively. The ci’s are called cylindrifications and the dij’s

are called diagonal elements. The Boolean reduct of A is BlA = 〈A,∨,∧,−, 0, 1〉.

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There are two construction methods which lead to important classes of cylindric

algebras, and illustrate the connection between cylindric algebras and logic. We

will refer to them as “algebraizing syntax” and “algebraizing semantics”.

Algebraizing Syntax. Suppose L is a first order language with equality. Two

formulas ϕ and ψ are defined to be equivalent mod Σ, in symbols ϕ ≡Σ ψ, iff

Σ |= ∀(ϕ ⇔ ψ), where ∀(ϕ ⇔ ψ) is the universal closure of ϕ ⇔ ψ. Clearly,

≡Σ is an equivalence relation on Form(L). Consider the following operations on

Form(L)/≡Σ:

(ϕ/≡Σ) · (ψ/≡Σ) = (ϕ ∧ ψ)/≡Σ, −(ϕ/≡Σ) = (¬ϕ)/≡Σ,

(ϕ/≡Σ) + (ψ/≡Σ) = (ϕ ∨ ψ)/≡Σ, ci(ϕ/≡Σ) = (∃viϕ)/≡Σ . (1)

It is easy to see that the above operations are well defined. We also define certain

distinguished elements of Form(L)/≡Σ as follows. (T and F denote the truth and

falsehood constants, that is, for every model A for L and every valuation σ, we

have A |= T [σ] and A|6= F [σ].)

1 = T/≡Σ, 0 = F/≡Σ, dij = (vi = vj)/≡Σ . (2)

By Theorem 1.1.10 of [10], 〈Form(L)/≡Σ,+, ·,−, 0, 1, ci, dij〉i,j<ω is a cylindric al-

gebra. It is denoted by CA(Σ), and referred to as the cylindric algebra of formulas

(in L) associated with Σ, or simply the cylindric algebra of Σ, or the Lindenbaum

algebra of Σ.

Algebraizing Semantics. Take an arbitrary set U and an ordinal α. Let ∪, ∩

and − denote set theoretical union, intersection and complementation w.r.t. αU,

and define the following operations on and distinguished elements of P(αU), for

all i, j < α :

CiX = {s ∈ αU : (∃u ∈ X)(u|α−i = s|α−i)} for all X ∈ P(αU),

Dij = {s ∈ αU : s(i) = s(j)}. (3)

A ⊆ P(αU) is said to be an α-dimensional cylindric field of sets iff it is closed under

the operations ∪, ∩, − and Ci (for all i < α) and contains the elements ∅, αU and

Dij (for all i, j < α). U is called the base of A. A cylindric set algebra of dimension

α with base U is an algebraic structure A = 〈A,∪,∩,−, ∅,α U,Ci, Dij〉i,j<α, where

A is a cylindric field of sets with base U . If A = P(αU), then A and A are called a

full cylindric field of sets and a full cylindric set algebra respectively. The class of

cylindric set algebras of dimension α is denoted by Csα. All cylindric set algebras

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of dimension α are CAα’s, according to Theorem 1.1.6 of [10].

Now, consider an L-model M = 〈U,R〉R∈L . For each formula ϕ of L, let us

denote by ‖ϕ‖M the “relation” defined by ϕ in M, i.e., ‖ϕ‖M = {s ∈ ωU : M |=

ϕ[s]}. Clearly, for all ϕ, ψ ∈ Form(L) and i, j < ω, the following hold:

‖ϕ ∧ ψ‖M = ‖ϕ‖M ∩ ‖ψ‖M, ‖T‖M = ωU,

‖ϕ ∨ ψ‖M = ‖ϕ‖M ∪ ‖ψ‖M, ‖F‖M = ∅, (4)

‖¬ϕ‖M = −‖ϕ‖M, ‖vi = vj‖M = Dij,

‖∃viϕ‖M = Ci ‖ϕ‖

M.

Consequently, A = {‖ϕ‖M : ϕ ∈ Form(L)} is a cylindric field of sets, and

Csω(M) = 〈A,∪,∩,−, ∅,α U,Ci, Dij〉i,j<α is a cylindric set algebra with base U .

2.2 Dimension Sets

Definition 2.2. Given an element x of an α dimensional cylindric algebra A, we

define the dimension set of x to be

∆x = {i < α : cix 6= x}.

If A is the Lindenbaum algebra of a theory Σ, then ∆ (ϕ/≡Σ) is the set of all

k < α for which Σ |6= ϕ ⇔ ∃vkϕ, or equivalently, Σ |6= ϕ ⇔ ∀vkϕ. Similarly, if

A = Csω(M), then ∆(‖ϕ‖M) is the set of all k < α such that M |6= ϕ⇔ ∃vkϕ.

The proof of the next proposition is routine computation, and is therefore not

included here.

Proposition 2.3. For an arbitrary A ∈ CAα, and x, y ∈ A and i < α, the

following are satisfied:

1. ∆0 = ∆1 = ∅,

2. ∆(−x) = ∆x,

3. ∆(x ∨ y) ⊆ ∆x ∪∆y,

4. ∆(x ∧ y) ⊆ ∆(x) ∪∆y,

5. ∆(cix) ⊆ ∆x−{i},

6. if dij 6= 1, then ∆dij = {i, j}.

Definition 2.4. An α dimensional cylindric algebra A is said to be locally finite

dimensional, in symbols A ∈ Lfα or simply A ∈ Lf, iff every element of A has a

finite dimension set.

Because each ϕ ∈ Form(L) has only finitely many free variables, CA(Σ) ∈ Lfω

and Csω(M) ∈ Lfω for an arbitrary theory Σ and an arbitrary model M.

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Definition 2.5. An element X of an α dimensional cylindric set algebra A is

defined to be regular iff for every s, z ∈ αU we have

s ∈ X and s|∆X = z|∆X imply z ∈ X.

A cylindric set algebra is regular iff all of its elements are regular. The class of α

dimensional regular cylindric set algebras is denoted by CsRegα , and the class of all

regular cylindric set algebras by CsReg .

Not all locally finite dimensional cylindric set algebras are regular. Here is a

well known counterexample.

Proposition 2.6. Consider the following subset of αα:

X = {s ∈ αα : s(i) 6= i only finitely many times}.

The cylindric set algebra A generated by X is locally finite dimensional, but not

regular.

Proof. It is clear that s ∈ X and s|α−{i} = z|α−{i} imply z ∈ X for all i < α.

Therefore ciX = X for all i < α, which means ∆X = ∅. Hence, by 2.3, A is an

Lfα. Suppose A is regular. Then, since ∆X = ∅ and X is regular, we have

(∀s ∈ X)(∀z ∈ αU)(s|∅ = z|∅ ⇒ z ∈ X).

This means, that X = ∅, or X = αU , an obvious contradiction.

If M is a model for L, then Csω(M) is regular, because if ϕ is a formula of L

and s ∈ ωU is a valuation (where U is the universe of M), then whether M |= ϕ[s]

holds or not depends only on the value of s at those k for which M |6= ∀(∃vkϕ ⇔

ϕ), (where ∀(∃vkϕ ⇔ ϕ) is the universal closure of ∃vkϕ ⇔ ϕ). Equivalently,

s ∈ ‖ϕ‖M depends only on s|∆(‖ϕ‖M). It is well known that, besides being locally

finite dimensional and regular, Csω(M) has no further properties. Below, we adapt

a proof of this statement to show that there is a one-one correspondence between

the models of a theory Σ and the homomorphisms from CA(Σ) onto locally finite

dimensional and regular cylindric set algebras.

Proposition 2.7. Suppose A ∈ Csω, and Σ is a theory in a language L (with

equality).

1. If A = Csω(M) for a model M of Σ, then A ∈ Lfω∩CsRegω , and h : CA(Σ) −→

A, ϕ/≡Σ 7→ ‖ϕ‖M is a surjective homomorphism.

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2. Conversely, if A ∈ Lfω∩CsRegω , and h : CA(Σ) −→ A is a surjective homomor-

phism, then there is a model M of Σ such that A = Csω(M). Furthermore,

for every ϕ ∈ Form(L) we have h(ϕ/≡Σ) = ‖ϕ‖M.

Proof. We have already seen that if A = Csω(M) for some model M |= Σ, then

A ∈ Lfω ∩ CsRegω . Suppose h(ϕ/≡Σ) = ‖ϕ‖M for each formula ϕ in L. h is well

defined on CA(Σ), because ϕ/≡Σ= ψ/≡Σ implies Σ |= ϕ ⇔ ψ, which implies

M |= ϕ⇔ ψ, that is ‖ϕ‖M = ‖ψ‖M. Now, it is easy to see that h : CA(Σ) −→ A

is a surjective homomorphism.

To see 2., suppose A ∈ Lfω ∩ CsRegω with base U , and h : CA(Σ) −→ A is a

surjective homomorphism. For each x ∈ A, define the set x′ = {s|∆x : s ∈ x}.

Then x′ is a ∆x-ary relation on U , and M =⟨

U, (h(R/≡Σ))′⟩

R∈Lis a model for

the language L. Furthermore, if R ∈ L and x = h(R/≡Σ), then we have

‖R‖M = {s : M |= R[s]} = {s : s|∆x ∈ x′} = x.

The last equation is true because on one hand, s ∈ x implies s|∆x ∈ x′, by defini-

tion. On the other hand, s|∆x ∈ x′ means that for some z ∈ x we have s|∆x = z|∆x,

and because of regularity, this implies s ∈ x. Thus, h(R/≡Σ) = ‖R‖M holds for all

R ∈ L. Induction on the complexity of formulas of L shows that h(ϕ/≡Σ) = ‖ϕ‖M

for all ϕ ∈ Form(L), and therefore A = Csω(M). Particularly, for all σ ∈ Σ, we

have ‖σ‖M = h(σ/≡Σ) = h(1) = ωU and consequently, M |= Σ.

2.3 Generalized Cylindrifications and Substitutions

Definition 2.8. Suppose A is an α dimensional cylindric algebra, and Γ is a finite

subset of α. The generalized cylindrification correlated to Γ is defined as follows:

for each x ∈ A, let

c(Γ)x =

{

x if Γ = 0,

ci0ci1 . . . cin−1x if Γ = {i0, i1, . . . , in−1}.

