Hedman. A First Course in Logic, 2004 (Oxford)

280 Models of countable theories

Proof Suppose M is prime. By the Downward L¨owenhiem–Skolem theorem, there exists a countable model of T . Since M can be elementarily embedded into this model, M must be countable. It remains to be shown that M is atomic. Let p be a nonisolated type in S(T ). By the Omitting Types theorem 6.13, there exists a model N of T that omits p. Since M can be elementarily embedded into N , M must also omit p (see Exercise 6.1). So M realizes only the isolated types in S(T ) and is atomic.

Now suppose that M is countable and atomic. Then every type realized in M is realized in every model of T . By Proposition 6.26, M can be elementarily embedded into any homogeneous model of T . In a similar manner, we show that, since M is atomic, it can be elementarily embedded into any model (homogeneous or not).

Since M is countable, we can enumerate the underlying set of M as UM = {a1, a2, a3, . . .}. (As usual, if M is finite, then this proposition is trivial.) For each n N, let a¯n denote the n-tuple (a1, . . . , an). Let N be an arbitrary model of T . Let b1 be an element of N that realizes the isolated type tpM (a1).

¯

θ(bn, bn+1) for some element bn+1 of N . Since there is only one type in Sn+1(T ) containing θ, tpN (b1, . . . , bn+1) = tpM (a1, . . . , an+1).

In this manner we can construct a sequence b1, b2, b3, . . . as in the proof of Proposition 6.26. Let function f defined by f (ai) = bi is an elementary embedding of M into N .

Since N was arbitrary, M is prime.

We summarize the results of this section. If T is a small theory, then there exists an atomic countable model M of T . This model is unique up to isomorphism, is homogeneous, and can be elementarily embedded into any model of T . In this sense, M is the smallest model of T . Countable atomic models also exist for theories that are not small (recall Example 6.23 and see Exercise 6.14). We now turn our attention to big countable models.

6.4 Big models of small theories

We define and investigate countable saturated models of a countable complete theory. We show that countable saturated models, like countable atomic models, are homogeneous and unique up to isomorphism. We also show that every countable model of a theory can be elementarily embedded into the countable saturated model (provided it exists). So countable saturated models are the largest countable models in the same sense that countable atomic models are

the smallest models. In the second part of this section, we extend the notion of saturation to apply to uncountable structures.

6.4.1 Countable saturated models. Before defining countable saturated models, we must introduce the concept of a type over a set. Let M be a V-structure having underlying set UM . Let A be a subset of UM . A type over A is a type that allows parameters from A. More specifically, an n-type over A is a set of V(A)- formulas in n free variables that is realized in some elementary extension of M .

Example 6.32 Consider the structure Q< = {Q| <}.

Let A be the set {1, 2, 3}. The three numbers in A break Q into four intervals. Each of these intervals corresponds to a 1-type over A. These types are isolated by the formulas

(x1 < 1), ¬(x1 < 1) ¬(x1 = 1) (x1 < 2),

¬(x1 < 2) ¬(x1 = 2) (x1 < 3), and ¬(x1 < 3) ¬(x1 = 3).

In addition, there are the three types over A isolated by the formulas x1 = 1, x1 = 2, and x1 = 3. So there are seven isolated types over A.

Let B be the natural numbers and let C be the set of all rational numbers. Then there are denumerably many types over B exactly one of which is nonisolated. The nonisolated type contains the formulas ¬(x1 < n) for each n B. This type is not realized in Q< but is realized in an elementary extension of Q<. As was shown in Example 6.19, there are 2 0 types over C.

We make formal the definition of a type over a set and introduce notation for this concept.

Definition 6.33 Let M be a V-structure having underlying set UM . For any

of b over A in M , denoted tpM (b/A), is the set of all V(A)-formulas having free

variables among x1, . . . , xn that hold in M when each xi in x¯ is replaced by bi.

¯

The types tpM (b/A) are called complete types over A. Let S(A) denote the set of all complete types over A. The subset of all n-types in S(A) is denoted by Sn(A). Since the theory T is not mentioned in this notation, S(A) is ambiguous when taken out of context. In Example 6.32, we said that S(A) contains seven types when A = {1, 2, 3}. If T is the theory of the rational numbers with addition and multiplication, then this is not true. We shall only use the notation S(A) when T is understood.

Definition 6.34 Let T be a complete theory. A countable model M of T is saturated if, for every finite subset A of the underlying set of M , every 1-type in S(A) is realized in M .

