Saturated Model Theory

The following sections are included:

• Contents

The intemperate young man who in 1972 wrote the Introduction below has vanished beyond recall. His successor has made numerous corrections, mostly typographical, some mathematical. The sweeping assertions and dubious jokes remain. Altering them would have been unjust to the author of long ago.

Ordinals are denoted by α,β,γ,δ,…; each ordinal is equal to the set of all lesser ordinals; thus α = {β|β < α}. 0 is the empty set. Cardinals are denoted by κ,ρ,μ, …; a cardinal is an ordinal that cannot be put into one-to-one correspondence with any lesser ordinal. The infinite cardinals in increasing order are: ω₀(=ω), ω₁,ω₂, … , ω_α,…; a set has cardinality κ if it can be put into one-to-one correspondence with κ. Card A is the cardinality of the set A. κ⁺ is the least cardinal greater than κ. A set is countable if it is finite or has cardinality ω. A successor ordinal is an ordinal of the form α+1. λ denotes a limit ordinal, i.e. an ordinal neither 0 nor a successor…

The coarsest classification of structures is by similarity type. Let N be the set of positive integers. A similarity type τ is a 5-tuple

such that θ: I → N and Ψ : J → N…

A monomorphism m of into , in symbols

is a one–one map (a₁ = a₂ iff ma₁ = ma₂) such that:…

Associated with each similarity type τ is a first order language . If is a structure of type τ, then each sentence of has a definite truth value in .…

Suppose τ = 〈I, J, K, β, θ〉 is the similarity type of 𝒜. Assume K ∩ A = 0. Define τA to be the similarity type

…

Let m be a map from A into B. m is an elementary monomorphism, in symbols

, means: iff for every formula F(x₁, …, x_n) of and every sequence a₁, … , a_n ∈ A.…

Let S be a set of sentences of some language ℒ_τ, and let F be a formula of ℒ_τ. F is a logical consequence of S, in symbols

, if F is among the formulas generated from S as follows:

(i) if F ∈ S, then S ⊢ F;

(ii) if F is a logical axiom, then S ⊢ F;

(iii) if S ⊢ F_i when 1 ≤ i ≤ n, and if F is the result of applying some logical rule of inference to the sequence F₁, …, F_n, then S ⊢ F…

The notion of model completeness was inspired by Hilbert's Nullstellensatz (9.2). A theory T is model complete (A. Robinson) if every monomorphism between models of T is elementary. The diagram of 𝒜, denoted by D𝒜, is the set of all atomic sentences and negations of atomic sentences true in 〈𝒜, a〉_a∈A. (If 𝒜 is a multiplicative group, then D𝒜 conveys the same information as the multiplication table for 𝒜.) The models of D𝒜 coincide with the extensions of 𝒜…

The theory of algebraically closed fields (ACF) extends the theory of fields (TF) by requiring that each nonconstant polynomial have a root. ACF is TF augmented by

fox each n > 0…

The direct limit operation is needed to erect structures, e.g. saturated models of ω-stable theories, whose existence is not immediate from 7.1…

Skolemization is a device for factoring monomorphisms.

Let ℒ be a language. Extend ℒ to ℒ^* by adding:

(i) a new individual constant C_F for each formula F(x₁) of ℒ;

(ii) a new n-place function symbol f_F for each formula F(x₁,…, x_n+1) of ℒ…

A. Robinson's notion of model completion is useful for theorizing about theories of fields and for resolving questions concerning the solvability of systems of equations. It will be used in Sec. 40 to justify the definition of differentially closed field and to derive Seidenberg's Nullstellensatz for differential fields…

A theory T is substructure complete if T ∪ D𝒜 is complete for every substructure 𝒜 of a model of T…

From now on every similarity type

will be countable; i.e. I, J and K will be countable sets. Thus every structure mentioned in every future section will have only countably many relations, functions and distinguished elements…

The notion of element type is needed to make a fine study of structures.