By Theorem 1.2.17 of [10], the operations c(Γ) are well defined (i.e., the value

of c(Γ)x does not depend on the order of the cik on the right hand side).In the

case of a CA(Σ) or a Csω(M) (for some theory Σ in L or some model M for L),

the operation c(Γ) takes the equivalence class of a formula ϕ (i.e., either ϕ/≡Σ or

‖ϕ‖M) to the equivalence class of ∃vi0∃vi1 . . . ∃vin−1ϕ.

The proof of the next proposition is a routine computation; hence, we omit it.

Proposition 2.9. Let A be a CAα. For all x, y ∈ A, i < α and Γ,∆ ∈ [α]<ω we

have

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1. c({i})x = cix,

2. c(∅)x = x,

3. c(Γ)0 = 0,

4. x ≤ c(Γ)x,

5. c(Γ)(x ∧ c(Γ)y) = c(Γ)x ∧ c(Γ)y,

6. c(Γ)c(∆)x = c(Γ∪∆)x.

Definition 2.10. If A is a cylindric algebra, then let, all x ∈ A and i, j < α

sijx =

{

x if i = j,

ci(x ∧ dij) if i 6= j.

Suppose A = CA(Σ) or a A = Csω(M), and for a formula ϕ in L, let ϕ(vi/vj)

denote the formula obtained by substituting vj for vi in ϕ (where both vi and vj are

assumed to be unbound variables in ϕ). Then |= ϕ(vi/vj) ⇔ ∃vi(ϕ∧ vi = vj), and

consequently, ϕ(vi/vj)/≡Σ= sij(ϕ/≡Σ) in CA(Σ), and ‖ϕ(vi/vj)‖M = sij(‖ϕ‖

M) in

Csω(M). Because of this, we call sij the j for i substitution, or just substitution.

We note, for later use, that cix = sup{sijx : j ∈ α−∆x} = sup{sijx : j < α}.

The proof of this statement can be found in 1.11.6 of [10], but it will also follow

from Proposition 2.15.

Next, we will recall the definition of a generalization of the operations sij in

Lfα’s where α ≥ ω. First, we will deal with the finite case: operations that

can be correlated not just with a pair of ordinals i, j < α, but with a pair of

finite sequences of ordinals less than α, 〈i0, i1, . . . , in−1〉 and 〈j0, j1, . . . , jn−1〉, or

equivalently, a finite transformation τ of α. By a finite transformation of α, we

mean a τ ∈ αα such that I = {i < α : τ(i) 6= i} is finite. Because τ|α−I = Idα−I ,

τ is uniquely determined by its values on I. Therefore if I = {i0, i1, . . . , in−1} and

τ(ik) = jk (for all k < n), then τ can be represented as [i0/j0, i1/j1, . . . , in−1/jn−1].

If we assume that i0 < i1 < · · · < in−1, then this representation is unique, and is

called the canonical representation of τ . We denote the set of finite transformations

of α by FTα.

Definition 2.11. Let A ∈ Lfα, and suppose τ is a finite transformation of α

with the canonical representation [i0/j0, i1/j1, . . . , in−1/jn−1]. Then the substitu-

tion correlated with τ is defined to be the operation sτ that takes each x ∈ A

to

sτx = sk0j0 sk1j1. . . s

kn−1

jn−1si0k0s

i1k1. . . s

in−1

kn−1x,

where k0, k1, . . . , kn−1 are the first k ordinals in α−(∆x ∪ Range i ∪ Range j).

Since CA(Σ) and Csω(M) are locally finite dimensional, the former definitions

can be applied in their case. Let τ = [i0/j0, i1/j1, . . . , in−1/jn−1] ∈αα. Similarly

9

to the sij case, sτ takes the equivalence class of a formula ϕ (that is either ϕ/≡Σ

or ‖ϕ‖M) to the equivalence class of the formula ψ obtained by simultaneously

substituting vj0 , vj1 , . . . , vjn−1for vi0 , vi1 , . . . , vin−1

in ϕ.

In the case A ∈ Csα ∩ Lfα with base U , SτX = {x ∈ αU : x ◦ τ ∈ X}.

Using the above finitary case, we will recall the generalized substitutions for an

arbitrary τ ∈ αα. This will turn out not to be much of a generalization however, as

the new operations will reduce to the old ones, due to the fact that every element

of A has a finite dimension set.

Definition 2.12. Given an arbitrary A ∈ Lfα and τ ∈ αα, the generalized sub-

stitution correlated with τ is defined to be the operation s+τ that takes each x ∈ A

to

s+τ x = sτ|∆x∪Idα−∆x.

The next proposition can be found as part of Theorem 1.11.14 of [10].

Proposition 2.13. Let A ∈ Lfα. For any x ∈ A, any i, j ∈ α, any Γ ∈ [α]<ω and

any τ, σ ∈ αα, the following statements hold:

1. if τ is finite, then s+τ = sτ ,

2. s+[i/j]x = sijx,

3. s+Idαx = x,

4. s+τ is an endomorphism of BlA,

5. s+σ◦τx = s+σ s+τ x,

6. if σ|α−Γ = τ|α−Γ, then s+σ c(Γ)x = s+τ c(Γ)x,

7. if σ|∆x = τ|∆x, then s+σ x = s+τ x,

8. if τ−1(Γ) = ∆ and τ|∆ is one-one, then c(Γ)s+τ x = s+τ c(∆)x,

9. s+τ dij = dτ(i)τ(j),

2.4 Algebraizing Logic Without Equality with Quasi-Polyadic

Algebras

Now we examine how first order logic without equality can be algebraized similarly

to the case with equality, using quasi-polyadic algebras.

Definition 2.14. Let α be any ordinal. By definition, a quasi-polyadic algebra of

dimension α, is an algebraic structure

A =⟨

A,∨,∧,−, 0, 1, c(Γ), sτ⟩

Γ∈[α]<ω ,τ∈FTα

10

such that 0 and 1 are distinguished elements of A, and −, c(Γ) and sτ are unary

operations on A (for all Γ ∈ [α]<ω and τ ∈ FTα), and ∨ and ∧ are binary opera-

tions on A, and the following axioms are satisfied for all x, y ∈ A, all Γ ∈ [α]<ω

and all τ ∈ FTα:

(Q0) 〈A,∨,∧,−, 0, 1〉 is a Boolean algebra,

(Q1) c(Γ)0 = 0,

(Q2) x ≤ c(Γ)x,

(Q3) c(Γ)(x ∧ c(Γ)y) = c(Γ)x ∧ c(Γ)y,

(Q4) c(∅)x = x,

(Q5) c(Γ)c(J)x = c(Γ∪∆)x,

(Q6) sIdαx = x,

(Q7) sσsτx = sσ◦τx,

(Q8) sτ (x ∧ y) = sτx ∧ sτy,

(Q9) sτ (−x) = −sτx,

(Q10) if σ|α−Γ = τ|α−Γ, then sσc(Γ)x = sτc(Γ)x,

(Q11) if ∆ = τ−1(Γ) and τ|∆ is one-one, then c(Γ)sτx = sτc(∆)x.

QPA and QPAα denote the class of all quasi-polyadic algebras and the class of all

α dimensional quasi-polyadic algebras respectively. The ci’s are called cylindrifi-

cations and the sτ ’s are called substitutions. The Boolean reduct of an A ∈ QPA

is the Boolean algebra BlA = 〈A,∨,∧,−, 0, 1〉.

We note that another axiomatization of quasi-polyadic algebras was provided

by I. Sain and R. Thompson in [16].

Because of Propositions 2.9 and 2.13, “all locally finite dimensional CA’s are

also QPA’s”. By saying this, we mean that the operations c(Γ) and sτ are term

definable in Lf’s so that axioms (Q0) to (Q11) are satisfied.

Observe that by (Q5), the equation c(Γ)x = c({i0}) . . . c({in−1})x is valid in any

QPAα for any finite Γ = {i0, . . . , in−1} ⊆ α. Consequently, the set of operations

of QPA is redundant in the sense that for any finite Γ, the operation c(Γ) is term

definable from {c{i} : i < α}. Hence throughout this work we assume that the set

of basic operations of QPA consists of the Boolean operations, sτ for finite τ ’s and

c({i}) for i ∈ α, only. Furthermore, we will use the notation ci instead of c({i}), and

sij instead of s[i/j].

The concepts of the dimension set ∆x of an element x of a QPA, of locally

finite dimensional quasi-polyadic algebras (LfQPA’s), of regular quasi-polyadic set

algebras (QsReg ’s) and of the generalized substitution operation s+τ correlated with

an arbitrary τ ∈ αα are defined exactly as their cylindric counterparts. The state-

ments made previously in this section can also be carried over to the quasi-polyadic

11

setting (with the obvious exception of the ones concerning dij’s). Specifically, we

have the next two propositions.

Proposition 2.15. Take an ordinal α ≥ ω and a locally finite dimensional A ∈

QPAα. For any x ∈ A we have

cix = sup{sijx : j ∈ α−∆x} = sup{sijx : j < α}.

Proof. Suppose i, j < α and x ∈ A. By taking Γ = {i} and σ = [i/j] and τ = Idα

in (Q10), and using (Q6), we arrive at sijcix = sIdαcix = cix. Using (Q2) and

the fact that sij is a Boolean endomorphism (which follows from (Q8) and (Q9)),

we obtain cix = sijcix ≥ sijx. Therefore cix ≥ sup{sijx : j < α} ≥ sup{sijx : j ∈

α−∆x}.

To see the other direction, suppose y ≥ sijx for all j ∈ ω−∆x and take a k ∈

α−(∆x∪∆y), (such a k exists because ∆x and ∆y are finite and α ≥ ω). Denoting

by s[l,j] = s[l/j,j/l], we have, for this k, sikx = sikckx

(Q10)= s[i,k]ckx = s[i,k]x. Therefore

cksikx = cks[i,k]x

(Q11)= s[i,k]cix = s[i,k]cickx

(Q5),(Q10)= sIdα

cickx = cickx = cix. And

so, y ∨ cix = cky ∨ cksikx = ck(y ∨ s

ikx) = cky = y, that is, y ≥ cix. Consequently,

cix ≤ sup{sijx : j ∈ α−∆x}.