Example 6.35 Let T be the VE -theory defined in Example 6.8. Recall that M is the model of T having exactly one equivalence class of size n for each n N. It was shown that there exists a type in S1(T ) that is not realized in M . So this structure is not saturated. Let Nm be the model of T having exactly m infinite equivalence classes. Let A = {a1, . . . , am} be a set of elements from each of these infinite classes. The type over A saying that x1 has an infinite class but is not equivalent to ai for each i is not realized in Nm. The only countable saturated model of T is the countable model containing denumerably many infinite equivalence classes.

Example 6.36 Let T be the theory defined in Example 5.56. The model containing countably many copies of Z in each equivalence class is the only saturated model of T .

As with atomic models, countable saturated models exist for small theories. Unlike the atomic models, these are the only theories having countable saturated models.

Proposition 6.37 A complete theory T is small if and only if it has a countable saturated model.

Proof Suppose first that T has a countable saturated model M . Then every type in S(T ) is realized by some tuple of M . Since M is countable, S(T ) must be countable also and T is small.

Conversely, suppose that T is small. Let M1 be a countable model of T . We define an elementary chain of countable models M1 M2 M3 . . .

Suppose that countable Mn has been defined. Let A be a finite subset of the underlying set of Mn. If S1(A) is uncountable, then so is Sk+1(T ) where k = |A|. Since T is small, this is not the case. So we can enumerate S1(A) as the possibly finite set {p1, p2, . . .}. For each pi in this set, there exists an elementary extension of Mn realizing pi. By the Downward L¨owenhiem–Skolem theorem, there exists a countable elementary extension Ni of Mn that realizes pi. By Proposition 4.37, there exists a countable model MA of T such that Mn and each Ni can be elementarily embedded into MA. Since Mn is countable, there are countably many finite subsets of Mn. Again applying Proposition 4.37, there exists a countable model Mn+1 of T such that MA can be elementariliy embedded into Mn+1 for each finite subset A of Mn.

Let M be the limit of the elementary chain M1 M2 M3 · · · . Then M is a countable model of M . Any finite subset A of the universe of M is in the universe of Mn for some n N. By the definition of Mn+1, every type in S1(A) is realized in Mn+1. Since Mn+1 M , every type in S1(A) is realized in M and M is saturated.

Proposition 6.38 Countable saturated models are homogeneous.

Proof Let M be a countable saturated model of a complete theory T and let

¯

a¯ = (a1, . . . , an) and b = (b1, . . . , bn) be two n-tuples of M that realize the same type in Sn(T ). Let c be an element of M . Let p1(x1) = tpM (c/a¯). Let p2 be

¯

the type over b obtained by replacing each occurrence of ai in p1 with bi (for i = 1, . . . , n).

Claim p2(x1) is realizable.

shows that any finite subset of p2(x1) is realizable. The claim then follows from Proposition 6.6.

Since M is saturated, p2(x1) is realizable in M . Let d be an element of M

¯

that realizes this type. Then tpM (b, d) = tpM (¯a, c) and M is homogeneous.

So countable saturated models, like atomic models, are homogeneous. From this fact we can immediately deduce two more properties of countable saturated models. They are universal and unique.

Definition 6.39 Let T be a theory and let M be a countable model of T . If every countable model of T can be elementarily embedded into M , then M is said to be a universal model of T .

Corollary 6.40 Countable saturated models are universal.

Proof This follows from Propositions 6.38 and 6.26.

The converse of Corollary 6.40 does not hold. Exercise 6.26 provides an example of a countable universal model that is not saturated. For the universal model to be saturated, it must be homogeneous.

Proposition 6.41 Let T be a small theory. A countable model M of T is saturated if and only if it is universal and homogeneous.

Proof A countable saturated model is universal and homogeneous by Corollary 6.40 and Proposition 6.38. We must prove the converse. Suppose that M is a countable model of T that is both universal and homogeneous. Let A be a finite subset of M and let p be a type in S1(A). We must show that there exists an element d so that tpM (d/A) = p.

The type p is realized in some elementary extension of M . By the Downward L¨owenhiem–Skolem theorem, p is realized in some countable model N containing A. So tpN (c/A) = p for some element c of N . Since M is universal, N can be elementarily embedded into M . Let f : N → M be elementary. Let B be

284 Models of countable theories

{f (a)|a A}. Note that B does not necessarily equal A, but it does have the

¯

same type as A in M . That is, tpM (¯a) = tpM (b), where (a1, . . . , ak) is some

¯

enumeration of A and b = (f (a1), . . . , f (ak)). Since M is homogeneous, there

¯

exists d so that tpM (d, a¯) = tpM (f (c), b). For all formulas ϕ(x1) in p, since N |= ϕ(c) and f : N → M is elementary, we have M |= ϕ(d). So tpM (d/A) = p and p is realized in M . Since p is arbitrary, M is saturated.

