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

5.4.Show that the following theories are not finitely axiomatizable:

(a)The theory Ts of the integers with a successor function.

(b)The theory TRG of the random graph.

(c)The theory TACF 0 of algebraically closed fields of characteristic 0.

5.5.Let T be a complete VE -theory that contains the theory of equivalence relations TE . Show that T is finitely axiomatizable if and only if T has a finite model.

5.6.Let Γ1 be the set of V<-sentences that hold in every finite model of TLO. Let Γ2 be the set of sentences saying that there exist at least n elements for each n N. Let TF LO be the set V<-sentences that can be derived from Γ1 Γ2.

(a)Show that TF LO is a theory.

(b)Show that TF LO is quasi-finitely axiomatizable.

(c)Show that TF LO is not κ-categorical for any κ.

5.7.Let T1 and T2 be bi-definable theories each having finite vocabularies.

(a)Show that T1 is complete if and only if T2 is.

(b)Show that T1 is finitely axiomatizable if and only if T2 is.

(c)Show that T1 is quasi-finitely axiomatizable if and only if T2 is.

(d)Show that T1 is κ-categorical if and only if T2 is.

(e)Show that T1 is strongly minimal if and only if T2 is.

5.8.Let VP be the vocabulary consisting of a single unary relation P . Let T be a complete VP -theory having infinite models.

(a)Show that T is countable categorical.

(b)Give examples showing that T may or may not be totally categorical.

5.9.Show that there exists a complete quasi-finitely axiomatizable V-theory having infinite models for every finite vocabulary V.

5.10.For any first-order sentence ϕ, let Spec(ϕ) denote the finite spectrum of ϕ (as defined in Exercise 2.3). Show that either Spec(ϕ) or Spec(¬ϕ) is cofinite. (Hint: Use the previous exercise.)

5.11.Let VE be the vocabulary consisting of a single binary relation E. Let M be an infinite VE -structure that interprets E as an equivalence relation. Suppose that each equivalence class of M has the same size.

(a)Show that T h(M ) is countably categorical.

(b)Show that T h(M ) is uncountably categorical if and only if the equivalence classes are finite.

(c)Show that T h(M ) has quantifier elimination.

z(¬f (z) = y). Let M + be an expansion of M to a V+-structure that interprets f as a one-to-one and onto function and models ϕ+.

(d)Show that T h(M +) is finitely axiomatizable.

(e)Show that T h(M +) is not κ-categorical for any κ.

5.17.Let M and M + be as in Exercise 5.16.

(a)Show that M is minimal but not strongly minimal.

(b)Show that M + is not minimal.

5.18.Complete the proof of Proposition 5.58 by showing that T has quantifier elimination if and only if condition (ii) holds.

5.19.Let T be a countable complete theory. Show that T has quantifier elimination if and only if condition (ii) from Proposition 5.58 holds for all countable models M of T .

5.20.Let B be the set of all finite sequences of 0s and 1s (including the empty sequence). Let M = (B|S) be the structure in the vocabulary of a single binary relation S that interprets S as follows. For sequences s1 and s2 in B, M |= S(s1, s2) if and only if s2 is obtained by adding a 0 or a 1 to the end of s1. So S is a successor relation and every element of B has exactly two successors and at most one predecessor.

(a)Show that T h(B) is bi-definable with a model-complete theory that has a finite relational vocabulary. (Include a constant for the element having no predecessor.)

(b)Show that any theory in a finite relational vocabulary that is bidefinable with T h(B) cannot have quantifier elimination.

(c)Show that B is a strongly minimal structure.

5.21.Let VP s be the vocabulary consisting of denumerably many unary relations

P1, P2, P3, . . . and let I and O be disjoint finite subsets of N. Let ϕI,O(x)

be the VP s-formula i I Pi(x) i O ¬Pi(x).

This formula says that x is in each of the sets defined by Pi for i I and outside each of the sets defined by Pi for i O. Let TP be the VP s- theory axiomatized by the sentences saying that there exist at least n elements satisfying ϕI,O for each n in N and any finite disjoint subsets I and O of N.

(a)Show that TP has quantifier elimination.

(b)Show that TP is not κ-categorical for any κ.

5.22.An automorphism of a structure M is an isomorphism f : M → M from M onto itself. Let T be a countable complete theory.

