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

systems of deduction such as resolution and formal proofs discussed in Chapter 3. In this chapter, we have shown that the rules of deduction for first-order logic entail many nice properties. These properties give rise to the model theory of the next two chapters. Because of these desirable properties, the language of first-order logic is necessarily weak. In particular, the Compactness theorem and Downward L¨owenheim–Skolem theorem impose limitations on the expressive power of first-order logic.

We claim that every property of first-order logic discussed in this chapter is a consequence of the Compactness theorem and the Downward L¨owenheim–Skolem theorem. The completeness of first-order logic can be deduced from compactness in the same manner that this is done in Theorem 1.80 for propositional logic. The theorems of Section 4.4 stating that infinite structures M and N can be amalgamated in some manner into structure D are direct consequences of compactness. The Downward L¨owenheim–Skolem theorem guarantees that there exists such D having the same size as M or N . Inspecting the proofs, we see that Robinson’s Joint Consistency lemma, the Beth Definability theorem, and the preservation theorems are consequences of compactness.

By compactness, there cannot exist a sentence of first-order logic that holds for infinite structures and only for infinite structures. By the Downward L¨owenheim–Skolem Theorem, there cannot exist a sentence of first-order logic that holds for uncountable structures and only uncountable structures. Because of these restrictions, there are basic concepts that first-order logic is incapable of expressing.

Example 4.73 In first-order logic, we cannot say that two definable subsets have the same size. To be precise, let V be a vocabulary that includes unary relations P and Q. For any V-structure M having underlying set U ,

let P (M ) = {a U |M |= P (a)} and let Q(M ) = {a U |M |= Q(a)}.

There is no set of V-sentences that says P (M ) and Q(M ) have the same size. In contrast, we can easily write sentences that say P (M ) and Q(M ) both have size n for any particular n. We can easily define a set of sentences that say P (M ) and Q(M ) are both infinite.

Note that V may contain symbols other than P and Q. For example, V may contain a unary function f . If this is the case, then we can write a V-sentence ϕf that says f is a one-to-one correspondence between P (M ) and Q(M ). The existence of such a bijection is precisely what it means for P (M ) and Q(M ) to have the “same size.” So if M |= ϕf , then |P (M )| = |Q(M )|. But the converse of this is not true. There exists N |= ¬ϕf such that |P (N )| = |Q(N )|. Likewise, there is no V-sentence (nor set of V-sentences) that holds if and only if P and Q define subsets of equal size.

To verify this, let Γ be a set of V-sentences. Suppose that |P (M )| = |Q(M )| for any model M of Γ. We show that |P (N )| = |Q(N )| for some V-structure N which does not model Γ. Let N0 be any V-structure such that P (N0) is uncountable and Q(N0) is denumerable. Then N1 |= ¬γ for some γ Γ. Let X be a subset of the universe U of N1 such that both X ∩ P (N1) and X ∩ Q(N1) are denumerable. By the Downward L¨owenheim–Skolem Theorem, there exists a countable elementary substructure N of N1 that contains X in its universe. Since N N1, we have N |= ¬γ. So N is a V-structure that does not model Γ for which |P (N )| = |Q(N )| = 0.

Example 4.74 Let G be a graph. Recall that a path in G from vertex a to vertex b is a sequence of adjacent vertices beginning with a and ending with b. The length of the path is one less than the number of vertices in the sequence (that is, the number of edges in the path). By Exercise 2.13, there exist formulas dn(x, y) expressing the existence of a path between vertices x and y of length n. In contrast, we claim that the concept of a path cannot be expressed in first-order logic. Whereas we can say there is a path of some specified length, we cannot say there is a path of arbitrary length. Suppose to the contrary that we have a formula φ(x, y) that holds of any vertices x and y in any graph G if and only if there exists a path from x to y in G. Consider the following set of sentences in a vocabulary containing R and constants a and b:

x yφ(x, y), ¬d1(a, b), ¬d2(a, b), ¬d3(a, b), . . .

The first sentence says that there is a path between any two vertices. This sentence holds in a graph if and only if the graph is connected. Since the other sentences assert that there is no path between a and b, this set of sentences is contradictory. However, any finite subset of these sentences is satisfiable. This contradicts the Compactness theorem. We conclude that the formula φ(x, y) cannot exist.