Next we show that the saturated model of a theory is unique up to isomorphism. This fact, like Corollary 6.40, is an immediate consequence of Proposition 6.38.

Corollary 6.42 Let T be a complete small theory. Any two countable saturated models of T are isomorphic.

Proof This follows immediately from Propositions 6.38 and 6.27.

We summarize. Let T be a small theory. Then T possesses both a countable atomic model M and a countable saturated model N . Each of these is unique up to isomorphism. The countable atomic model is the smallest model in the sense that it can be elementarily embedded into any model of T . The countable saturated model M is the biggest countable model of T in the sense that every countable model of T can be elementarily embedded into M . Countable saturated models are characterized by this property together with homogeneity. Likewise, countable atomic models are characterized as prime models (which are necessarily homogeneous).

We turn to theories that are not small in the next section. We close the present subsection by extracting the following characterization of 0-categorical theories from the above results.

Proposition 6.43 A theory is 0-categorical if and only if it has an atomic model and a countable saturated model that are isomorphic.

Proof Suppose T is 0-categorical. Since 0-categorical theories are small, T possesses a countable atomic model and a countable saturated model. These models must be isomorphic since T only has one countable model up to isomorphism.

Conversely, suppose that T has an atomic model N and a countable satur-

ated model M with N M . Since M is universal, any countable model of T can

=

be elementarily embedded into M . Since N M , any countable model of T can

=

be elementarily embedded into the atomic model. It follows that every countable model of T realizes only isolated types. So every type in S(T ) must be isolated and T is 0-categorical by Proposition 6.15.

as homogenous models and universal models, are countable. The following definitions extend these notions to uncountable structures.

Definition 6.44 Let M be a V-structure having universe U and theory T = T h(M ). Let κ be an infinite cardinal.

We say that M is κ-saturated if, for each A U with |A| < κ, every type in S1(A) is realized in M . We simply say that M is saturated if M is |M |-saturated.

We say that M is κ-universal if every model N of T with |N | < κ can be elementarily embedded into M . We simply say that M is universal if M is |M |-universal.

To extend the notion of notion of a homogeneous model, recall from Section 5.7 the definition of an M -elementary function. Note that a countable model M is homogeneous if and only every finite M -elementary function can be extended.

Definition 6.45 Let M be a structure and let κ be an infinite cardinal. We say that M is κ-homogeneous if, for each A U with |A| < κ and each a U , every M -elementary function f : A → U extends to an M -elementary function g : A {a} → U . We simply say that M is homogeneous if M is |M |-homogeneous.

Proposition 6.46 A structure M is κ-saturated if and only if M is both κ-homogeneous and κ-universal.

Proof This can be proved in the same manner as Proposition 6.41. We leave this as Exercise 6.30.

In particular, a model is saturated if and only if it is both homogeneous and universal. As with countable saturated models, we can use the homogeneity of saturated models to show that any two elementarily equivalent saturated models of the same cardinality must be isomorphic (see Exercise 6.31).

Now let T be a theory and suppose we wish to analyze the collection M od(T ) of all models of T . Suppose that we only care to consider models in M od(T ) of size less than κ. Since κ may be a ridiculously large cardinal, this is a reasonable assumption. If M is a saturated model of T of size κ, then we can replace the collection M od(T ) with the model M . Every structure in M od(T ) that we care to consider is an elementary substructure of M (since M is κ-universal). Moreover, any isomorphism between substructures of these models extends to an automorphism of M (by homogeneity and Exercise 6.20). Rather than considering the elements of M od(T ) as separate entities, the saturated model M allows us the convenience of working within a single model. Such a model is referred to as a monster model. Model theorists often use the preamble “we work inside of a monster model M . . . .”

Unfortunately, saturated models of large cardinalities may not exist. To guarantee the existence of arbitrarily large saturated models, we must assume set theoretic hypotheses beyond ZFC such as the General Continuum Hypothesis or (less severely) the existence of inaccessible cardinals. If we want to avoid such considerations, then we must settle for κ-saturated models instead of saturated models. Since they are both κ-universal and κ-homogeneous, κ-saturated models may serve as monster models. Although they are not necessarily homogeneous, κ-saturated models possess the fortunate property of existence. We prove this as the following proposition. The proof of this proposition also shows why saturated models may not exist. The κ-saturated model we construct is much larger than κ and so is not necessarily saturated.