Let T be a complete theory. For each n > 0, let F_nT be the set of all formulas in the language of T whose free variables lie among x₁,…, x_n. Two formulas, F and G, of F_nT are called equivalent if T ⊢ F ↔ G. Let [F] be the equivalence class of F. B_nT is a Boolean algebra whose members are the [F]'s…

Saturated structures are useful when devising model theoretic versions of syntactic notions, as in the characterization of model complete theories afforded by 17.1, and when studying categoricity, as in the characterization of ω-categoricity provided by 18.2…

One of the classical applications of model theory to algebra is Tarski's elimination of quantifiers for the theory of real closed fields. Tarki's proof was based on an extension of Sturm's algorithm. The proof below is necessarily less constructive, since it is based on a saturated model criterion for model completeness, but it does give the algebraic details an organization that can be imposed on other theories of fields…

A structure 𝒜 is said to omit an n-type p if no member of Aⁿ realizes p in 𝒜. The results of Sec. 17 were straightforward because of the great ease with which structures can be extended to realize n-types. Another class of results, necessarily deeper, employs constructions in which structures are extended and selected n-types are omitted. Clearly it is more difficult to omit than it is to realize, since the act of omitting requires that every element of the extension be worried over. (A not well known model theorist once remarked: “Any fool can realize a type, but it takes a model theorist to omit one.”) In this and future sections a variety of techniques for omitting a type will be developed…

Let S₁𝒜 denote S₁T(〈𝒜, a〉_a∈A). Let T be a countable theory without any finite models. T is κ-stable (M. Morley) if card S₁𝒜 = κ whenever 𝒜 is a model of T of cardinality κ. The notion of ω-stability is absolute; in fact it is …

Let 𝒜 be infinite, X, Y ⊂ 𝒜, and f: X → Y be onto. f is an elementary partial automorphism of 𝒜 if

. Card f is defined to be card X…

Throughout this section T is a complete, countable theory without finite models. Let n(T) be the number of isomorphism classes of countable models of T. For each cardinal

, there is a T such that n(T) = κ…

Let ℒ be a countable language, and let R(x) be a distinguished formula of ℒ. If has the same similarity type as ℒ, then the two-cardinal type of is 〈κ,ρ〉, where , , and . (It is customary to call the distinguished subset of .)…

Let ℒ be a countable language, and let R(x) be a distinguished formula of ℒ. Suppose that B has the same similarity type as ℒ, and that card B > card R^B ≥ ω. By 22.6 there is a C ≡ B such that card C = ω₁ and card R^C = ω. Let κ be a regular uncountable cardinal such that 2^ρ ≤ κ for all ρ < κ. The argument below furnishes a C ≡ B such that card C = κ⁺ and card R^C = κ. The assumptions on κ are needed to obtain certain saturated structures of cardinality κ, structures that will be utilized to omit a type as were the countable homogeneous structures in the proof of 22.4…

Yet another way of omitting a type is by means of the completeness theorem (24.2) for ω-logic.

Let τ be a similarity type. Conjoined with τ is the first order language ℒ_τ defined in Sec. 4…

The Morley analysis of 1-types begins in Sec. 27. Its essential properties can be clearly stated in the language of categories and functors; so it is not surprising that similar analyses ([Sh1], [Sh2]) of 1-types have similar properties. The Morley derivative will be presented in Sec. 29 as an operation that acts on certain contravariant functors. The present section reviews all the category theoretic notions needed in Sec. 29…

The Morley analysis of l-types is centered on the notion of inverse limit. The key property of the Morley derivative, to be established in Sec. 29, is its commutativity with the inverse limit operation…

Let T be a substructure complete theory, and let Κ(T) be the category of all substructures of all models of T and all monomorphisms. Suppose 𝒜 ⊂ B ∈ Κ(T). B is a simple extension of 𝒜 if there is a b ∈ B such that B is the least substructure of B whose universe contains A ∪ {b}, in symbols 𝒜(b) = B…

The Morley derivative is an improved version of the Cantor–Bendixson derivative, improved in a sense to be made precise at the beginning of Sec. 29…

The Cantor–Bendixson derivative d is imperfect in the sense that the conclusion of 28.2 cannot be improved to read f[dY] = dX. The Morley derivative D is designed so that f[DY] = DX whenever f: Y → X belongs to ℋ…

Throughout this section κ is a category that admits direct limits and filtrations with amalgamation, and F: κ → ℋ is a contravariant functor that preserves limits. Fix 𝒜 ∈ κ. The computation of D^αF𝒜, the α-th Morley derivative of F𝒜, is in practice cumbersome, because its definition ranges over all of κ. In this section and the next conditions are given under which the value of D^αF𝒜 depends only on κ^*, a small full subcategory of κ containing 𝒜. The reduction of κ to κ^* is inspired by the downward going Skolem–Löwenheim theorem. The reduction will be at its most extreme in Sec. 31, where it will be seen that for many 𝒜's, the α-th Morley derivative of S𝒜 equals the α-th Cantor–Bendixson derivative of S𝒜 for all α…