So there is no first-order formula that defines the concept of a path. Likewise, there is no first-order sentence that holds in a graph if and only if it is connected. Another basic graph-theoretic property is k-colorability. A graph is said to be k-colorable if the vertices of the graph can be colored with k colors in such a way that no two vertices of the same color share an edge. There does not exist a first-order sentence ϕk such that G |= ϕk if and only if G is a k-colorable graph. First-order logic cannot even say that there exists an even number of vertices in a finite graph. This is a consequence of the 0–1 law for first-order logic that is a subject of Section 5.4 of the next chapter.

This first-order impotence is by no means limited to graph theory. We list some of the many fundamental concepts from various areas of mathematics that first-order logic is incapable of expressing.

•Linear orders: there is no first-order sentence that holds for well ordered sets and only well ordered sets.

•Group theory: there is no first-order sentence that holds for simple groups and only simple groups.

•Ring theory: there is no first-order sentence that holds for Noetherian rings and only Noetherian rings.

•Metric spaces: there is no first-order sentence that holds for complete metric spaces and only complete metric spaces. In particular, the notion of a Cauchy sequence cannot be defined

To express these and other concepts, we must extend the logic. In Chapter 9, we consider extensions of first-order logic such as infinitary logics and second-order logic. Infinitary logics permit as formulas infinite conjunctions and disjunctions of first-order formulas. For example, consider the disjunction

want to say that two definable subsets have the same size as in Example 4.73. Second-order logic can express this. This logic allows quantification over subsets of the universe. Second-order logic is extremely powerful and can express each of the properties mentioned above.

Extending first-order logic comes at an expense. Since it can express the concept of a path, Lω1ω must not have compactness. Likewise, since secondorder logic can say that two definable sets have the same size, the Downward L¨owenheim–Skolem theorem must fail for this logic. Moreover, both compactness and completeness fail for second-order logic. Unlike first-order logic, we cannot list a set of rules from which we can deduce all truths of second-order logic. In this sense, the expressive power of second-order logic is too great.

The Compactness theorem and the Downward L¨owenheim–Skolem theorem make first-order logic the primary language of model theory. Model theory considers the relationship between a set of sentences T and the set of structures Mod(T ) that model T . Just as first-order logic can describe any finite structure up to isomorphism (by Proposition 2.81), infinitary logics and second-order logic can describe any countable structure up to isomorphism. This makes for an uninteresting model theory. If T is the second-order theory of a countable structure M , then M is the only structure in Mod(T ). Moreover, by the failure of completeness, we have no way to determine which sentences are in T .

Although there are basic concepts that cannot be defined in first-order logic, there are many concepts that can be defined. Moreover, we claim that those

properties that are first-order definable form a natural class of mathematical objects. The language of first-order logic, containing , , , and ¬ is a natural mathematical language to consider. First-order theories, which are the topic of the next two chapters, are natural objects of study. Since the Compactness and Downward L¨owenheim–Skolem theorems are central to model theory, we should consider the most powerful logic possessing these properties. By Lindstr¨om’s theorem, which we shall prove in Section 9.4, first-order logic is this logic. This theorem states that any extension of first-order logic for which both the Compactness and Downward L¨owenheim–Skolem theorems hold must be equivalent to first-order logic itself. So in some precise sense, first-order logic is the most powerful logic that possesses the properties discussed in this chapter.

Exercises

4.1.Let T be an incomplete countable theory. For each of the following, either prove the statement or provide a counter example.

(a)If T has an uncountable model, then T has a countable model.

(b)If T has arbitrarily large finite models, then T has a denumerable model.

(c)If T has finite models and a denumerable model, then T has arbitrarily large finite models.

4.2.Let T be an incomplete theory in an uncountable vocabulary. Repeat (a) and (b) from Exercise 4.1.

4.3.Let T1 be a complete V1-theory and let T2 be a complete V2-theory. Show

that T1 T2 is consistent if and only if ϕ1 ϕ2 is satisfiable for every

ϕ1 T1 and ϕ2 T2.

4.4.Let ϕ be a first-order sentence that is not contained in any complete theory. Show that {ϕ} ¬ϕ.

4.5.Let ϕ(x) be a quantifier-free V-formula. Let C = {c1, c2, c3, . . .} be a denumerable set of constants that do not occur in V. Let V(C) = V C. Show that the sentence xϕ(x) is a tautology if and only if the sentence

ϕ(t1) ϕ(t2) · · · ϕ(tn) is a tautology for some n N and V(C)-terms t1, . . . , tn.