Proposition 6.47 Let T be a complete theory having infinite models. Let κ be a cardinal. There exists a κ-saturated model of T .

Proof As in the proof of Proposition 6.37, we define an elementary chain of models M1 M2 M3 · · · . To begin, let M1 be any model of T . Given Mi, let Mi+1 be a model of T that realizes every type over every subset of the universe of Mi. If δ is a limit ordinal, let Mδ be the union of the chain of Mβ for β < δ. Consider the model Mα where |α| = κ. Any subset of size κ of the universe of Mα must also be a subset of Mβ for some β < α. Every type over A is realized in Mβ+1 Mα.

6.5 Theories with many types

Let T be a theory that is not small. By definition, |S(T )| is uncountable. We show that, in fact, |S(T )| = 2 0 . We use this fact to show that T has the maximal number of nonisomorphic countable models.

Lemma 6.48 Let T be a countable complete theory. Let P be an uncountable subset of S(T ). There exists a formula ψ such that both ψ and ¬ψ are contained in uncountably many types of P .

Proof Let V be the vocabulary of T . Let F (T ) denote the set of all V-formulas that occur in some p in S(T ). That is, F (T ) is the set of formulas that are realized in some model of T . Each formula in F (T ) is either contained in uncountably many types of P or countably many (possibly zero) types of P . Let {ϕ1, ϕ2, ϕ3, . . .} be the (possibly finite) set of those formulas in F (T ) that occur in only countably many types of P .

Let Pi be the set of all types in P that contain the formula ϕi. Then Pi is countable. Let Pϕ be the union of all the Pis. Since it is a countable union of countable sets, Pϕ is countable (by Proposition 2.43).

Now, Pϕ is the set of all types in P that contain ϕi for some i. Since P is uncountable, there must be uncountably many types in P that are not in Pϕ. Suppose it were the case that, for every formula ψ F (T ), either ψ Pϕ or ¬ψ Pϕ. Then there would be at most one type of in P not contained in Pϕ (namely, the type consisting of ¬ϕi for each i). So this cannot be the case and there must exist some formula ψ such that neither ψ nor ¬ψ is in Pϕ. By the definition of Pϕ, both ψ and ¬ψ are contained in uncountably many types of P .

Proposition 6.49 Let T be a countable complete theory. If T is not small, then |S(T )| = 2 0 .

Proof First, we show that |S(T )| ≤ 2 0 . Since T is countable, the set of all formulas in the vocabulary of T can be placed into one-to-one correspondence with N (by Proposition 2.47). Since each type is a set of formulas, |S(T )| ≤ |P(N)| = 2 0 .

Now suppose that T is not small. We show that |S(T )| ≥ 2 0 .

By definition, 2 0 is the cardinality of the set of all functions from N to the set {0, 1}. Each such function can be viewed as a denumerable sequence of 0s and 1s. For each of these sequences, we define a distinct type in S(T ).

Since T is not small, S(T ) is uncountable. By Lemma 6.48, there exists a formula ψ such that both ψ and ¬ψ are contained in uncountably many types of S(T ).

Let χ0 be ¬ψ and χ1 be ψ.

Let s be a finite sequence of 0s and 1s. For either i = 0 or i = 1, let s i be the sequence obtained by adding an i to the end of sequence s.

Suppose that we have defined a formula χs that is contained in uncountably many types in S(T ). Let Ps be the set of types in S(T ) that contain χs. By Lemma 6.48, there exists a formula ψ such that both ψ and ¬ψ are contained in uncountably many types of Ps.

Let χs 0 be χs ¬ψ and χs 1 be χs ψ.

In this manner we define a formula χs for each finite sequence s of 0s and 1s. By design, we have both T χs 0 → χs and T χs 1 → χs. Moreover, χs 0 and χs 1 cannot both be realized by the same elements of a model of T since one formula implies ψ and the other implies ¬ψ.

Let {0, 1}ω denote the set of all denumerable sequences of 0s and 1s. For t {0, 1}ω and n N, let t|n denote the first n terms of the sequence t. Let Γt be the set of all formulas χs such that s = t|n for some n N.

Claim For each t {0, 1}ω, Γt is realizable.