Proposition 2.16. Suppose A ∈ LfQPAα. For any x ∈ A, any i, j ∈ α, any

Γ ∈ [α]<ω and any τ, σ ∈ αα, the following statements hold:

1. if τ is finite, then s+τ = sτ ,

2. s+[i/j]x = sijx,

3. s+Idαx = x,

4. s+τ is an endomorphism of BlA,

5. s+σ◦τx = s+σ s+τ x,

6. if σ|α−Γ = τ|α−Γ, then s+σ c(Γ)x = s+τ c(Γ)x,

7. if σ|∆x = τ|∆x, then s+σ x = s+τ x,

8. if τ−1(Γ) = ∆ and τ|∆ is one-one, then c(Γ)s+τ x = s+τ c(∆)x,

The previous proposition corresponds to Proposition 2.13 and can be proven

similarly. (The proof of 2.13 can be found in [10].)

We algebraize syntax and semantics in the equality free case very similarly

to the case with equality. Note that although CA(Σ)’s and Csω(M)’s are QPA’s,

they were defined only in the case of languages containing the equation symbol.

Therefore we will not use them here.

Let L be a language that does not contain the equality symbol. Let Σ be a

theory in L. We define equivalence mod Σ (the relation ≡Σ) on Form(L) just like

12

when algebraizing the syntax of first order logic with equality, in Subsection 2.1:

ϕ ≡Σ ψ iff Σ |= ϕ ⇔ ψ. Just like before, it will be an equivalence relation.The

distinguished elements 0 and 1 of Form(L)/≡Σ and the operations +, ·, − and

ci are defined exactly as in the definitions in 1 and 2 of Subsection 2.1. For each

τ ∈ FTω and ϕ ∈ Form(L), let

sτ (ϕ/≡Σ) = ψ/≡Σ,

where ψ denotes the formula obtained from ϕ by simultaneously substituting

vτ(i) for vi in the case of all i ∈ ω such that τ(i) 6= i. The operations sτ

are all well defined (as are the other operations). Similarly to proving that

CA(Σ)’s are cylindric set algebras (see Theorem 1.1.10 of [10]), one can prove that

〈Form(L)/≡Σ,+, ·,−, 0, 1, ci, sτ 〉i<ω,τ∈FTωis a QPAω by checking that the postu-

lates (Q0) to (Q11) hold. We will call this algebra the quasi-polyadic algebra of

formulas (in L) associated with Σ and denote it by QPA(Σ).

Let α be any ordinal, and U be any set. Consider the following operations Sτ

on the set P(αU) for all τ ∈ FTα:

SτX = {x ∈ αU : x ◦ τ ∈ X} for all X ∈ P(αU).

Recall the definition of the operations Ci on and the distinguished elements Dij

of P(αU) from 3 of Subsection 2.1. Note that, denoting Sij = S[i/j] we have for all

i, j < α and X ⊆ ωU and τ ∈ FTα,

SijX = Ci(X ∩Dij),

SτDij = Dτ(i)τ(j).

By definition, an α-dimensional quasi-polyadic field of sets with base U is a

subset A of P(αU) that is closed under the operations ∪, ∩, −, Ci (for all i < α),

and Sτ (for all τ ∈ FTα) and contains the elements ∅ and αU . We define a

quasi-polyadic set algebra of dimension α with base U to be an algebraic structure

A = 〈A,∪,∩,−, ∅,α U,Ci, Sτ 〉i<α,τ∈FTα, where A is a quasi-polyadic field of sets

with base U . We will denote the class of quasi-polyadic set algebras of dimension

α by Qsα. It is not difficult to see that the postulates (Q0) to (Q11) are satisfied in

a Qsα, and so all quasi-polyadic set algebras are quasi-polyadic algebras, similarly

to the cylindric case (see Theorem 1.1.6 of [10]). The concepts of a full quasi-

polyadic field of sets and a full quasi-polyadic set algebra are defined just as their

cylindric counterparts.

Suppose A is a quasi-polyadic set algebra. Observe that A is a cylindric set

algebra iff it contains the elements Dij (for all i, j < ω) iff it contains D01. The

13

latter is true because Dij = S[0/i,1/j]D01 for arbitrary i, j < ω, and therefore

D01 ∈ A implies Dij ∈ A for all i, j < ω. For example, a full quasi-polyadic set

algebra is also a (full) cylindric set algebra.

Suppose M = 〈U,R〉R∈L is a model for the language L. Similarly to the

algebraization of the semantics of first order logic with equality, we define ‖ϕ‖M =

{s ∈ ωU : M |= ϕ[s]}. The equations in 4 of Subsection 2.1 will hold in this case

as well, with the obvious exception of the one concerning the formula vi = vj. We

also have

Sτ (‖ϕ‖M) = ‖ψ‖M,

where ψ is the formula obtained from ϕ by simultaneously substituting vτ(i) for vi in

the case of all i such that τ(i) 6= i. Consequently, A = {‖ϕ‖M : ϕ ∈ Form(L)} will

be a quasi-polyadic field of sets, and so, Qs(M) = 〈A,∪,∩,−, ∅,α U,Ci, Sτ 〉i<ω,τ∈FTω

will be a quasi-polyadic set algebra.

Observe that D01 ∈ Qs(M) iff there is a formula ǫ such that ‖ǫ‖M = D01 iff

there is a formula ǫ such that for all valuations s ∈ ωU we have

M |= ǫ[s] ⇔ s(0) = s(1).

Because Dij = S[0/i,1/j]D01 for all i and j less than ω, D01 ∈ Qs(M) implies

that there is formula ǫ such that for all i, j < ω and valuations s we have M |=

ǫ(v0/vi, v1/vj)[s] iff s(i) = s(j).

Just as in the cylindric case, there is a one-one correspondence between the

models of a theory Σ, and the homomorphisms from CA(Σ) onto locally finite

and regular quasi-polyadic algebras, as the next proposition shows. Its proof is a

straightforward adaptation of the proof of Proposition 2.7.

Proposition 2.17. Suppose A ∈ Qsω, and Σ is a theory in a language L without

equality.

1. If A = Qs(M) for a model M of Σ, then A ∈ LfQPAω∩QsRegω . Furthermore,

h : QPA(Σ) −→ A, ϕ/≡Σ 7→ ‖ϕ‖M is a surjective homomorphism.

2. If A ∈ LfQPAω ∩QsRegω and h : QPA(Σ) −→ A is a surjective homomorpsim,

then there is a model M of Σ such that A = Qs(M) and h(ϕ/≡Σ) = ‖ϕ‖M

for every ϕ ∈ Form(L).

14

3 Representations of Certain Cylindric and

Quasi-Polyadic Algebras

Suppose Σ is a theory in a language L either with or without equality. Due to

Propositions 2.7 and 2.17, there is a one-one correspondence between the models

of Σ and the homomorphisms from CA(Σ) onto elements of Lfω ∩CsRegω in the case

with equality, and from QPA(Σ) onto elements of LfQPAω ∩ QsRegω in the equality

free case. Consequently, the problem of finding all the possible countable models

of Σ is equivalent to finding all homomorphisms from either CA(Σ) or QPA(Σ)

onto some locally finite dimensional and regular set algebra with a countable base.

In this section, we will deal with the corresponding special case of the repre-

sentation problem. Representation theory in the case of cylindric algebras deals

with finding homomorphisms from CA’s to cylindric set algebras (or sometimes to

elements of a class that is a generalized version of Cs). For details, see Chapter 3

of [11]. The representation problem in the case of QPA’s is similar.

3.1 Constructing Representations

3.1.1 The Equality-Free Case

A method of constructing homomorphisms from an Lfω with a countable universe

onto some Lfω ∩ CsRegω was introduced by H. Andreka and I. Nemeti in [1]; this

method can also be found in Remark 3.2.9 of [11]. In [17], it is shown that all

such homomorphisms can be obtained as a result of this construction. We adapt

this argument for quasi-polyadic algebras below. To do so, we need some further

preparation.

With each Boolean algebra B, one can associate a topological space B∗ =

〈U(B), T 〉, called its Stone space. Here, U(B) denotes the set of ultrafilters on B,

and T is the topology generated by the basis {Nx : x ∈ B}, where for each x ∈ B,

Nx = {F ∈ U(B) : x ∈ F}.

By a theorem of Stone’s (see Theorem IV.4.6 of [5]), B∗ is a compact Hausdorff

space with a clopen basis. Stone’s representation theorem tells us that B is iso-

morphic to the algebra of the clopen subsets of its Stone space; the isomorphism

takes each x ∈ B to Nx.

Suppose that for some element x and subset Y of B we have x = supY . We

will say an ultrafilter F ∈ U(B) preserves Y iff x ∈ F implies y ∈ F for some

y ∈ Y .

15

Let A be an LfQPAω that has a countable universe.

Definition 3.1.

1. We denote by Ui,x(A) the set of ultrafilters on A that preserve {sijx : j < ω},

that is,

Ui,x(A) = {F ∈ U(A) : cix ∈ F implies sijx ∈ F for some j < ω}

= N−cix ∪⋃

j<ω

Nsijx.

2. Let H(A) denote the set of those ultrafilters of A that preserve {sijx : j < ω}

for all x ∈ A and i < ω, that is

H(A) =⋂

i<ω,x∈A

Ui,x(A).

For completeness, we note that H(A) is nonempty. This follows from the Baire

category theorem and the fact that the for each i, x, the complement of Ui,x(A) is a

nowhere dense subset of A∗. We do not prove this, because it follows immediately

from Lemma 3.4.

Definition 3.2. Take any F ∈ H(A), and for each x ∈ A, let repF take x to the

following subset of ωω:

repF(x) = {τ ∈ ωω : s+τ x ∈ F}.

In the next two lemmas, we show that repF is a homomorphism from A onto

some locally finite dimensional and regular Qsω, and that the converse is also true:

any such homomorphism can be obtained as a result of this construction.

Lemma 3.3. For an arbitrary F ∈ H(A), repF is a homomorphism onto some

LfQPAω ∩QsRegω with base ω. Furthermore, for all F0,F1 ∈ H(A) we have repF0

=

repF1iff F0 = F1.

Proof. In order to prove the proposition, we will first show (i) and (ii) below. (i)

repF(x) is regular for each x ∈ A. (ii) repF preserves −, ∧, ci for all i < ω and sσ

for all finite transformations σ of ω.

• Suppose that for some σ, τ ∈ ωω we have τ ∈ repF(x), and τ|∆x = σ|∆x.

Then by 7. of 2.16, we have s+σ x = s+τ x ∈ F . Consequently, σ ∈ repF(x).