4.6. Let V be a vocabulary containing denumerably many constants {c1, c2, c3, . . .}. Let T be a V-theory having the following two properties.

•If T |= xθ(x), then T |= θ(ci) for some i N.

•T |= ci = cj for any i, j N with i = j.

Show that T is complete.

4.7.Let T be an incomplete V-theory and let θ be a V-formula. Suppose that for each M |= T there exists a V-formula ϕM such that M |= θ ↔ ϕM .

Show that there exists finitely many V-formulas ϕ1, . . . , ϕn such that T

n

i=1(θ ↔ ϕi).

4.8.Let V be a vocabulary that contains only constants (and neither functions nor relations). Let M and N be two infinite V-structures. Using the TarskiVaught Criterion, show that if M N , then M N .

4.9.Let R be the structure (R|+, ·, 0, 1, <) having the real numbers as an underlying set that interprets the vocabulary in the usual manner.

(a)Show that there exists an elementary extension M of R that has infinitesimals (an element c is an infinitesimal if 0 < c < 1/n for each n N).

(b)Let UM be the underlying set of M . Show that the set of infinitesimals in UM has the same size as the set of infinite elements in UM (an element c is infinite if n < c for each n N).

4.10.Let N be the V-structure (N|+, ·, 1) from Exercise 2.7. By part (c) of Exercise 2.7, there exists a V-formula λ(x, y) such that, for any a and b in N, N |= λ(a, b) if and only if a < b. By the Upward L¨owenheim–Skolem theorem, N has an elementary extension M of cardinality 1.

(a)Let c be in the universe of M . Show that c is not in N if and only if M |= λ(n, c) for each n N. Call such an element c “infinite.”

(b)Show that there is no least infinite number in the universe of M . (That is, for every infinite c, there exists an infinite d such that M |= λ(d, c).)

(c)By part (b) of Exercise 2.7, there exists a V-formula π(x) such that, for any n N, N |= π(n) if and only if n is prime. Show that M |= π(c) for some infinite c. Call such a c an “infinite prime.”

(d)Show that there cannot be two consecutive infinite primes in the universe of M . (a and b are consecutive if a + 1 = b.)

(e)Let ϕ(x) be a V-formula. Show that the following are equivalent:

(i)N |= ϕ(n) for infinitely many n N.

(ii)M |= ϕ(c) for some infinite c.

(iii)There exists an elementary extension M1 of M such that

M1 |= ϕ(a) for 23 many elements a in its universe.

4.11.A graph is said to be k-colorable if the vertices can be colored with k di erent colors in such a way that no two vertices of the same color share an edge.

A graph is said to be planar if it can be drawn on the Euclidian plane in such a way that no two edges cross each other. The Four Color Theorem states that any planar graph is four-colorable. This famous theorem was

proved by Appel and Haken in 1976. Assuming that this theorem is true for finite graphs, prove that it is true for infinite graphs.

(Hint: Given an infinite planar graph G, consider the union of D(G) and a suitable set of V -sentences where V an expansion of VR containing unary relations representing each of the colors.)

4.12.The relation < is a partial order on a set A if

1.for all a and b in X, at most one of the following hold: either a < b, b < a, or a = b, and

2.for all a, b and c in X, if a < b and b < c then a < c.

If it is also true that either a < b or b < a for distinct a and b in A, then the partial order is a linear order. Using the compactness of first-order logic, show that any partial order on a set A can be extended to a linear order on A.

(Hint: First use induction to show that this is true for finite A.)

4.13.Let T be the set of all sentences in the vocabulary VR that hold in every connected graph. Show that there exists a model G of T that is not a connected graph.

4.14.Derive the Compactness theorem from the Completeness theorem.

4.15.Let T be the set of all sentences in the vocabulary V< = {<} that hold in every well ordered set. Show that there exists a model M of T that does not interpret < as a well ordering of the underlying set of M .

4.16.Let M be a V-structure having underlying set U . For any n-tuple a¯ of elements from U , let a¯ be the substructure of M generated by a¯ as defined in Exercise 2.34. Show that M can be embedded into a model of a theory T if and only if a¯ can be embedded into a model of T for every finite tuple a¯ of elements from U .

4.17.Let F be a set of formulas having an infinite vocabulary V. Show that

|F | = |V|.