Proof By Proposition 6.6, it su ces to show that any finite subset of Γt is realizable. If ∆ is a finite subset of Γt, then, for some m N and every χs in ∆,

288 Models of countable theories

the sequence s has length less than m. Then T χt|m → χs for each χs in ∆. By definition, χt|m is contained in uncountably many types of S(T ). It follows that ∆ is contained in uncountably many types of S(T ). In particular, ∆ is realizable.

Let pt be a type in S(T ) containing Γt. If t1 and t2 are distinct sequences in {0, 1}ω, then pt1 and pt2 are distinct types in S(T ) since there exists a formula ψ such that ψ is contained in one of these types and ¬ψ is contained in the other. It follows that |S(T )| ≥ |{0, 1}ω| = 2 0 .

So, for any countable T , there are only two possibilities for |S(T )|. Either |S(T )| = 0 or |S(T )| = 2 0 . This is true even if the continuum hypothesis is false. Even if there exists cardinal numbers between 0 and 2 0 , the set S(T ) is forbidden from having such cardinalities.

We now show that if T is not small, then T has the maximal number of nonisomorphic models of size 0. First we compute this maximal number.

Proposition 6.50 Let T be a countable V-theory and let κ be an infinite cardinal. There exist at most 2κ nonisomorphic models of T of size κ.

Proof Let U be any set of size κ. Let us count the number of V-structures having U as an underlying set. Suppose we wish to define such a V-structure.

Given any constant c in V, we may interpret c as any element of U . There are |U | = κ many possibilities.

We may interpret each n-ary relation in V as any subset of U n. There are |P(U n)| = 2κ possible choices.

Finally, there are κκ = 2κ functions from U n to U . We may interpret each n-ary function in V as any one of these functions.

So each symbol in V can be interpreted in at most 2κ di erent ways on the set U . Since V is countable, there are at most 0 · 2κ = 2κ ways to interpret this vocabulary on U . That is, there are at most 2κ V-structures having underlying set U .

Let M be a model of T of size κ. Then there is a one-to-one correspondence between U and the underlying set of M . So, with no loss of generality, we may assume that M has underlying set U . It follows that there are at most 2κ models of T of size κ.

So a countable theory T can have at most 2 0 countable models. If T is not small, then it attains this maximal number.

Proposition 6.51 Let T be a countable theory. If |S(T )| = 2 0 then the number of nonisomorphic countable models is 2 0 .

Proof Suppose |S(T )| = 2 0 . Since each type in S(T ) is realized in some countable model (by the Downward L¨owenhiem–Skolem theorem), there must be at

least 2 0 countable models. By the previous proposition there are also at most this many countable models of T .

6.6 The number of nonisomorphic models

Let T be a theory. For any infinite cardinal κ, let I(T , κ) denote the number of nonisomorphic models of T of size κ. The function I(T , x) = y, restricted to infinite cardinals x, is called the spectrum of T . When restricted to uncountable cardinals, this function is called the uncountable spectrum of T . The spectra provide a natural classification of the class of first-order theories. For example, totally categorical theories are the theories having the constant function I(T , x) = 1 as a spectrum. We have also seen uncountably categorical theories T having spectrum

0, x = 0

I(T , x) =

1, x > 0.

The Baldwin–Lachlan theorem states that every uncountably categorical theory that is not totally categorical has this function as a spectrum.

Of course, there are many possible spectra. Let T be a countable complete theory that has infinite models. For any infinite cardinal κ, 1 ≤ I(T , κ) ≤ 2κ. The lower bound of 1 follows from the L¨owenhiem–Skolem theorems and the upper bound is from Proposition 6.50. It may seem that the possibilities for I(T , κ) are endless. It is a most remarkable fact that we can list the possible uncountable spectra for T .

Largely due to the work of Shelah, the uncountable spectra for the seemingly boundless and unmanageable class of countable first-order theories have been determined. Moreover, the work of Shelah shows that the spectrum of a given theory has structural consequences for the models of the theory. If a theory T has an uncountable spectrum other than the maximal I(T , κ) = 2κ, then the models of T have an inherent notion of independence. We defined “independence” for strongly minimal structures in Section 5.7. By Theorem 5.100, any strongly minimal T has uncountable spectrum I(T , κ) = 1. The notion of independence for strongly minimal theories generalizes to a class of theories known as the simple theories that includes all theories having uncountable spectra other than I(T , κ) = 2κ. For these theories, the notion of independence give rise to a system of invariants (analogous to dimension) that determine up to isomorphism the models of the theory. As humbling as it sounds, simple theories are beyond the scope of this book (as are supersimple theories). In the next section, we shall say a little more about simple theories and other classes of theories and provide references.