• repF(−x) = −repF(x) because τ ∈ repF(−x) iff F ∋ s+τ (−x)2.16= −s+τ x iff

s+τ x /∈ F iff τ ∈ −repF(x).

16

• τ ∈ repF(x ∧ y) iff F ∋ s+τ (x ∧ y)2.16= s+τ x ∧ s+τ y iff s+τ x ∈ F and s+τ y ∈ F

iff τ ∈ repF(x) ∩ repF(y).

• τ ∈ repFsσx iff F ∋ s+τ sσx2.16= s+τ◦σx iff τ ◦ σ ∈ repF(x) iff τ ∈ {ρ ∈ ωω :

ρ ◦ σ ∈ repF(x)} = SσrepF(x).

• To see that CirepF(x) = repF(cix), first suppose that τ ∈ CirepF(x). This

means that for some σ ∈ ωω, we have σ|ω−{i} = τ|ω−{i} and s+σ x ∈ F . By

4. and 6. of 2.16, this implies that s+σ x ≤ s+σ cix = s+τ cix, and so, s+τ ∈ F .

Hence τ ∈ repF(cix).

For the other direction, suppose τ ∈ repF(cix). First, take a ρ ∈ ωω such

that

ρ(j) =

τ(j) if j ∈ ∆x−{i}

0 if j ∈ ω−(∆x ∪ {i})

l for j = i,

where l ∈ ω−(τ ∗∆x ∪ {0}). We have ρ|∆x−{i} = τ|∆x−{i}, and so, ρ|∆cix =

τ|∆cix, by 5. of Proposition 2.3. Also, τ−1{l} = {i}. Hence, by 7. and 8. of

2.16 we have

s+τ cix = s+ρ cix = cρ(i)s+ρ x = cls

+ρ x.

Since cls+ρ x = s+τ cix ∈ F and F ∈ H(A), there is a k ∈ ω such that

slks+ρ x ∈ F , that is [l/k] ◦ ρ ∈ repF(x).

Let σ = ([l/k] ◦ ρ)|∆x ∪ Idω−∆x. Notice that ∆repF(x) ⊆ ∆x. (This is

because repF(x) ⊆ CirepF(x) ⊆ repF(cix) for all i ∈ ω. So, if i /∈ ∆(x), then

it follows that repF(x) = CirepF(x).) Therefore, since repF(x) is regular,

σ ∈ repF(x). Hence, we have

CirepF(x) ∋ σ|ω−{i} ∪ Id{i} = τ|∆x−{i} ∪ Id (ω−∆x)∪{i}.

Since repF(x) is regular, so is CirepF(x), and so τ ∈ CirepF(x).

Consequently, repF is a homomorphism from A onto some quasi-polyadic set al-

gebra B with base ω. B is locally finite dimensional because A is, and is regular,

because each of its elements are.

To prove the second statement, suppose that F0 6= F1. Then (taking−x instead

of x if necessary) there is an x ∈ A such that x ∈ F0 and x /∈ F1. Consequently,

Idω ∈ repF0(x)−repF1

(x), implying that repF06= repF1

.

Next, we show that every homomorphism fromA onto a locally finite QsRegω with

base ω can be obtained as a result of the previous construction.

17

Lemma 3.4. Let B ∈ LfQPAω ∩ QsRegω with base ω. For any homomorphism h

from A onto B, there is an F ∈ H(A) such that h = repF .

Proof. Consider the following subset of A:

F = {x ∈ A : Idω ∈ h(x)}.

It is not hard to check that F is an ultrafilter. We will show that F ∈ H(A), and

that repF = h.

Let x ∈ A and i ∈ ω. Suppose we have cix ∈ F . This means that Idω ∈

h(cix) = Cih(x). Therefore [i/k] ∈ h(x) for some k ∈ ω, implying that

Idω ∈ {σ : σ ◦ [i/k] ∈ h(x)} = Sikh(x) = h(sikx)

and so sikx ∈ F . Consequently, F ∈ Ui,x(A). Because the above x and i were

chosen arbitrarily, F ∈ H(A).

For any x ∈ A and τ ∈ ωω, we have

τ ∈ repF(x) iff s+τ x ∈ F iff Idω ∈ h(s+τ x) = S+

τ h(x)

iff Idω ◦ τ ∈ h(x) iff τ ∈ h(x).

This means that h = repF , as desired.

3.1.2 The Case with Equality

For the rest of this subsection, let A be a locally finite cylindric algebra with a

countable universe. Suppose F ∈ H(A). Since A is also a quasi-polyadic algebra

(in the sense described after Definition 2.14), repF : x 7→ {τ : s+τ x ∈ F} will be

a quasi-polyadic homomorphism onto some locally finite B ∈ QsRegω with base ω,

by Lemma 3.3. Below, we show that if F satisfies an additional condition, namely

that dij ∈ F iff i = j (iff dij = 1), then repF will be a cylindric homomorphism

(and B will be a cylindric set algebra). Let H′(A) be the set of such ultrafilters:

Definition 3.5.

H′(A) = H(A)−⋃

i 6=j∈ω

Ndij = H(A) ∩⋂

i 6=j∈ω

N−dij .

The following lemmas can be considered cylindric algebraic analogs of Lemmas

3.3 and 3.4.

Lemma 3.6. If F ∈ H′(A), then repF is a homomorphism from A onto some

Lfω ∩ CsRegω with base ω. In addition, repF0

= repF1iff F0 = F1 for all F0,F1 ∈

H′(A).

18

Proof. Due to Lemma 3.3, all that remains to be seen is that repF preserves dij

for all i, j ∈ ω. This is so, because

τ ∈ repF(dij) iff F ∋ s+τ dij2.13= dτ(i)τ(j)

iff τ(i) = τ(j)

iff τ ∈ Dij.

Hence Dij ∈ B for all i, j < ω, meaning that B is also a – regular and locally

finite dimensional – cylindric set algebra. The second statement also follows from

Lemma 3.3.

Lemma 3.7. Suppose B ∈ Lfω ∩ CsRegω with base ω, and suppose h is a homomor-

phism from A to B. Then there is an F ∈ H′(A) such that h = repF .

Proof. As we have seen in the proof of Lemma 3.4, for

F = {x ∈ A : Idω ∈ h(x)},

we have F ∈ H(A) and h = repF .

Take any i 6= j ∈ ω. Then we have Idω /∈ Dij = h(dij). Therefore dij /∈ F , or

equivalently, F /∈ Ndij . Because i and j were chosen arbitrarily, F is in H′(A).

Remark 3.8. The construction of homomorphisms from an A ∈ Lfω with a count-

able universe onto Lfω ∩ CsRegω ’s described in [17] is not exactly the same as our

construction. There, H(A) is defined as the set of those ultrafilters that preserve

{sijx : j ∈ ω −∆x} for all x ∈ A and i ∈ ω (instead of preserving {sijx : j ∈ ω}).

Given an arbitrary F ∈ H(A), an equivalence relation E = {〈i, j〉 : dij ∈ F} is

defined in [17] and is called the kernel of F . Using this E, repF(x) is defined there

as

repF(x) = {τ/E : s+τ x ∈ F}.

It is plain to see that if we ignore the difference in the definitions of H(A), then

H′(A) is the set of ultrafilters of H(A) whose kernel is the identity relation, and

that for these ultrafilters, the definition of repF in [17] coincides with our definition.

3.2 On Non-Isomorphic Representations

When proving Morley’s theorem, we are interested in the countable models of a

consistent theory Σ only up to isomorphism; that is, we do not want to distinguish

between two isomorphic models – or their corresponding ultrafilters. In our variant,

we don’t want to distinguish between two models if they are elementarily bi-

embeddable. Our goal therefore is to characterize the ultrafilters that lead to

19

isomorphic models (resp. elementarily bi-embeddable models); this will be done in

Theorem 3.9 in the equality-free case, and Theorem 3.11 in the case with equality.

3.2.1 The Equality-Free Case

Suppose A = QPA(Σ) for some consistent theory Σ in a countable language L

without equality. Note that QPA(Σ) is locally finite and has a countable universe.

Let

KΣ = {M |= Σ : the universe of M is ω}.

Due to Proposition 2.17 and Lemmas 3.3 and 3.4, there is a one-one correspondence

between the models in KΣ and the ultrafilters in F ∈ H(A). More specifically, if

M ∈ KΣ, then there is an F ∈ H(A) such that repF is a homomorphism from

A onto Qs(M). Conversely, if F ∈ H(A), then repF is a homomorphism from A

onto some B ∈ LfQPAω ∩ QsRegω with base ω, and so B = Qs(M) for some model

M of Σ with universe ω. Note that in both cases, we have repF(ϕ/≡Σ) = ‖ϕ‖M.

Consequently, instead of the models of Σ with base ω, we can work with H(A).

Now, we will determine which of its elements lead to “the same” models.

Suppose ρ is a bijection between the sets U and W . For every X ⊆ αU let

f(X) = {ρ ◦ σ : σ ∈ X}.

If B ∈ Qsα with base U , then f|B is clearly an isomorphim onto some Qsα. Iso-

morphisms of this form are called base isomorphisms.

Suppose B and C are α dimensional quasi-polyadic set algebras with bases U

and V , and U ⊇ V . Suppose Φ is an isomorphism between them. If for every

X ∈ B we have

Φ(X) = X ∩ αV

then Φ is called an ext.-isomorphism. We call a function h = f ◦ Φ an ext.-base-

isomorphism iff f is a base isomorphism, and Φ is an ext.-isomorphism. (We note

that a slightly different, although equivalent definition of ext.-base-isomorphisms

is given in 3.1.41 of [11].)

In the case of cylindric algebras (i.e., the case of languages with equality), the

characterization of isomorphism and elementary embeddability with the previous

concepts is well known. (Base isomorphisms and ext.-isomorphisms between cylin-

dric set algebras are defined similarly to the quasi-polyadic case.) Suppose M and

N are models for a language L that contains the equality symbol.

• M ∼= N iff Csω(M) is base isomorphic to Csω(N ) with the natural generators

preserved, i.e., the image of ‖R‖M is ‖R‖N for all R ∈ L.

20

• M < N iff Csω(M) is ext.-isomorphic to Csω(N ) with the natural genera-

tors preserved, (see 10. of Remark 4.3.86 of [11]). Thus N is elementarily

embeddable into M iff Csω(M) and Csω(N ) are ext.-base-isomorphic with

the natural generators preserved.