4.18. Show that the order defined in the proof of Theorem 4.15 makes δ × δ a well ordered set.

4.19. For any set A of cardinals, let supA denote the least cardinal λ such that κ ≤ λ for each κ A. Let α be an infinite ordinal and let {κι | ι < α} be

aset of cardinals. Show that Σι<ακι = sup{|α|, κι|ι < α}.

4.20.Show that the following equalities hold for any ordinal α and any car-

dinal κ,

·|α|, and

ι<α Πι<ακ = κ|α|.

4.21.Prove that there are uncountably many countable ordinals.

4.22.Let α1 > α2 > α3 > · · · be a descending sequence of ordinals. Show that there can be only finitely many ordinals in this sequence.

4.23.Let T be a complete theory. Let α be a nonzero ordinal. For each β < α, let Mβ be a model of T .

(a)Show that there exists a model D of T such that each Mβ can be elementarily embedded into D.

(b)Show that we can find D in part (a) so that |D| ≤ |α|·|Mβ | for each β.

4.24.Prove Theorem 4.61.

4.25.Let T be a V-theory. Let T be the set of all universal sentences ψ such that T ψ. Let M be a V-structure that models T . Show that M can be embedded into a model of T .

4.26.Let T be a V-theory and let ϕ(x) and ψ(x) be two V-formulas. Suppose that, for any models M and N of T with N M ,

if M |= ϕ(a) then N |= ψ(a)

for any element a in the universe of N .

Show that there exists a universal V-formula θ(x) such that T ϕ(x) → θ(x) and T θ(x) → ψ(x).

4.27. Let T be an incomplete V-theory and let ϕ(¯x) be a V-formula having n free variables (for n N). Let M be a model of T having underlying set UM .

| | ¯

(a) Suppose that M = ϕ(¯a) if and only if M = ϕ(b) for any n-tuples a¯

¯ V

and b of elements of UM that satisfy the same atomic -formulas in M . Show that M |= ϕ(¯x) ↔ ψ(¯x) for some quantifier-free V-formula ψ(¯x).

(b)Show that ϕ(¯x) is not necessarily T -equivalent to a quantifier-free formula by providing appropriate example.

4.28.Let T be a V-theory and let ϕ(x1, . . . , xn) be a V-formula. Prove that the following are equivalent:

(i)ϕ(x1, . . . , xn) is T -equivalent to a quantifier-free formula.

(ii)For any model M of T and any V-structure C, if f : C → M and g : C → M are two embeddings of C into M , then

M |= ϕ(f (c1), . . . , f (cn)) if and only if M |= ϕ(g(c1, . . . , cn))

for any n-tuple of elements from the underlying set of C. (Hint: see Exercise 2.34.)

4.29.Prove Proposition 4.70.

4.30.For any V-theory T , let T be the set of 2 V-sentences that can be derived from T . Prove that the following are equivalent:

(i)T T .

(ii)If M is the union of a chain of models of T , then M |= T .

(iii)Let M be a V-structure having underlying set U . If for every a U , there exists N M such that a is in the universe of N and N |= T , then M |= T .

4.31.Let V be a vocabulary and let R be an n-ary relation not in V. Let T be an incomplete theory in the vocabulary V {R}. Suppose that, for each M |= T , there exists a V-formula ϕM (¯x) such that M |= R(¯x) ↔ ϕM (¯x). Prove that R is explicitly defined by T in terms of V. (Hint: see Exercise 4.7.)

4.32.(Lyndon) Refer to Exercise 2.33. A formula is said to be positive if it does not contain the symbols ¬, →, nor ↔. Let T be a V-theory and let ϕ be a V-formula. Show that the following are equivalent:

(i)ϕ is T -equivalent to a positive formula.

(ii)ϕ is preserved by every homomorphism f : M → N that is onto where both M and N are models of T .

4.33.(Lyndon) Let ϕ and ψ be V-sentences in conjunctive prenex normal form. A relation R is said to occur negatively in ϕ if ¬R occurs as subformula. Prove that if |= ϕ → ψ then there exists a V-sentence θ in conjunctive prenex normal form such that |= ϕ → θ, |= θ → ψ, and every relation that occurs negatively in θ also occurs negatively in both ϕ and ψ. (Hint: Modify the proof of Theorem 4.65.)