Throughout this section T is a complete, substructure complete theory. Let S: κ(T) → ℋ be the contravariant Stone functor defined in Sec. 27. Thus for each 𝒜 a substructure of a model of T, S𝒜 is the Stone space whose points correspond to the isomorphism types over 𝒜 of the simple extensions of 𝒜. For each ordinal α, define D^αS, the α-th Morley derivative of S, as in Sec. 29. If there exists an α such that p ∈ D^αS𝒜 − D^α+1S𝒜, then p is a ranked point of S𝒜 of rank α. In symbols: rank p = α…

Throughout this section T is a complete, substructure complete theory. Let 𝒜 ∈ K(T). B is a prime model extension of 𝒜 if 𝒜 ⊂ B, B is a model of T, and the following diagram can be completed as shown whenever C is a model of T…

Morley created Theorem 33.1 in order to omit a type, but it turned out that Corollary 32.4 sufficed. It seems likely that 33.1 will have an essential use someday. If not, its existence is justified by its internal beauty…

Let T be a theory with an infinite model. A seemingly innocent question about T arising from quotidian experiences is: does T have an infinite model with a nontrivial automorphism? The answer, found by Ehrenfeuct and Mostowski, is yes. Their argument was based on the notion of indiscernibility. Later Morley discovered that indiscernibles could be used to omit a type, as in the proof of 37.2. They can also be used to construct isomorphisms, as in the proof of 36.1, and to define a notion of dimension for models of ω₁-categorical theories, as in Secs. 38 and 39. In this section they are needed to show that every theory categorical in some uncountable power is ω-stable…

The elements of I are indiscernible in if I ⊂ A and

whenever {i₁, …, i_n} and {i′₁,…, i′_n} are n-element subsets of I and F(x₁, …, x_n) is a formula in the language underlying . Suppose is an uncountable model of an ω-stable theory. Then for each regular , contains a set of indiscernibles of cardinality κ according to 35.7 and 31.6. If is a prime model extension of , then every set of indiscernibles in is countable by 35.9. The proofs of 35.9 and 35.10 demonstrate the power of the rank and degree machinery of Sec. 29…

Suppose T is totally transcendental, 𝒜 ∈ κ(T), and B₁ and B₂ are prime model extensions of 𝒜. Then B₁ and B₂ are isomorphic over 𝒜 according to Theorem 36.2. The isomorphism between B₁ and B₂ owes its existence principally to Lemma 35.9, which says no prime model extension of 𝒜 contains an uncountable set of indiscernibles over 𝒜. The definition of the isomorphism is by induction on a slight modification of Morley rank…

Morley's categoricity theorem (37.4) follows from a type omitting lemma (37.2) based on a downward going Skolem–Lőwenheim result (37.1)…

Minimal generators are the source of a dimension theory for models of ω₁-categorical theories. (The concept of minimal generator is a reformulation of W. Marsh's concept of minimal formula [Mal].) It will be seen in the next section that every model 𝒜 of every ω₁-categorical theory has a well-defined dimension in the same sense that every algebraically closed field has a dimension equal to the cardinality of any of its transcendence bases. Vaught's two-cardinal theorem is the key that unlocks the dimensionality of 𝒜…

Let T^* be a finitely generated extension of T as defined in Sec. 38. T^* is principal if T^* = T ∪ q, where q is a principal n-type of T. The next lemma is the essential modification of 38.1 needed to develop a dimension theory for ω₁-categorical theories that lack minimal generators…

The language of differential fields is the language of fields augmented by a l-place function symbol D. The theory of differential fields of characteristic 0 (DF₀) is the theory of fields of characteristic 0 increased by two axioms that relate to the derivative D:

…

Morley's rank and degree machinery can be applied directly to the theory of differentially closed fields of characteristic 0 (DCF₀), because 40.2 implies DCF₀ is complete and substructure complete…

The most promising area of model theory neglected in this book is infinitary logic and in particular ℒ_ω1,ω. The formulas of ℒ_ω1,ω are generated according to the rules of Sec. 4 and one further rule: if {F_i|i < ω} is a sequence of formulas, then &{F_i | i < ω} is a formula. Although ℒ_ω1,ω allows countably infinite conjunctions and disjunctions, it forbids infinitely long quantifier prefixes. (It follows that some formulas lack prenex normal equivalents.) The axioms and rules of inference of ℒ_ω1,ω are those alluded to in Sec. 7 augmented by an ineluctable infinitary rule: if F_i is a consequence of S for each i < ω, then &{F_i|i < ω} is a consequence of S…

The following sections are included:

• References

• Notation Index

• Index