In Definition 3.1 and Theorem 3.2 of [17], a characterization of ultrafilters that

correspond to isomorphic models is given (although only in the case with equality).

The next theorem is motivated by this characterization, and the above. (A is still

QPA(Σ), where Σ is a theory in an equality-free language L.)

Theorem 3.9. Let M0,M1 ∈ KΣ. Suppose F0,F1 ∈ H(A) are such that repFiis a

homomorphism from A onto Qsω(Mi) (i = 0, 1). Suppose ρ : ω −→ ω is injective,

and let U = Range ρ. For each X ∈ Qsω(M0) let fρ(X) = {ρ ◦ σ : σ ∈ X}, and

for each X ∈ Qsω(M1) let Φ(X) = X ∩ ωU . Then, the following are equivalent:

1. ρ : M0 4 M1,

2. fρ ◦ repF0= Φ ◦ repF1

,

3. For each x ∈ A we have x ∈ F0 iff s+ρ x ∈ F1, (that is, F0 = (s+ρ )−1F1).

The statement remains true if we replace “injective” by “bijective, and “4” by

“∼=”.

Proof. It is enough to prove the first part of the theorem, since the part concerning

isomorphism obviously follows. To see that 1. and 2. are equivalent, we will first

prove the claim below.

Claim 3.10. ρ : M0 4 M1 iff for every formula ϕ of L we have fρ(‖ϕ‖M0) =

Φ(‖ϕ‖M1).

Proof of Claim 3.10

ρ : M0 4 M1 iff

M0 |= ϕ[σ] iff M1 |= ϕ[ρ ◦ σ] for every σ ∈ ωω and ϕ ∈ Form(L) iff

σ ∈ ‖ϕ‖M0 iff ρ ◦ σ ∈ ‖ϕ‖M1 for every σ ∈ ωω and ϕ ∈ Form(L) iff

σ ∈ ‖ϕ‖M0 iff ρ ◦ σ ∈ ‖ϕ‖M1 ∩ ωU for every σ ∈ ωω and ϕ ∈ Form(L) iff

fρ(‖ϕ‖M0) = Φ(‖ϕ‖M1) for every ϕ ∈ Form(L).

This completes the proof of Claim 3.10.

Because we have ‖ϕ‖Mi = repFi(ϕ/≡Σ) (for i = 0, 1) for every formula ϕ, and

21

because the universe of A is A = {ϕ/≡Σ: ϕ ∈ Form(L)} we obtain, by Claim 3.10,

ρ : M0 4 M1 iff

fρ(‖ϕ‖M0) = Φ(‖ϕ‖M1) for every ϕ ∈ Form(L) iff

fρ(repF0(ϕ/≡Σ)) = Φ(repF1

(ϕ/≡Σ)) for every ϕ ∈ Form(L) iff

fρ(repF0(x)) = Φ(repF1

(x)) for every x ∈ A iff

fρ ◦ repF0= Φ ◦ repF1

.

Now suppose the second statement holds, that is fρ ◦ repF0(x) = Φ ◦ repF1

(x)

for every x ∈ A. Then, for all x ∈ A we have,

x ∈ F0 iff

s+Idωx ∈ F0 iff

Idω ∈ repF0(x) iff

ρ ◦ Idω ∈ fρ ◦ repF0(x) iff

ρ ◦ Idω ∈ Φ ◦ repF1(x) = repF1

(x) ∩ ωU iff

ρ ∈ repF1(x) iff

s+ρ x ∈ F1.

Therefore the second statement implies the third. To see the other direction,

suppose x ∈ F0 iff s+ρ x ∈ F1 for all x ∈ A. Then, for all τ ∈ ωω we have

τ ∈ repF0(x) iff

s+τ x ∈ F0 iff (by 5. of 2.16)

s+ρ◦τx ∈ F1 iff

ρ ◦ τ ∈ repF1(x) iff

ρ ◦ τ ∈ repF1(x) ∩ ωU = Φ ◦ repF1

(x) iff (since fρ : P(ωω) −→ P(ωU) is a bijection)

τ ∈ fρ−1 ◦ Φ ◦ repF1

(x).

Thus we have repF0= fρ

−1 ◦Φ ◦ repF1, or equivalently, fρ ◦ repF0

= Φ ◦ repF1.

Note that Claim 3.10 implies that, (using the notation of the previous theorem,)

ρ : M0 4 M1 iff f−1ρ ◦ Φ is an ext.-base-isomorphism.

3.2.2 The Case with Equality

Let A = CA(Σ) for some theory Σ in a countable language L with equality. Again,

we let KΣ = {M |= Σ : the universe of M is ω}. Due to Lemmas 3.6 and 3.7,

there is a one-one correspondence between KΣ and H′(A), just as in the case

22

without equality. The next theorem tells us that ultrafilters of H′(A) that lead to

pairwise non elementarily embeddable models are characterized exactly as in the

case without equality (see Theorem 3.9). Its proof is no different from the proof

of Theorem 3.9.

Theorem 3.11. Let M0 and M1 be models in KΣ, and let F0,F1 ∈ H′(A).

Suppose repFiis a homomorphism from A onto Csω(Mi) (i = 0, 1). Let ρ :

ω −→ ω be injective, and let U = Range ρ. For each X ∈ Csω(M0), define

fρ(X) = {ρ ◦ σ : σ ∈ X}, and for each X ∈ Csω(M1) let Φ(X) = X ∩ ωU . Then

the following are equivalent:

1. ρ : M0 4 M1,

2. fρ ◦ repF0= Φ ◦ repF1

,

3. F0 = (s+ρ )−1F1.

The statement remains true if we replace “injective” by “bijective, and “4” by

“∼=”.

23

4 A New Proof for Morley’s Original Theorem

In this section, we give a new proof of Morley’s original result in Theorem 4.4,

which is based on the representation theory of cylindric algebras, and some results

from descriptive set theory. A proof of the corresponding theorem for languages

without equality is easily obtained as a straightforward adaptation of our proof.

We then prove our variant about elementary bi-embeddability, in Theorem 4.5.

Definition 4.1. We say a topological space is Polish if it is separable and com-

pletely metrizable.

Notice that a countable product of Polish spaces is Polish. Thus, because any

countable discrete space is Polish, ω2 and ωω are Polish. It is also easy to see that

closed subsets of a Polish space are Polish. In fact, any Gδ subset of a Polish space

is Polish (by Theorem 3.11 of [13]).

The Stone space of a Boolean algebra B is Polish if its universe B is countable.

(Specifically, the Stone space of the Lindenbaum algebra of a theory Σ in a count-

able language is Polish.) This is because B∗ is homeomorphic to a subspace of the

Polish space B2, the homeomorphism being U(B) −→ B2; F 7→ {x ∈ B2 : x(b) =

1 iff b ∈ F}. Since B∗ is compact, its homeomorphic image is a closed subspace ofB2, implying that B∗ is also Polish.

Definition 4.2. A subset A of a Polish space X is analytic if there is a Polish

space Y and a continuous function f : Y −→ X such that f ∗Y = A.

Before proving Morley’s theorem, we state (a consequence of) Burgess’s the-

orem on analytic equivalence relations (see Theorem 9.1.5 of [7]). Just like in

Morley’s original proof, this result plays a key part in our proof.

Theorem 4.3 (Burgess). Let ∼ be an analytic equivalence relation on a Polish

space X (i.e., ∼ is analytic as a subspace of X × X). If ∼ has more than ℵ1

equivalence classes, then it has 2ℵ0 many equivalence classes.

Theorem 4.4.

1. (Morley) Suppose Σ is a theory in a countable language with equality. Then

I (Σ,ℵ0) > ℵ1 implies I (Σ,ℵ0) = 2ℵ0.

2. The same holds for countable languages without equality.

Proof. We will only prove 1., since the proof of 2. is a straightforward adaptation

of this proof. Let Σ be a theory in a countable language with equality, and let

A = CA(Σ). From its definition, it is obvious that H′(A) is a Gδ subset of the

24

Polish space A∗, and is therefore a Polish space. Thus, H = H′(A)×H′(A) is also

a Polish space.

Consider the following equivalence relation ∼ on H′(A). For F1,F2 ∈ H′(A),

let F1 ∼ F2 iff they correspond to isomorphic models. It is enough to show that

∼ is analytic, due to the previous theorem. By Theorem 3.11,

∼= {(F1,F2) ∈ H : F1 = (s+ρ )−1F2 for some ρ ∈ Sym(ω)}.

Notice that

Sym(ω) =⋂

m 6=n∈ω

{ρ ∈ ωω : ρ(n) 6= ρ(m)} ∩⋂

n∈ω

{ρ ∈ ωω : n ∈ ρ∗ω}

is a Gδ subset ofωω. Hence, H× Sym(ω) is a Polish space. Consider

C = {(F1,F2, ρ) ∈ H × Sym(ω) : x ∈ F1 iff s+ρ x ∈ F2 for all x ∈ A}.

We will show that C is a closed subset of H × Sym(ω), implying that C is a

Polish space. This is enough, as ∼ is the image of C under the natural projection

π : C −→ H; (F1,F2, ρ) 7→ (F1,F2), which is a continuous map.

For each x ∈ A, let

Cx = {(F1,F2, ρ) ∈ H × Sym(ω) : x ∈ F1 iff s+ρ x ∈ F2}.

We will show that Cx is a closed subset of H × Sym(ω), by showing that its

complement is open. Suppose (F1,F2, ρ) ∈ (H× Sym(ω))− Cx. Since Cx = C−x,

we can assume that x /∈ F1 and s+ρ x ∈ F2. Then

N = (N−x ∩H′(A))× (Ns+ρ x ∩H′(A))× {τ ∈ Sym(ω) : τ|∆x = ρ|∆x}

is a basic open set for which (F1,F2, ρ) ∈ N ⊆ (H× Sym(ω))−Cx. And so, Cx is

closed, implying that C =⋂

x∈ACx is also closed in H× Sym(ω).

4.1 A Variant about Elementary Bi-Embeddability

Recall from the introduction that, following the terminology of [8], we defined

two models M and N to be elementarily bi-embeddable iff M is elementarily

embeddable into N and N is elementarily embeddable into M. Elementary

bi-embeddability is an equivalence relation on the countable models of Σ, and

I ′(Σ,ℵ0) denotes the number of its equivalence classes.

Theorem 4.5.