4.34. Let V1 and V2 be two vocabularies. Let V = V1 ∩ V2. Let M be a V1-structure, N be a V2-structure, C be a V-structure. Let f1 : C → M and f2 : C → N be V-elementary embeddings. Show that there exist

(V1 V2)-structure D,

V1-elementary embedding g1 : M → D, and V2-elementary embedding g2 : N → D

such that g1(f1(c)) = g2(f2(c)) for each c in the underlying set of C.

4.35.Derive Robinson’s Joint Consistency lemma from Compactness and Craig’s Interpolation theorems.

4.36.Show that the Beth Definability theorem holds for functions as well as relations.

4.37.Let M be the structure (Z|S) that interprets the binary relation S as the successor relation on the integers. Let N = (Z|S, <) be the expansion of M that interprets the binary relation < as the usual order. Let

T = T h(N ).

(a)Show that N is the only expansion of M to a the vocabulary {S, <} that models T .

(b)Show that < is not explicitly defined by T in terms of {S}.

5First-order theories

We continue our study of Model Theory. This is the branch of logic concerned with the interplay between sentences of a formal language and mathematical structures. Primarily, Model Theory studies the relationship between a set of first-order sentences T and the class Mod(T ) of structures that model T .

Basic results of Model Theory were proved in the previous chapter. For example, it was shown that, in first-order logic, every model has a theory and every theory has a model. Put another way, T is consistent if and only if Mod(T ) is nonempty. As a consequence of this, we proved the Completeness theorem. This theorem states that T ϕ if and only if M |= ϕ for each M in Mod(T ). So to study a theory T , we can avoid the concept of and the methods of deduction introduced in Chapter 3, and instead work with the concept of |= and analyze the class Mod(T ). More generally, we can go back and forth between the notions on the left side of the following table and their counterparts on the right.

Progress in mathematics is often the result of having two or more points of view that are shown to be equivalent. A prime example is the relationship between the algebra of equations and the geometry of the graphs defined by the equations. Combining these two points of view yield concepts and results that would not be possible in either geometry or algebra alone. The Completeness theorem equates the two points of view exemplified in the above table. Model Theory exploits the relationship between these two points of view to investigate mathematical structures.

First-order theories serve as our objects of study in this chapter. A firstorder theory may be viewed as a consistent set of sentences T or as an elementary class of structures Mod(T ). We shall present examples of theories and consider properties that the theories may or may not possess such as completeness, categoricity, quantifier-elimination, and model-completeness. The properties that a theory possesses shed light on the structures that model the theory. We analyze examples of first-order structures including linear orders, vector spaces, the random graph, and the complex numbers. In the final section, we use the modeltheoretic properties of the theory of complex numbers to prove a fundamental result of algebraic geometry.

As in the previous chapter, all formulas are first-order unless stated otherwise. In particular, all theories are sets of first-order sentences.

5.1 Completeness and decidability

We demonstrate several examples of theories in this section. Variations of these theories are used throughout this chapter to illustrate the concepts to be introduced. Although any consistent set of sentences forms a theory, we typically restrict our attention to those theories that are deductively closed.

Definition 5.1 Let Γ be a set of sentences. The deductive closure of Γ is the set of all sentences that can be formally derived from Γ. If Γ equals its deductive closure, then Γ is said to be deductively closed.

Given a deductively closed theory, we consider the question of whether or not the theory is complete. To show that a V-theory T is complete, we must show that, for every V-sentence ϕ, either ϕ T or ¬ϕ T . It is a much easier task to show that T is incomplete. To accomplish this, it su ces to produce only one sentence ϕ such that neither ϕ nor ¬ϕ is in T . Instead of considering V-sentences, we can consider V-structures. To show that T is incomplete, it su ces to find two models of T that are not elementarily equivalent. This is also a necessary condition for T to be incomplete.

Proposition 5.2 Let T be a deductively closed theory. Then T is incomplete if and only if there exist models M and N of T that are not elementarily equivalent.

Proof First suppose that T is incomplete. Then there exists a sentence ϕ such that neither ϕ nor ¬ϕ is in T . Since T is deductively closed, neither ϕ nor ¬ϕ can be derived from T . This happens if and only if both T {ϕ} and T {¬ϕ} are consistent. By Theorem 4.27, if these sets of sentences are consistent, then they are satisfiable. So if T is incomplete, then, for some V-sentence ϕ, there