(a)Suppose that, for any M |= T and tuples (a1, . . . , an) and (b1, . . . , bn) satisfying the same atomic formulas in M , there is an automorphism

f of M with f (ai) = bi for i = 1, . . . , n. Show that T has quantifier elimination.

(b)Suppose that T has quantifier elimination. Show that, for any M |= T and tuples (a1, . . . , an) and (b1, . . . , bn) satisfying the same atomic formulas in M , there exist an elementary extension N of M and an automorphism f of N with f (ai) = bi for i = 1, . . . , n.

5.23.Let TE = T h(M2) where M2 is the countable VE -structure defined in Example 5.18. Let Ts = T h(Zs) where Zs is the Vs-structure defined in Example 5.20.

We define a theory T that contains both of these theories. Let V be the vocabulary {E, s}. Let T be the set of all V-sentences that can be derived from the set TE Ts { x y(s(x) = y → E(x, y))}.

(a)Show that T is complete.

(b)Refer to Exercise 5.22. Demonstrate a model M of T and tuples (a1, . . . , an) and (b1, . . . , bn) from the universe of M such that

• (a1, . . . , an) and (b1, . . . , bn) satisfy the same atomic formulas in M , and

•there is no automorphism f of M for which f (ai) = bi for i = 1, . . . , n.

(c)Show that T has quantifier elimination.

5.24.Let T be a theory. Prove that T is model-complete if and only if, for any model M of T , T D(M ) is complete.

5.25.Let T be a theory. Let M be a model of T that can be embedded into any model of T . Show that if T is model-complete, then T is complete.

5.26.Show that T is model-complete if and only if, for any models M and N of T with M N , there exists an elementary extension M of M such that

M N M .

5.27.Let T be a model-complete theory. Let T be the set of all 2-sentences ϕ such that T ϕ. Show that M |= T if and only if M |= T .

(Hint: Show that every model of T has an elementary extension that is the union of a chain of models of T .)

5.28.Let T be a theory and let M be a model of T . Show that M is existentially closed with respect to T if and only if M is existentially closed with respect to T .

5.29.Show that the following theories have the amalgamation property:

(a)The theory of graphs TG.

(b)The theory of linear orders TLO.

(c)The theory of fields TF .

(a)Show that TRG is the model-companion of TG.

(b)Show that TDLO is the model-companion of TLO.

(c)Show that TACF is the model-companion of TF .

5.31.Refer to the previous two exercises. Prove that if T has the amalgamation property and T is the model-companion of T , then T has quantifier elimination.

5.32.Verify that aclM (aclM (A)) = aclM (A) for any structure M and any subset A of the underlying set of M .

5.33.Let M be a strongly minimal V-structure having underlying set U . Let ϕ(x, y) be a V-formula having two free variables. Show that there exists n N such that, for all a U : |ϕ(a, M )| is infinite if and only if |ϕ(a, M )| ≥ n. Show that this is not true for the minimal structure M from Exercise 5.17.

5.34.Let M be a structure and let f : A → B be an M -elementary function between subsets A and B of M . Show that A is algebraically closed if and only if B is algebraically closed.

5.35.Let G be a graph having a strongly minimal theory. Let a and b be vertices of G such that dimG(a, b) = 2. Let dG(a, b) be the length of the shortest path (in G) from a to b if such a path exists and ∞ otherwise. Prove that there are exactly three possible values for dG(a, b) (including ∞).

5.36.Let T be a strongly minimal theory. Show that the following are equivalent.

(i)T is locally modular.

(ii)If T is expanded by adding one constant to the vocabulary, then the result is modular.

(iii)Some expansion of T by constants is modular.

5.37.Let T be a strongly minimal theory. Show that the following are equivalent.

(i)T is modular.

(ii)If c aclM (A {b}), then c aclM ({a, b}) for some a A for any model M of T and any subset A {b} of the underlying set of M

with aclM (A) = A.

(Hint: To show (ii) implies (i) use induction on n = dimM (A).)

5.38.Let Rf = {R|f , <, +, ·, 0, 1} be an expansion of Ror where f is a unary function.

(a)Show that if Rf interprets f (x) as a polynomial, then Rf is o-minimal. (Use the fact that Ror is o-minimal.)

(b)Show that if Rf interprets f (x) as sin(x), then Rf is not o-minimal.

(c)For any real number x, the floor of x, denoted "x#, is greatest integer less than or equal to x. Show that if Rf interprets f (x) as "x#, then Rf is not o-minimal.