1. Suppose Σ is a theory in a countable language with equality. If we have

I ′(Σ,ℵ0) > ℵ1, then I ′(Σ,ℵ0) = 2ℵ0.

25

2. The same holds for countable languages without equality.

Proof. The proof of this theorem is very similar to our proof of Theorem 4.4. As

before, we will only prove 1. Suppose Σ is a theory in a countable language with

equality, and let A = CA(Σ). We have seen previously that H = H′(A) ×H′(A)

is a Polish space.

Let ∼b be the equivalence relation on H′(A) defined in the following manner.

If F1,F2 ∈ H′(A) then F1 ∼b F2 iff they correspond to elementary bi-embeddable

models. We will show that∼b is analytic. This is enough, by Theorem 4.3. Because

of Theorem 3.11, we have

∼b= {(F1,F2) ∈ H : F1 = (s+ρ )−1F2 for some ρ ∈ Inj (ω)

and F2 = (s+σ )−1F1 for some σ ∈ Inj (ω)},

where Inj (ω) denotes the set of injective funtions in ωω. Because Inj (ω) =⋂

m 6=n∈ω{ρ ∈ ωω : ρ(n) 6= ρ(m)}, it is a closed subset of ωω, and therefore Polish.

And so, H× Inj (ω)× Inj (ω) is a Polish space. An argument similar to the one in

the previous proof shows that the following subspace Cb of H× Inj (ω)× Inj (ω) is

closed:

Cb = {(F1,F2, ρ, σ) : x ∈ F1 iff s+ρ x ∈ F2 for all x ∈ A

and y ∈ F2 iff s+σ y ∈ F1 for all y ∈ A}.

Hence, ∼b is the image of the Polish space Cb under the continuous π : Cb −→ H;

(F1,F2, ρ, σ) 7→ (F1,F2), and so is analytic.

26

5 Some Examples of Models

In this section, we first give a simple example showing that elementary bi-embed-

dability does not imply isomorphism. Then, we will see that elementary embed-

dability is not “reversible” even for theories with some “nice” properties, (i.e., we

will give an example of an ω-stable, model complete theory in a finite language

that has “no algebraic formulas”, and has two models A and B such that B 4 A

but A is not elementarily embeddable into B.)

We will use the following notations in this section. Let L be a finite language.

When not stated specifically, we will assume neither that L is a language with

equality, nor that L is an equality-free language. As before, 〈vi : i ∈ ω〉 denotes

the sequence of variables of L. v will usually denote a finite sequence of variables.

By a ∈ A, we mean that a is a finite sequence of elements of A. The length of a is

l(a) = k iff a = 〈a0, . . . , ak−1〉. When it does not lead to confusion, we will write

a instead of Range a.

To keep notation simpler, if ϕ = ϕ(w, v), where the variables w are bound in

ϕ, we will write ϕ(w, a) instead of ϕ[a] or ϕ(w, v)[a]. For example, ∃v0ϕ(v0, a) will

be written instead of ∃v0ϕ(v0, v)[a].

Suppose A is a model for L, and X ⊆ A. Then LX denotes the language

obtained by adding constant symbols 〈cx : x ∈ X〉 to L, and AX denotes the LX-

model obtained from A by interpreting each cx as x. For the sake of simplicity,

we will not differentiate between the constant cx and the element x; for example,

instead of ϕ(v, cx0, . . . , cxn−1

), we will write ϕ(v, x0, . . . , xn−1) or just ϕ(v, x).

More generally, suppose we have the languages L ⊆ L+. If A is a model for L+,

then, by simply forgetting the interpretations of the symbols in L+−L, we obtain

an L-model. This model is called the L-reduct of A, in symbols, A|L. When A is

an L+-model and B is its L-reduct, we say that A is an expansion of B to L+.

5.1 Elementary Bi-Embeddability and Isomorphism

Two isomorphic models are obviously also elementarily bi-embeddable. The next

examples shows that the converse is not always true, neither for languages with or

without equality. We will make use of the following propositions, both of which

are proved easily by induction on the complexity of formulas.

Proposition 5.1 (Tarski-Vaught Criterion). Let L be a first order language (with

or without equality), and A and B be models for L. Then the following are equiv-

alent.

1. B is an elementary substructure of A.

27

2. B ⊆ A, and for every L-formula ϕ(v0, v) and every b ∈ B, if A |= ∃v0ϕ(v0, b),

then there exists an element d ∈ B such that A |= ϕ(d, b).

Proposition 5.2. Let L be a first order language (with or without equality). Sup-

pose A and B are models for L and f : A −→ B. If f is surjective, and

for each n > 0, each n-ary relation symbol R ∈ L and

each n-tuple a ∈ A we have a ∈ RA iff f(a) ∈ RB, (∗)

then f preserves those formulas of L which do not contain the equality symbol,

(i.e., for all formulas ϕ not containing the equality symbol and all tuples a ∈ A we

have A |= ϕ[a] iff B |= ϕ[f(a)]). Specifically, if L is a language without equality,

then f is an elementary map.

Note that in the case of equality-free languages, elementary maps are not nec-

essarily injective.

Let L = 〈E〉 be a language with equality, where E is a binary relation symbol.

Let Σ be the theory that says “E is an equivalence relation with infinitely many

equivalence classes, each of which are infinite”. Let A be the model of Σ consisting

of ℵ0 many equivalence classes of size ℵ1, and let B be the model of Σ consisting of

one equivalence class of size ℵ0 and ℵ0 many equivalence classes of size ℵ1. These

two models are obviously not isomorphic.

Proposition 5.3. The two models A and B defined above are elementarily bi-

embeddable.

Proof. Let 〈Ai : i ∈ ω〉 be an enumeration of the equivalence classes of EA, and

〈Bi : i ∈ ω〉 be an enumeration of the equivalence classes of EB such that B0 is

the equivalence class with ℵ0 elements.

Let f : B −→ A be such that f|Bi: Bi −→ Ai is a bijection for all i > 0,

and f|B0: B0 −→ A0 is injective. Denoting by B′ the substructure of A whose

universe is B′ = Range f , we have B ∼= B′ ⊆ A. Using the Tarski-Vaught Criterion

(Proposition 5.1), we will show that B′ 4 A, which will imply that f : B 4 A.

Suppose that for some formula ϕ(v0, v) ∈ Form(L) and b ∈ B′, we have A |=

∃v0ϕ(v0, b). Then for some element c ∈ A, we have A |= ϕ(c, b). If c ∈ B′, we

are done. If c ∈ A−B′ = A0−B′0 (where B′

0 = f ∗B0), then take any d ∈ B′0− b.

The permutation of A switching d and c, and leaving the rest of the elements of

A fixed, is an automorphism of A. Therefore, A |= ϕ(d, b).

Now take a g : A −→ B such that g|Ai: Ai −→ Bi+1 is a bijection for all i,

and let A′ be the substructure of B whose universe A′ is the range of g. Again, we

28

will use the Tarski-Vaught Criterion. If we have B |= ∃v0ϕ(v0, a) for some formula

ϕ(v0, v) and some a ∈ A′, then for some element d of B, B |= ϕ(d, a). Suppose

d ∈ B−A′ = B0. Take a 0 < j < ω such that Bj ∩ a = ∅, and let B′j be a

countable subset of Bj. Define B′′ to be the substructure of B whose universe is

B′′ = B′j ∪

⋃

i∈ω−{j}Bi. Note that B′j ⊆ A′, and d, a ∈ B′′. Then, similarly to

the argument in the paragraph above, one can show that B′′ 4 B. Furthermore, a

permutation ρ of B′′ which maps the elements of B0 bijectively onto B′j and the

elements of B′j bijectively onto B0, and leaves the rest of the elements fixed, is an

automorphism of B′′ which leaves a fixed. We therefore have

B |= ϕ(d, a) iff B′′ |= ϕ(d, a) iff B′′ |= ϕ(ρ(d), a) iff B |= ϕ(ρ(d), a).

Now ρ(d) ∈ B′j ⊆ A′ and B |= ϕ(ρ(d), a). Thus, by the Tarski-Vaught Criterion,

A′ 4 B, implying that g is an elementary embedding of A into B.

The above example can be easily adapted to provide elementarily bi-embeddable

non-isomorphic models of size κ for any uncountable cardinal κ.

Consider the case of a language L′ = 〈E〉 without equality, where E is a binary

relation symbol, and the theory Σ′ that says “E is an equivalence relation”. Note

that if A′ and B′ are the equality-free reducts of the (uncountable) models A and

B above, then, because A and B are bi-elementarily embeddable, so are A′ and

B′, and obviously, they are not isomorphic. However, motivated by the above

example, we will now construct two countable models of Σ which are elementarily

bi-embeddable and nonisomorphic.

Let A be the model of Σ consisting of ℵ0 many equivalence classes of size ℵ0,

and let B be the model of Σ consisting of one finite equivalence class and ℵ0 many

equivalence classes of size ℵ0.

Proposition 5.4. The models A and B are elementarily bi-embeddable.

Proof. To show that B is elementarily embeddable into A, let 〈Ai : i ∈ ω〉 be an

enumeration of the equivalence classes of EA and 〈Bi : i ∈ ω〉 be an enumeration of

the equivalence classes of EB. Take a function f : A −→ B such that f|Ai: Ai −→

Bi is surjective for all i ∈ ω. Then f is surjective and satisfies (∗) from Proposition

5.2, and so, f is an elementary map from A to B, that is, for all formulas ϕ and

tuples a ∈ A, we have

A |= ϕ[a] iff B |= ϕ[f(a)].

Note that f is not necessarily injective, since L′ is a language without equality.

Now let A′ ⊆ A be such that A′ contains exactly one element from f−1({b}) for

29

all b ∈ B, (i.e., f|A′ : A′ −→ B is a bijection), and let g = (f|A′)−1. Then, for all

formulas ϕ and all tuples b ∈ B we have

B |= ϕ[b] iff A |= ϕ[g(b)],

and so, g is an elementary embedding of B into A.

The other direction is easily obtained by adapting the argument found in the

second part of the proof of Proposition 5.3.

Constructing countable elementarily bi-embeddable non-isomorphic models is

not so simple. Two such models are given in Example 1.4.4 of [8].

5.2 Reversibility of Elementary Embeddings

Suppose Σ is a first order theory such that the following is satisfied: if A and B

are (countable) models of Σ and B 4 A, then A is elementarily embeddable into

B. Then for A,B ∈ KΣ, one is elementarily embeddable into the other iff they

are bi-elementarily embeddable. And so, by Theorem 4.5, if Σ has more than ℵ1

many pairwise non elementarily embeddable models, then Σ has 2ℵ0 such models.

In this subsection, we will see that even some “nice” properties of a theory are

not enough to guarantee the “reversibility” of elementary embeddability discussed

in the paragraph above. More specifically, a theory will be given, which is model

complete, ω-stable, has a finite language and has no algebraic formulas (in the

sense of Definition 5.9 below), but has two countable models such that one is

elementarily embeddable into the other, but not vice versa.

5.2.1 Basic Definitions and Lemmas

Definition 5.5. A theory Σ is model complete if every embedding between models

of Σ is elementary.

Definition 5.6. Let v = 〈v0, . . . , vn−1〉, and let A be a model for L and X ⊆ A.

A set p of LX-formulas in free variables v0, . . . , vn−1 is called an n-type over X in

A if the following are satisfied:

1. for every p0 ⊆ω p we have AX |= ∃v0 . . . ∃vn−1

∧

p0,

2. for every ϕ ∈ Form(LX) in free variables v0, . . . , vn−1, we have either ϕ ∈ p

or ¬ϕ ∈ p.

The set of all n-types over X in A is denoted by SAn (X).

30

Let A be a model for L, a ∈ A, X ⊆ A and Γ ⊆ Form(L). The type of a over

X (with respect to A) is

tpA(a/X) = {ϕ(v) ∈ Form(LX) : ϕ ∈ Γ or ¬ϕ ∈ Γ, and AX |= ϕ[a]},

(where v = 〈v0, . . . , vl(a)−1〉). It is obviously an l(a)-type over X in A.

Note that if B 4 A and X ⊆ B, then SAn (X) = SB

n (X). Suppose p is an n-type

over X in B, and A is an elementary extension of B. We will say a ∈ A realizes p

in A if AX |= ϕ[a] for all ϕ ∈ p.

Definition 5.7. Let Σ be a consistent theory and κ be an infinite cardinal. We

say that Σ is κ-stable if for all models A of Σ and all X ∈ [A]κ we have

|SA1 (X)| ≤ κ.

A theory Σ is called stable if it is κ-stable for some κ.

We note that Σ is κ-stable iff for all models A of Σ, all X ∈ [A]κ and all n < ω

we have |SAn (X)| ≤ κ, and that if Σ is an ω-stable theory in a countable language,

then it is κ-stable for all infinite cardinals κ. (See for example Lemma 6.7.4 and

Theorem 6.7.5 of [12].)

Lemma 5.8. A consistent theory Σ is κ-stable iff the following is satisfied: for all

A |= Σ and X ∈ [A]κ we have

∣

∣{tpA(a/X) : a ∈ A}∣

∣ ≤ κ

Proof. To see the nontrivial direction, suppose B |= Σ and X ∈ [B]κ. Let L

be the language of Σ. Expand LX to the language L+ by adding new constant

symbols {dp : p ∈ SB1 (X)}, and let

Σ+ = Σ ∪ {ϕ(dp) : p ∈ SB1 (X), ϕ ∈ p}.

For every Γ ⊆ω Σ+, there is an expansion BΓ of BX which satisfies Γ. By the

Compactness Theorem, there is a model A+ of Σ+ which is an ultraproduct of

{BΓ : Γ ⊆ω Σ+}. Then the L-reduct A of A+ is an elementary extension of B in

which all p ∈ SB1 (X) are realized. And so, we have

∣

∣SB1 (X)

∣

∣ =∣

∣{tpA(a/X) : a ∈ A}∣

∣ ≤ κ.

Definition 5.9. Suppose A is a model for L and X is a subset of A.

31

1. An element a ∈ A is called algebraic over X if there is a ϕ ∈ Form(LX) such

that AX |= ϕ[a], and ‖ϕ‖AX is finite.

2. The algebraic closure of the set X in A is

acl(X) = {a ∈ A : a is algebraic over X}.

3. We will say that a theory Σ has no algebraic formulas iff for all models B of

Σ and all Y ⊆ B we have acl(Y ) = Y .

Note that a theory Σ has no algebraic formulas iff the following holds: for all

B |= Σ, all Y ⊆ B, all b ∈ B−Y and all ϕ ∈ Form(LX),

if BY |= ϕ[b], then ‖ϕ‖BY is infinite.

5.2.2 Our Counterexample

Let L be a language with equality which contains two binary relation symbols F

and E, and let the Σ be the L-theory which says (S0) – (S4) below; (an intuitive

explanation will be given right away).

(S0) E is an equivalence relation whose equivalence classes are infinite,

(S1) ∀v0∀v1∀v2∀v3(F (v0, v2) ⇒

[v0Ev1 ⇒ F (v1, v2)] ∧ [v2Ev3 ⇒ F (v0, v3)]),

(S2) ∀v0∀v1∀v2∀v3

F (v0, v2) ∧ F (v1, v3) ⇒ [v0Ev1 ⇔ v2Ev3],

(S3)

∀v0∃v1F (v0, v1) ∧ ∀v1∃v0F (v0, v1),

(S4) for all n ∈ N+, we have

∀v0∀v1[F◦n(v0, v1) ⇒ ¬(v0Ev1)],

where we write vEw instead of E(v, w), and F ◦n(v0, vn) denotes the formula

∃v1 . . . ∃vn−1(F (v0, v1) ∧ · · · ∧ F (vn−1, vn)) for n ∈ N+. If we interpret F (v, w)

as having a (directed) edge from v to w, then (S1) – (S4) say the following: “the

edges go between equivalence classes; there is exactly one edge beginning in each

equivalence class and one arriving at each equivalence class; there are no loops”.

(Note that if we assume (S0) – (S3), then (S4) is equivalent to saying that for

all n ∈ N+ we have ∀v0[¬F

◦n(v0, v0)].)

32

In any model A of Σ, we call two elements a, b ∈ A connected if either A |= aEb

or for some m > 0 we have A |= F ◦m(a, b) ∨ F ◦m(b, a). This is an equivalence

relation, and we call its equivalence classes the connected components of A. Notice

that a connected component is determined up to isomorphism, (but a countable

model can have any countable number of components).

Let B be the countable model of Σ consisting of one connected component, and

let A be the countable model of Σ consisting of two connected components. Then

the function which maps B isomorphically onto one component of A is an embed-

ding, and because of the model completeness of Σ (which we will see in Proposition

5.14 below), this embedding is elementary. So B is elementarily embeddable into

A, but the converse is not true, as we will now see.

Proposition 5.10. If A and B are defined as above, then A is not (elementarily)

embeddable into B.

Proof. Suppose that f is an embedding of A into B. Let a and b be elements of

A that are not connected in A. Then

A |= ¬(aEb), and A |= ¬F ◦m(a, b) ∧ ¬F ◦m(b, a)

holds for all m > 0. But f(a) and f(b) must be connected in B, and therefore we

have either

B |= f(a)Ef(b) or B |= F ◦m(a, b) ∨ F ◦m(b, a)

for some m > 0. This is obviously a contradiction.

It remains to be seen that Σ has the “nice” properties mentioned above. First

of all, the language of the theory Σ is finite. The propositions below show that Σ

has no algebraic formulas, is stable, and is model complete.

Proposition 5.11. Σ has no algebraic formulas (in the sense of Definition 5.9).

Proof. Take any model A of Σ, any X ⊆ A, any b ∈ A−X and any ϕ = ϕ(v, x) ∈

Form(LX) (where x ∈ X). Suppose AX |= ϕ(b, x). By the remark after Definition

5.9, it is enough to show that ‖ϕ‖AX is infinite. Since x is finite, there is an infinte

set {bi : i < ω} of elements contained in the EA-equivalence class of b, but not

contained in x. For all i < ω, the permutation of A switching b and bi, but leaving

the rest of the elements of A fixed, is an automorphism of Ax. And so, we have,

for all i < ω,

AX |= ϕ(b, x) iff Ax |= ϕ(b, x) iff Ax |= ϕ(bi, x) iff AX |= ϕ(bi, x).

This means that the elements of {bi : i ∈ ω} realize ϕ, and thus, ‖ϕ‖AX is infinite.

33

Proposition 5.12. The theory Σ is ω-stable.

Proof. For all models A of Σ, all X ⊆ A and all a ∈ A, define the following:

d+(a,X) = min{n ∈ N ∪ {∞} : for some x ∈ X we have A |= F ◦n(a, x)},

d−(a,X) = min{n ∈ N ∪ {∞} : for some x ∈ X we have A |= F ◦n(x, a)},

where F ◦0(v0, v1) denotes the formula v0Ev1. Let the distance from a to X be

d(a,X) =

{

d+(a,X) if d+(a,X) ≥ d−(a,X),

−d−(a,X) if d−(a,X) > d+(a,X).

Note that d(a,X) = ∞ iff there is no x ∈ X such that a and x are connected. If

d(a,X) ∈ Z, then let mX(a) denote an element of X such that

d(a,X) = d(

a, {mX(a)})

.

Note that mX(a) is determined up to EA-equivalence.

To show that Σ is ω-stable, suppose A |= Σ and X ∈ [A]ℵ0 . Notice that A

has an extension A′ ⊇ A that is a model of Σ, such that all of the equivalence

classes of EA′are the same size. By the model completeness of Σ (to be shown

in Proposition 5.14 below), we have A 4 A′. This implies that SA1 (X) = SA′

1 (X).

Therefore, we can assume that all of the equivalence classes of A are the same size.

First, suppose that for a, b ∈ A−X we have d(a,X) = d(b,X) ∈ Z, and that

A |= mX(a)EmX(b). Then A |= aEb, and the permutation of A switching a and

b, but leaving the rest of the elements of A fixed, is an automorphism of AX . This

implies that

tpA(a/X) = tpB(b/X).

Now suppose d(a,X) = d(b,X) = ∞. If a and b are connected, then there

is an automorphism of A mapping a to b and leaving the elements not in the

connected component of a fixed. (Here, we use the fact that all of the equivalence

classes of EA are the same size.) Because d(a,X) = ∞, this map leaves all of the

elements of X fixed, and so, it is also an automorphism of AX . If a and b are

not connected, then, (using the fact that all the EA equivalence classes are the

same size,) there is an automorphism of A mapping a to b, mapping the connected

component of a bijectively onto the connected component of b and vice versa,

and leaving those elements of A not connected to either a or b fixed. Because

d(a,X) = d(b,X) = ∞ all of the elements of X remain fixed, implying that this

map is also an automorphism of AX . We therefore have, in either case,

tpA(a/X) = tpB(b/X).