5.39.Let M be a V-structure and let A be a subset of the universe U of M . The definable closure of A in M , denoted dclM (A), is the set of all d U such that M |= x(x = d ↔ ϕ(x)) for some V(A)-formula ϕ(x). (The formula ϕ(x) is said to define the unique element d over A.) Show that if M is o-minimal, then dclM (A) = aclM (A) for all A U . Show that this is not necessarily true if M is strongly minimal.

5.40.Show that TRG is not uncountably categorical.

5.41.We randomly construct a graph having vertices V = {v1, v2, v3, . . .}. For each pair of vertices vi and vj , we flip a coin. If the coin lands heads up, vi and vj share an edge. Otherwise, they do not share an edge. Suppose that our coin is unfair. Say that our coin lands heads up only 1 out of 1000 times. Show that (after flipping the coin infinitely many times) the resulting random graph will be isomorphic to GR (with probability 1).

5.42.We define a graph having N as vertices. Any natural number n can

be uniquely factored as pa11 · pa22 · pa33 · · · pamm where the pis are distinct primes. We say that each of the exponents ai in this factorization are “involved in n.” We now define our graph: two natural numbers a and b share an edge if and only if either a is involved in b or b is involved in a. Show that the resulting graph is isomorphic to the random graph.

5.43. Show that for every substructure A of the random graph G , either G

R R =

A or GR (GR − A) (where (GR − A) is the substructure having the

=

vertices that are not in A as an underlying set).

5.44.Show that the 0–1 law fails for vocabularies that are not relational. (Hint: Consider the sentence xf (x) = x.)

5.45.Let TACF p be the Var-theory of algebraically closed fields of characteristic p (for prime p). Prove that, for any Var-sentence ϕ, the following are equivalent:

(i)TACF 0 |= ϕ,

(ii)TACF p |= ϕ for su ciently large primes p, and

(iii)TACF p |= ϕ for arbitrarily large primes p.

5.46.Algebraically closed fields of any characteristic are necessarily infinite. However, every finite subset of TACF p has arbitrarily large finite models for any prime p. Using this fact (and the previous exercise) prove Ax’s theorem.

Ax’s theorem: Let f (x) be a polynomial having complex coe cients. If f : C → C is one-to-one, then f is onto.

6Models of countable theories

We define and study types of a complete first-order theory T . This concept allows us to refine our analysis of M od(T ). If T has few types, then M od(T ) contains a uniquely defined smallest model that can be elementarily embedded into any structure of M od(T ). We investigate the various properties of these small models in Section 6.3. In Section 6.4, we consider the “big” models of M od(T ). For any theory, the number of types is related to the number of models of the theory. For any cardinal κ, I(T , κ) denotes the number of models in M od(T ) of size κ. We prove two basic facts regarding this cardinal function. In Section 6.5, we show that if T has many types, then I(T , κ) takes on its maximal possible value of 2κ for each infinite κ. In Section 6.6, we prove Vaught’s theorem stating that I(T , 0) cannot equal 2.

All formulas are first-order formulas. All theories are sets of first-order sentences. For any structure M , we conveniently refer to an n-tuple of elements from the underlying set of M as an “n-tuple of M .”

6.1 Types

The notion of a type extends the notion of a theory to include formulas and not just sentences. Whereas theories describe structures, types describe elements within a structure.

Definition 6.1 Let M be a V-structure and let a¯ = (a1, . . . , an) be an n-tuple of M . The type of a¯ in M , denoted tpM (¯a), is the set of all V-formulas ϕ(¯x) having free variables among x1, . . . , xn that hold in M when each xi in x¯ is replaced by ai. More concisely, but less precisely: tpM (¯a) = {ϕ(¯x)|M |= ϕ(¯a)}.

If a¯ is an n-tuple, then each formula in tpM (¯a) contains at most n free variables but may contain fewer. In particular, the type of an n-tuple contains sentences. For any structure M and tuple a¯ of M , tpM (¯a) contains T h(M ) as a subset. The set tpM (¯a) provides the complete first-order description of the tuple a¯ and how it sits in M . This description is not necessarily categorical; many tuples within the same structure may have the same type.