34

Thus, for any a ∈ A−X, the type of a over X is completely determined by

d(a,X) and, in the finite case, mX(a). Hence, using the fact that X is countable,

∣

∣{tpA(a/X) : a ∈ A}∣

∣ ≤ ℵ0.

This implies, by Lemma 5.8, that Σ is ω-stable.

We will need the following lemma when proving the model completeness of Σ.

Lemma 5.13 (Relativisation Theorem). Suppose L and L+ are languages such

that L ⊆ L+ and P is a unary relation symbol in L+−L. Then for every ϕ ∈

Form(L), there exists a ϕP ∈ Form(L+) such that the following holds:

if A is a model for L+ and we have B ⊆ A|L and PA = B, then for every b ∈ B

B |= ϕ[b] iff A |= ϕP [b]. (∗∗)

Proof. We define ϕP by induction on the complexity of ϕ.

1. Let ϕP be ϕ when ϕ is atomic.

2. Let (∧n

i=1 ϕi)Pbe

∧ni=1 ϕ

Pi and let (

∨ni=1 ϕi)

Pbe

∨ni=1 ϕ

Pi .

3. Let (¬ϕ)P be ¬ϕP .

4. Let(

∀v0ϕ(v0, v))P

be ∀v0(

P (v0) ⇒ ϕP (v0, v))

and let(

∃v0ϕ(v0, v))P

be

∃v0(

P (v0) ∧ ϕP (v0, v)

)

.

Then (∗∗) is easily seen by induction on the complexity of ϕ.

Proposition 5.14. Σ is a model complete theory.

Proof. Suppose, seeking a contradiction, that there exist models A and B of Σ

such that B ⊆ A, but B is not an elementary substructure of A. Then for some

formula ϕ ∈ Form(L) and b ∈ B we have A |= ϕ[b] and B |= ¬ϕ[b]. Now extend

L to a language L+ by adding a new unary relation symbol P , and let A+ be the

expansion of A to L+ for which PA+

= B. Let Σ+ = Th(A+). Then, by the

Relativisation Theorem, we have

Σ+ |= ∃v(

P (v) ∧ ϕ(v) ∧ ¬ϕP (v))

.

Now extend L+ to a language L′ by adding countably many constant symbols

〈ci : i < ω〉, and let

Σ′ = Σ+ ∪ {P (ci) : i ∈ ω} ∪ {¬(ciEcj) : i, j ∈ ω, i 6= j}∪

∪ {¬F ◦m(ci, cj) : m ∈ N+, i, j ∈ ω, i 6= j}.

35

If Γ is a finite subset of Σ′, then there is an expansion AΓ of A+ such that AΓ |= Γ.

Therefore, by the Compactness Theorem and the Downward Lowenheim-Skolem

Theorem, there is a countable model of Σ′; let C+ be its L+-reduct. Then C+ is a

countable model of Σ+ such that P C+

has countably many connected components,

(and therefore, so does C+). Let C = C+|L, and let D be the L-structure such that

D = P C+

and D ⊆ C. By Lemma 5.15 below, we then have D 4 C; specifically,

D |= ϕ[d] iff C |= ϕ[d] for every d ∈ D. And so, by the Relativisation Theorem,

C+ |= ∀v(

P (v) ⇒ (ϕ(v) ⇔ ϕP (v)))

,

contradicting the fact that C+ is a model of Σ+.

Lemma 5.15. If C and D are both countable models of Σ with countably many

connected components and D ⊆ C, then D 4 C.

Proof. We will use the Tarski-Vaught Criterion (Proposition 5.1). Suppose that

for some ϕ(v0, v) ∈ Form(L) and d ∈ D, we have C |= ∃v0ϕ(v0, d), that is, C |=

ϕ(c, d) for some c ∈ C.

If d(c, d) ∈ Z, then, since the ED-equivalence classes are all infinite and d is

finite, we can choose an element b ∈ D such that d(b, d) = d(c, d) and md(b) =

md(c). Then C |= bEc, and so, the permutation of C switchig b and c but leaving

all the other elements of C fixed is an automorphism of Cd, giving us C |= ϕ(b, d).

If d(c, d) = ∞, then, because d is finite and D has infinitely many components,

there is a b ∈ D such that d(b, d) = ∞. Using the fact that the equivalence classes

of EC are all the same size, (countably infinite,) we can obtain an automorphism

of Cd that takes c to b, exactly as in the proof of Proposition 5.12. Therefore

C |= ϕ(b, d).

We note that the idea of the proof of Proposition 5.14 is the same as the idea

found in Example 8.3.1. of [12], and that this proof is very similar to the proof of

Lindstrom’s Test (found in Theorem 8.3.4. of [12]).

5.2.3 Our Counterexample in the Equality-Free Case

Let L′ = 〈F,E〉 be a language without equality, where F and E are binary relation

symbols. Take the L′-theory Σ′ which says (S1) – (S4) and, instead of (S0), says

(S′0) E is an equivalence relation.

Suppose A′ and B′ are the equality-free reducts of the models A and B defined

in the previous subsubsection, i.e., B′ consists of one countable connected com-

ponent in which all the EB′-equivalence classes are (countably) infinite, and A′

36

consists of two such connected components. Then B′ is elementarily embeddable

into A′ (because, for example, B is elementarily embeddable into A). The fact

that A′ is not even embeddable into B′ can be seen exactly as in the proof of

Proposition 5.10.

Obviously, it will not be true that Σ′ has no algebraic formulas, but if C ′ |= Σ′

and all of the EC′-equivalence classes are infinite, then acl(X) = X for all X ⊆ C ′.

Specifically, this holds for the models A′ and B′ defined above. The proof of this

statement is no different from the proof of Proposition 5.11.

Proposition 5.16. The theory Σ′ is model complete.

Proof. Suppose D′ ⊆ C ′ and they are models of Σ′. Extend D′ and C ′ to the

Σ′-models D and C by adding infinitely many elements to each ED′-equivalence

class. In other words, D and C are models of Σ, such that

• D′ ⊆ D and C ′ ⊆ C,

• D ⊆ C,

• if X1 ⊆ X0 are such that X1 is an ED-equivalence class and X0 is an EC-

equivalence class, then we have X0−C′ = X1−D

′, and this set is infinite.

Expand the models C and D to C+ and D+ by adding a = symbol to the

language, and interpreting =C+

and =D+

as equality. D+ ⊆ C+ and are models of

the theory Σ (of the previous subsection), and so, by Proposition 5.14, we have

D+ 4 C+. Therefore D 4 C.

There exist surjective functions f : C −→ C ′ and g : D −→ D′ that satisfy (∗)

from Proposition 5.2 such that

• f|C′ = IdC′ and g|D′ = IdD′ ,

• f|D = g.

Thus, because D 4 C, and by Proposition 5.2, we have

D′ |= ϕ[d] iff D |= ϕ[d] iff C |= ϕ[d] iff C ′ |= ϕ[d]

for all ϕ ∈ Form(L′) and all d ∈ D′. This implies that D 4 C. Therefore Σ′ is

model complete.

Proposition 5.17. The theory Σ′ ω-is stable.

Proof. The proof of this proposition is a straightforward adaptation of the proof

of Proposition 5.12, except for the following step. If we are given an arbitrary

model C of Σ′, we have to take an extension C ′ ⊇ C that is a model of Σ′ (and

therefore C 4 C ′) such that all of the equivalence classes of EC′are infinite and

are the same size.

37

References

[1] H. Andreka, I. Nemeti, A Simple, Purely Algebraic Proof of the Com-

pleteness of Some First Order Logics, Algebra Universalis, 5(1975) 8-15.

[2] E. Bouscaren, D. Lascar, Countable Models of Non-Multidimensional

ℵ0-stable Theories, Journal of Symbolic Logic, 48(1983) 197-205.

[3] S. Buechler, Vaught’s conjecture for superstable theories of finite rank, An-

nals of Pure and Applied Logic, 155(2008) 135-172.

[4] S. Buechler, Essential Stability Theory, Springer, (1996).

[5] S. Burris, H.P. Sankappanavar, A Course in Universal Algebra,

Springer–Verlag (1981).

[6] C. C. Chang, H. J. Keisler, Model Theory, North–Holland, Amsterdam

(1973).

[7] S. Gao, Invariant Descriptive Set Theory, Chapman & Hall/CRC, (2009).

[8] J.R. Goodrick, When are elementarily bi-embeddable models isomorphic?,

Ph. D. dissertation, University of California, Berkeley, (2007).

[9] A. Hajnal, P. Hamburger, Set Theory, Cambridge University Press,

(1999).

[10] L. Henkin, J.D. Monk, A. Tarski, Cylindric Algebras Part 1, North–

Holland, Amsterdam (1971).

[11] L. Henkin, J.D. Monk, A. Tarski, Cylindric Algebras Part 2, North–

Holland, Amsterdam (1985).

[12] W. Hodges, Model theory, Cambridge University Press, (1997).

[13] S. Kechris, Classical Descriptive Set Theory, Springer, (1995).

[14] D. Marker, Model theory: An Introduction, Springer, (2002).

[15] M. D. Morley, The Number of Countable Models, Journal of Symbolic

Logic, 35(1970) 14-18.

[16] I. Sain, R. Thompson, Finite Schema Axiomatization of Quasi-Polyadic

Algebras, (in Coll. Math. Soc. J. Bolyai, Algebraic Logic, 54. North Holland),

539- 571, 1991.

38

[17] G. Sagi, Vaught’s Conjecture from the perspective of Algebraic Logic,

Preprint (2008).

[18] S. Shelah, L. Harrington, M. Makkai A Proof of Vaught’s Conjecture

for ω-Stable Theories, Isr. J. Math. 49(1984), 259-280.

[19] S. Shelah, Classification theory, North–Holland, Amsterdam (1990).

[20] D. Sziraki, Applying Algebraic Logic to Vaught’s Conjecture and Related

Problems, student research (TDK) paper, 2010.

[21] R. Vaught, Denumerable Models of Complete Theories, in: Infinitistic

Methods, pp. 303-321, Pergamon, London, (1961).

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