Example 6.2 Let Q< be the structure (Q| <) that interprets < as the usual order on the rational numbers. This structure is a model of the theory TDLO of dense linear orders discussed in Section 5.4. Consider the four-tuple (−2, −1, 1, 2). The

268 Models of countable theories

type tpQ< (−2, −1, 1, 2) contains the formulas x1 < x2, x2 < x3, and x3 < x4. Since TDLO has quantifier elimination, for any four-tuple a¯ = (a1, a2, a3, a4) of rational numbers, if a1 < a2 < a3 < a4 then tpQ< (¯a) is the same as tpQ< (−2, −1, 1, 2).

Definition 6.3 Let Γ be a set of formulas having free variables among x1, . . . , xn. A structure M realizes Γ if Γ is a nonempty subset of tpM (¯a) for some tuple a¯ of M . Otherwise, M is said to omit Γ. The set Γ is realizable if it is realized in some structure.

Note the distinction between the terms realizable and satisfiable. The set tpM (¯a) is realizable by definition, but rarely is tpM (¯a) satisfiable (see Exercise 6.4). This is because tpM (¯a) contains formulas that are not sentences. Recall that a formula ϕ(¯x) is equivalent to the sentence xϕ¯ (¯x). So when we say that a formula ϕ(x1, . . . , xn) is satisfiable, we mean that it holds for all n-tuples of a structure. When we say that ϕ(x1, . . . , xn) is realizable, we mean that it holds for some n-tuple of a structure.

We now define the key concept of this chapter.

Definition 6.4 An n-type is a realizable set of formulas having free variables among x1, . . . , xn. A type is an n-type for some n.

The sets tpM (¯a) are examples of types. Moreover, these are the only examples we need to consider. Every type is a subset of tpM (¯a) for some M and a¯. The types tpM (¯a) are called complete types. Types that are not complete are called partial types. We typically use p, q, and r to denote types (Γ is used to denote arbitrary sets of formulas). We often write a type with its free variables as p(x1, . . . , xn). The notation p(t1, . . . , tn) represents the set of formulas obtained by replacing each xi with the term ti.

Since types are generally not satisfiable, they are not consistent. This is unfortunate. Much of the previous chapters has been devoted to consistent sets of formulas. We can recover results from the previous chapters and apply them to types by making the following observation: the formula ϕ(x) is realizable if and only if the sentence ϕ(c) is satisfiable for some constant c. We state this more generally as the following proposition.

Proposition 6.5 Let Γ(x1, . . . , xn) be a set of formulas having free variables among x1, . . . , xn. Let c1, . . . , cn be constants not in the vocabulary of Γ. Then Γ(x1, . . . , xn) is realizable if and only if Γ(c1, . . . , cn) is satisfiable.

Proof Γ(x1, . . . , xn) is realizable if and only if

Γ(x1, . . . , xn) is a subset of tpM (¯a) for some M and a¯.

M that interprets the constants c1, . . . , cn as the tuple a¯ of M .

So for any realizable set of formulas, there is a closely related set of sentences that is satisfiable. This allows us to apply properties regarding satisfiability to types that are not satisfiable. In particular, the Compactness theorem remains true when “satisfiable” is replaced with “realizable.”

Proposition 6.6 Let Γ(x1, . . . , xn) be a set of formulas having free variables among x1, . . . , xn. Every finite subset of Γ is realizable if and only if Γ is realizable.

Proof Let c1, . . . , cn be constants not in the vocabulary of Γ.

By Proposition 6.5, Γ(x1, . . . , xn) is realizable if and only if Γ(c1, . . . , cn) is satisfiable.

By the Compactness theorem, Γ(c1, . . . , cn) is satisfiable if and only in every finite subset of Γ(c1, . . . , cn) is satisfiable.

Finally, again by Proposition 6.5, every finite subset of Γ(c1, . . . , cn) is satisfiable if and only if every finite subset of Γ(x1, . . . , xn) is realizable.

Let T be a complete theory. Any type that is realized in a model of T , whether it is partial or complete, is called a type of T . The set of all complete types of T is denoted S(T ). Equivalently, S(T ) is the set of all complete types that contain T as a subset. We denote by Sn(T ) the set of all n-types in S(T ).

Corollary 6.7 Let T be a complete theory and let Γ be a set of formulas having free variables among x1, . . . , xn. If each finite subset of Γ is a type of T , then Γ is an type of T .

Proof Apply Proposition 6.6 to the set Γ T .

Example 6.8 Let VE be the vocabulary consisting of a single binary relation E. Let M be the V-structure that interprets E as an equivalence relation having exactly one equivalence class of size n for each n N and no other equivalence classes. Let T = T h(M ). We depict M as follows: