# Recursive functions and existentially closed structures

The purpose of this paper is to clarify the relationship between various conditions implying essential undecidability: our main result is that there exists a theory T in which all partially recursive functions are representable, yet T does not interpret Robinson's theory R. To this end, we borrow tools from model theory---specifically, we investigate model-theoretic properties of the model completion of the empty theory in a language with function symbols. We obtain a certain characterization of ∃∀ theories interpretable in existential theories in the process.

## Authors

• 8 publications
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• ### On Rudimentarity, Primitive Recursivity and Representability

It is quite well-known from Kurt Gödel's (1931) ground-breaking result o...
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• ### On Stoltenberg's quasi-uniform completion

In this paper, we give a new completion for quasi-uniform spaces which g...
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• ### Sheaf-Theoretic Stratification Learning

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• ### Polymorphic Automorphisms and the Picard Group

We investigate the concept of definable, or inner, automorphism in the l...
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## 1 Introduction

First-order theories studied by logicians may be broadly divided in two classes. One class comprises theories of “arithmetical strength”, such as various fragments and extensions of Peano arithmetic, or set theories. They are distinguished by their great expressive power that, on the one hand, allows them to work with all kinds of objects from mathematical practice in suitable encoding (indeed, some of these theories are designed to serve as foundation for all of mathematics, e.g., ZFC), and on the other hand, makes them subject to Gödel’s incompleteness theorems and related phenomena. The other class are theories of “tame” structures, for example algebraically closed or real closed fields, vector spaces, generic structures such as the random graph, etc. These theories have low expressive power (often manifested in classification of definable sets stemming from partial quantifier elimination), and consequently their models have a manageable structure of geometric nature. Tame theories tend to be decidable.

The borderline between arithmetical and tame theories is not sharply demarcated, but one typical feature of arithmetical theories is their essential undecidability, meaning that all consistent extensions of the theory are undecidable. This notion was isolated by Tarski, Mostowski, and Robinson [9]. This classic monograph also includes convenient methods for proving essential undecidability of a theory , which can be viewed as stand-alone properties implying essential undecidability. In order of increasing strength, these are:

• can represent all partially recursive functions (prf; see below for a precise definition).

• can interpret Robinson’s theory .

• can interpret Robinson’s arithmetic , or equivalently, the adjunctive set theory.

An even stronger condition is that of being an sequential theory [7, 11].

Recall that Robinson’s is, essentially, a theory axiomatizing the true  sentences of the standard model of arithmetic ; while it is in some ways less convenient to work with than the better-known arithmetic  (e.g., is not finitely axiomatizable), it is distinguished by its interpretability properties—see Visser [12].

The above-mentioned conditions on theories form an increasing chain. For most of the inclusions in this chain, it is clear (or at least, reasonably well known) that the inclusions are strict: in particular, there are theories interpreting  that are not sequential (in fact, itself is such a theory [10]), does not interpret  (as is a finitely axiomatized theory with no finite model, whereas is locally finitely satisfiable), and there are essentially undecidable recursively axiomatized theories that do not represent prf. However, one of these inclusions is not as easy to resolve, leading to the question that motivated this paper:

###### Question 1.1

þ If a theory represents all partial recursive functions, does it interpret Robinson’s theory ?

This may look plausible at first sight: is a very weak theory that only fixes the values of elementary arithmetic operations on standard natural numbers, and requires virtually nothing else from the rest of the model. Now, the definition of representability of prf does provide for natural number constants and definable functions on them that behave like elementary arithmetic operations as these operations are prf, so everything seems to be in order.

Despite this, the answer turns out to be negative. The devil is in the “virtually nothing else”: does, after all, involve universally quantified conditions that may look innocuous (in our favourite formulation of , these universal quantifiers are bounded by a constant, hence ostensibly “finite”), but actually turn out to be crucially important. Using Visser’s [12] characterization, interprets nontrivial universal theories such as the theory of infinite discrete linear order. In contrast, prf can be represented in a theory axiomatized purely by quantifier-free sentences, with no universal quantifiers lurking behind.

We are going to prove that consistent theories with quantifier-free—or even existential—axioms cannot interpret infinite linear orders and a couple of similar universal theories, and a fortiori, cannot interpret . This is not easy to work out directly: the weakness of existential theories—which should intuitively be the reason for nonexistence of such interpretations—backfires in that we have absolutely no control over the complexity of formulas that make up potential interpretations, and over the sets they define in models.

Our strategy to solve this problem is to consistently extend the interpreting theory to a theory with quantifier elimination, using the fact that the empty theory in an arbitrary language  has a model completion (which we denote , being the theory of existentially closed -structures). This fact is well known for relational languages, in which case is the theory of the “random -structure”. However, we need it for languages with function symbols, which case is mostly neglected in common literature, though the existence of was proved in full generality already by Winkler [14].

It follows that if a theory is interpretable in a consistent quantifier-free or existential theory, it is weakly interpretable in for some , and the interpretation can be taken quantifier-free. In order to see that this is heading in the right direction, we establish a converse result: if an theory is weakly interpretable in , it is interpretable in a quantifier-free theory.

We proceed to prove that does not, actually, weakly interpret various theories of interest. At this point, we are heading further and further into model theory, having left the realm of arithmetical theories. It turns out that our non-interpretability results can be naturally expressed in the language of classification theory. Arising through the work of Shelah [8], classification theory studies the landscape of “dividing lines” between tame and wild theories, and their structural consequences. Many dividing lines have the following form: a theory is wild if it has a model that contains a certain complex combinatorial arrangement. Usually, conditions of this form can be reformulated as (weak) interpretability of a specific  theory. For a concrete example, a theory  has the strict order property () if there exists a model , a formula , and tuples for such that defines in  a strict partial order, and whenever . Otherwise, is said to have the no-strict-order property ().

We observe that theories that can represent recursive functions, as well as consistent extensions of for sufficiently rich languages , are moderately wild in that they always have the tree property . However, we will prove that (for arbitrary ) has certain tameness properties: specifically, it has the no-strong-order property (which implies ), and it has elimination of infinity. Using a characterization of theories by Chernikov and Ramsey [2], we show that it even has the property. On the other hand, theories interpreting  are firmly on the wild side of all generally considered dividing lines.

For completeness, the paper also includes discussion of basic model-theoretic properties of  in the appendix.

## 2 Preliminaries

Let us first agree on a few bits of general notation. We will use and  more or less interchangeably to denote the set of nonnegative integers; may also denote the standard model of arithmetic . We denote sequences by angle brackets, and consider them indexed starting from ; tuples of finite-but-unspecified length will be denoted by placing a bar over a variable name, so that may stand for the -tuple .

We will write to denote that is a partial function from to . (We use this notation in the context of partial recursive functions, so virtually always we will have , .)

The notation means that and  are syntactically identical terms; we may also apply it to formulas and other syntactic objects.

### 2.1 Theories and interpretations

In this paper, a language consists of an arbitrary number of relation and function symbols of arbitrary finite arity (including : nullary functions are constants, nullary relations are propositional variables). A theory is a deductively closed set of sentences in a particular language. A theory in language  is also called an -theory. We will often consider theories specified by a set of axioms, in which case the theory is taken to be their deductive closure; we will frequently omit outer universal quantifiers from axioms. We will generally employ a form of first-order logic that allows empty models.

Many considerations in this paper revolve around the notion of interpretation of one theory in another, so we need to be somewhat specific about its meaning. However, since a precise technical definition of interpretations would get quite lengthy, we advise the reader to consult e.g. Visser [12, §2] for the details if necessary; we will only indicate the main distinctive features below.

Let be a theory in a language , and a theory in a language . In its most simple form, a translation of language into language  is specified by:

• An -formula denoting the domain of .

• For each relation symbol of , as well as the equality relation , an -formula of the same arity.

• For each function symbol of  of arity , an -formula of arity .

If is an -formula, its -translation is an -formula constructed as follows: we rewrite the formula in an equivalent way so that function symbols only occur in atomic subformulas of the form , where are variables; then we replace each such atomic formula with , we replace each atomic formula of the form with , and we restrict all quantifiers and free variables to objects satisfying . We take care to rename bound variables to avoid variable capture during the process.

A translation of into  is an interpretation of in  if proves:

• For each function symbol of , the formula expressing that is total on :

 ∀x0,…,xk−1(δI(x0)∧⋯∧δI(xk−1)→∃y(δI(y)∧FI(¯¯¯x,y))). (1)
• The -translations of all axioms of , and axioms of equality.

It follows that proves the -translations of all sentences provable in .

The simplified picture of translations and interpretations above actually describes only one-dimensional, parameter-free, and one-piece translations. In the full generality, we allow the following:

• Translations may be multi-dimensional. That is, we use -tuples of -objects to represent -objects (where is a fixed natural number, called the dimension of the translation): thus, has  free variables, has  free variables for a -ary relation , and similarly for functions; and when constructing , each quantifier is replaced with a block of quantifiers.

• Translations may use parameters. This means that the formulas , , and may include parameter variables that are assumed distinct from any proper variables used in the target formulas, and the specification of includes an -formula that describes which parameters are admissible. Parameters carry through the translation unchanged, so they appear as free variables in . The definition of interpretation is modified so that proves for each axiom , and likewise for (1).

• Translations may be piece-wise: the interpreted domain of -objects may be stitched together from finitely many pieces (possibly of different dimensions, and possibly overlapping). Each piece has its own formula, there is a separate formula for each choice of a sequence of pieces for the arguments of , etc.

A translation  is called unrelativized if, on each piece, is a tautologically true formula, and it has absolute equality if, on each piece, is the formula .

Under suitable conditions, we do not need the full generality of interpretations:

• Assume that proves the existence of at least two distinct objects. Then whenever has an interpretation in , it also has a one-piece interpretation. (The new interpretation may have larger dimension, but needs no extra parameters.) This can be achieved by using the pattern of equalities on an extra tuple of variables to distinguish pieces. For this reason, we will mostly think of interpretations as one-piece, to avoid unnecessary technical baggage.

• If has a definable object, then an interpretation of in  may be converted to an unrelativized interpretation by “equating” tuples outside the original domain with the definable object. If we do not mind using extra parameters, the same can be achieved even if just proves the existence of at least one object. This construction may not be always desirable, hence relativized interpretations will remain the norm for us.

• A theory has (non-functional) pairing if there is a formula such that proves

 π(x,y,z)∧π(x′,y′,z)→x=x′∧y=y′ ∀x∀y∃zπ(x,y,z).

If has an interpretation in a theory with pairing, it also has a one-dimensional interpretation, as we can use single elements to code tuples.

If is a translation of language into , and a translation of language into , the composition is a translation of into , and it is defined in an expected way. Note that if is an interpretation of a theory in , and an interpretation of in , then is an interpretation of in .

Let and  be theories. Some variants on the notion of interpretation of in  are:

• A weak interpretation of in  is an interpretation of in a consistent extension of  (in the same language as ), or equivalently, in a completion of .

• A cointerpretation of in  is a translation of language  into  (sic!) such that implies for every -sentence .

• A faithful interpretation of in  is an interpretation of in  that is at the same time a cointerpretation of in .

A theory is interpretable (weakly interpretable, cointerpretable) in a theory  if there exists an interpretation (weak interpretation, cointerpretation, resp.) of in .

If and  are complete theories, a translation of in is an interpretation of in  iff it is a weak interpretation iff (assuming is parameter-free) it is a cointerpretation of in .

###### Lemma 2.1

þ If is a weak interpretation of in , and a cointerpretation of in , then is a weak interpretation of in .

An interpretation of in , as defined, is a syntactic transformation of formulas provable in  into formulas provable in . However, it can be also viewed semantically: it provides a uniform way of building “internally definable” models of  out of models of .

Assume first is a parameter-free one-piece interpretation with absolute equality, and let . We construct a model as follows: if is -dimensional, the domain of is ; a -ary relation symbol is realized in  by , and similarly, a -ary function symbol is realized by the function whose graph is the subset of defined in  by the formula .

Next, if does not have absolute equality, we build the structure as before, and let be its quotient by the binary relation defined on it by the formula ; this relation is in fact a congruence, as proves the translations of equality axioms.

If is a piece-wise interpretation, we construct the domain of as the disjoint union of the finitely many pieces, each defined as above; we define relations and functions in the appropriate way.

Finally, if is an interpretation with parameters, we will not obtain a single model , but one model for each choice of parameters: that is, if is a tuple such that , then is a model of  built from the expanded structure by the procedure above.

### 2.2 Representation of recursive functions

The notion of representable111In the terminology of [9], definable. We reserve the latter word for something else, in accordance with current standard usage. predicates and functions in first-order theories was introduced in [9]. We summarize it below, with a few inessential modifications. (Warning: we are going to relax the definition a bit later in this section.)

###### Definition 2.2

þ Let be a theory in a language , and a fixed sequence of numerals: i.e., a sequence of closed terms  such that

 T⊢n––≠m––

for , .

A recursive predicate (rp) is represented in  w.r.t.  by a formula if

 ⟨n0,…,nk−1⟩∈P ⟹T⊢¬φ(n0–––,…,nk−1–––––), ⟨n0,…,nk−1⟩∉P ⟹T⊢¬φ(n0–––,…,nk−1–––––)

for all .

A partial recursive function (prf) is represented w.r.t.  by a formula if

 T⊢φ(n0–––,…,nk−1–––––,y)↔y=m––

whenever are such that .

A set of prf and rp is representable in  if there exists a sequence of numerals  such that each member of  is representable in  w.r.t. .

In fact, [9] only consider representation of total recursive functions (trf), but it can be obviously generalized to partial functions in the indicated fashion. Likewise, we can generalize representation of rp to representation of disjoint pairs of r.e. predicates (dprp): such a disjoint pair , where , is represented by a formula  if

 ⟨n0,…,nk−1⟩∈P+ ⟹T⊢¬φ(n0–––,…,nk−1–––––), ⟨n0,…,nk−1⟩∈P− ⟹T⊢¬φ(n0–––,…,nk−1–––––)

for all . We identify any relation with the disjoint pair .

Notice that a representation of a rp

is essentially the same as a representation of its characteristic function

; likewise for disjoint pairs (their characteristic functions are partial). Consequently, representability of all prf in  implies representability of all trf and representability of all dprp; in turn, either of the latter two properties implies representability of all rp.

The definition of representation of functions does not demand anything from when is not one of the tuples in the domain of the original function. However, if represents a partial function in , we may define

 φ′(¯¯¯x,y) ≏∀z(φ(¯¯¯x,z)↔z=y), φ′′(¯¯¯x,y) ≏φ′(¯¯¯x,y)∨(y=0–∧¬∃zφ′(¯¯¯x,z)).

Then and also represent in ; moreover, is -provably a partial function, and is -provably a total function. Thus, we could have included either condition in the definition with no ill effects.

A desirable condition that we did not include in the definition is that the sequence of numerals  be recursive: that is, we can compute the term on input . For most purposes, this is actually redundant if can represent recursive functions with respect to : using a formula representing the (recursive) successor function , we can build a recursive sequence of formulas  that define .

þ2.2 formally makes sense for representation of arbitrary predicates or partial functions in . However, there is little point in that: if is recursively axiomatizable, and the given numeral sequence is recursive (or if we can represent ), then all predicates and total functions represented in  are actually recursive, and each partial function represented in  extends to a partial recursive function represented in . (This is not necessarily true for non-recursive numeral sequences, see þC.2.)

The primary reason for discussing representability of recursive functions in [9] is that it implies essential undecidability. We include the argument below for completeness.

###### Proposition 2.3

þ If the set of all unary rp is representable in a theory  w.r.t. a recursive sequence of numerals, then is essentially undecidable.

Proof:   Let be decidable. This makes the predicate

 P(n)⟺n is the G\"{o}del number of a formula φ(x) s.t.\ S⊢¬φ(n––)

recursive, hence is represented in  by a formula . Let be its Gödel number. If , then by representability, hence by the definition of , which is a contradiction. Thus, . Then by the definition of , and by representability, hence is inconsistent.

Again, the assumption of recursivity of the numeral sequence in þ2.3 may be replaced with representability of . However, it cannot be dropped entirely, as shown in the appendix (þLABEL:prop:nonrecnumseq,prop:nonrec-all).

Likewise, it is essential in þ2.3 that all unary rp are representable at once: we show in þC.1 that any finite (or uniformly recursive) set of rp and trf is representable in a decidable theory. In contrast, there is one fixed unary dprp (or: prf) whose representability in a theory w.r.t. a recursive numeral sequence implies essential undecidability: in fact, any recursively inseparable pair has this property.

The reader may have realized that representation of recursive functions and predicates in  amounts to an interpretation of a particular theory in . We now make this connection explicit.

###### Definition 2.4

þ Let be a set of prf and dprp. The language consists of constants , function symbols of appropriate arity for every prf , and likewise relation symbols for every dprp . The theory in language is axiomatized by

 n––≠m––

for ;

 F––(n0–––,…,nk−1–––––)=m––

for each -ary function , and such that ; and for each -ary disjoint pair , the axioms

 P––(n0–––,…,nk−1–––––)

for , and

 ¬P––(n0–––,…,nk−1–––––)

for . This definition also applies to rp using their identification with dprp .

Note that that the theory is axiomatized by open (= quantifier-free) sentences.

Let , , , and denote the sets of all prf, trf, dprp, and rp, respectively (where we consider and ). Since is included in an extension of by quantifier-free definitions, we will use as a proxy for .

For convenience, we also consider a finite-language formulation of . Let be the prf defined by

 U(0,m) =m+1, U(n+1,m+1) =[n,m], U([n,m]+1,0) ≃ϕn(m),

where denotes a recursive bijective pairing function (e.g., the Cantor pairing function ), and an efficient numeration of unary prf. Let be the fragment of  in the language ; it can be axiomatized by

 0–≠S(0–), U––(Sn(0–),Sm(0–))=Sk(0–)

for all such that , where denotes .

###### Lemma 2.5

þ is included in an extension of  by definitions of function symbols by terms, thus a theory interprets iff it interprets .

Proof:   We can read the definition of backwards to obtain definitions of , , and in terms of and : , , . Then any prf can be written in the form for a suitable .

Using the above-mentioned fact that representations of (partial) functions may be assumed to be actual definable functions, we see:

###### Observation 2.6

þ A set of prf and dprp is representable in a theory  according to þ2.2 iff is interpretable in  by a one-piece one-dimensional parameter-free interpretation  with absolute equality such that each is definable in  by a closed term.

Now, the restrictions on the interpretation in þ2.6 are mostly irrelevant and arbitrary; as we are looking at the concept of representations from the viewpoint of interpretability, it seems we obtain a cleaner concept if we just drop them:

###### Definition 2.7

þ A loose representation of in a theory  is an interpretation of in .

In particular, a theory loosely represents all prf iff it interprets the theory .

### 2.3 The theory R

Robinson’s theory was originally defined in [9]. Some inessential variants (mutually interpretable) of the theory appear in the literature; we prefer the following form in this paper.

###### Definition 2.8

þ Let denote the theory in the language axiomatized by

 n––+m––=n+m–––––––, (2) n––⋅m––=nm––––, (3) x

for all , where .

(In particular, note that axiom (4) for states .) It is easy to show that implies for distinct .

Observe that an -structure is a model of  iff it contains the standard model  as an initial (i.e., closed downward under ) substructure.

As usual, bounded quantifiers are introduced in  as the short-hands

 ∃y

where is a term not containing the variable . An -formula is (or bounded) if all quantifiers in  are bounded. A formula is if it consists of a block of existential quantifiers followed by a  formula.

###### Proposition 2.9

þ proves all  sentences true in the standard model . Conversely, it can be axiomatized by a set of true (universal)  sentences.

As already proved in [9] (for the original, slightly stronger definition of the theory), can represent recursive functions. We briefly sketch the argument below for completeness.

###### Proposition 2.10

þ Every prf is representable in  by a  formula w.r.t. the usual sequence of numerals as in þ2.8.

Proof:   The graph is definable in  by a  formula of the form , where . Put

 α(w) ≏0

One can check

 R⊢α(w)→w=1–∨⋯∨w=r–∨–r

for any .

We claim that represents in . Assume . On the one hand, is a true  sentence, and as such it is provable in . On the other hand, fix that witnesses the quantifier in . Working in , assume , we need to show . Let witness the existential quantifiers in . Using (5), either equals a standard numeral, or . In the latter case, implies as needed. In the former case, are also standard. It again follows that , as otherwise would be a true sentence, thus provable in .

Consequently, is essentially undecidable.

It is easy to see that (therefore any theory interpretable in ) is locally finitely satisfiable, i.e., every finite subset has a finite model: indeed, if we identify all elements of  above , we obtain a model satisfying (2), (3), and (4) for . Visser [12] proved a striking converse to this observation:

###### Theorem 2.11

þ Every locally finitely satisfiable, recursively axiomatizable theory in a finite language is interpretable in , using a one-piece one-dimensional parameter-free interpretation.

Since relational  sentences have the finite model property, this in particular implies that interprets any consistent theory axiomatized by a recursive set of  sentences in a finite relational language.

### 2.4 Model theory

Since this paper is intended to be accessible to a non-model-theoretic audience (and the author is not a model theorist either), it will only assume modest prerequisites in model theory—mostly common knowledge among logicians. The material needed should be covered by a textbook such as [3], except that we will also need a few concepts from classification theory. We will review a few selected topics in more detail below.

First, let us start with a few basic conventions. Recall that we allow models to be empty, and that we denote finite tuples as . For any structure , we denote by its diagram: the set of quantifier-free sentences true in  in the language of  augmented with constants for each element of . By a slight abuse of language, we will also use this notation to denote the set of quantifier-free sentences true in  in its original language, if every element of  is the value of a closed term (i.e., if is -generated).

Even though we normally work with one-sorted logic, the following construction is best thought of as yielding a multi-sorted structure. For any structure , let be the structure that has itself as one of its sorts, and for each equivalence relation on  definable without parameters in , it has a sort whose elements are the equivalence classes of ; the structure includes the projection function to this sort from . It is easy to see that each such equivalence relation is definable in  by a formula that provably defines an equivalence relation in predicate logic; thus, the following makes sense: for any theory , let be the multi-sorted theory whose models are exactly the structures for . (Officially, and can be coded in a suitable one-sorted language.) Note that is interpretable in , and any interpretation of another theory in  can be made into an interpretation with absolute equality of in .

Since we will work a lot with model completions, let us recall the related background. Let be a class of structures in the same language. A model is existentially closed (e.c.) in  if for every model such that , we have : i.e., every existential formula with parameters from  which is satisfied in  is already satisfied in . We will often speak of (absolutely) e.c. models without reference to , in which case it is understood that is the class of all models in the given language. An e.c. model of a theory  is an e.c. structure in the class of models of . If is a -axiomatized theory, then every model embeds in an e.c. model of . (More generally, this holds for any class closed under limits of chains.)

A theory  is model-complete if all models are e.c. models of ; this implies the stronger condition that for all , implies . Equivalently, is model-complete iff every formula  is in  equivalent to an existential formula; it is enough to test this for universal formulas . A stronger condition is that has quantifier elimination, meaning that every formula  is in  equivalent to a quantifier-free formula; it is enough to test this for existential formulas  with only one quantifier. Any model-complete theory  is axiomatizable by  sentences.

Theories and  in the same language are companions if every model of embeds in a model of , and vice versa; equivalently, , where denotes the universal fragment of . A model companion of a theory  is a model-complete theory  that is a companion of . There are theories with no model companion (e.g., the theory of groups), but if a theory  has a model companion , it is unique: the models of  are exactly the e.c. models of . A theory has a model companion iff the class of e.c. models of  is elementary. Notice that a model companion of  is the same thing as a model companion of , hence we can as well restrict attention to universal theories .

If is a theory, and an existential formula, the resultant is the set of all universal formulas such that . For any structure and , we then have: iff embeds in a model of  satisfying . It follows that the class of e.c. models of is axiomatized by the following set of infinitary formulas:

 T+{∀¯¯¯x(⋀ResTφ(¯¯¯x)→φ(¯¯¯x)):φ∈∃1}. (6)

A universal theory has a model companion  iff the resultants are finitely axiomatizable (over ) for all , in which case (6) provides an explicit axiomatization of .

A model completion of a theory  is a model companion of  such that for every , the theory is complete. Equivalently, a model companion of  is a model completion of  iff has the amalgamation property (cf. þB.4). If is a universal theory (which is the case we are primarily interested in), a companion of  is a model completion of  iff has quantifier elimination. A universal theory  has a model completion iff the resultant is equivalent to a quantifier-free formula over  for every .

For completeness, we also mention that every theory  has a unique Kaiser hull , which is a largest -axiomatized companion of . Any e.c. model of  is also a model of . If has a model companion , then .

A convenient trick when studying models of a complete theory  is to use monster models. A monster model of  is a model sufficiently rich so that all models we need to discuss can be assumed to be submodels of ; in order for this to work, we make highly saturated: to be specific, let us posit that is -saturated (i.e., every type over parameters from  is realized in ) and strongly -homogeneous (i.e., every partial elementary self-map of  of size extends to an automorphism of ), where is a “large” cardinal number (in particular, larger than the size of the language, as well as any models of  that we are going to encounter during the argument). This also implies that is -universal (every model of  of size elementarily embeds in ). (If it were not for foundational issues that we prefer not to be dragged into, we could even take as an “-saturated” model: a proper class model of  saturated w.r.t. types over any set of parameters.) Having fixed the monster model , a small set is a subset of  of size (likewise for sequences and other similar objects); a small model is an elementary submodel of  of size .

Finally, let us introduce a few notions from classification theory. Classification theory was developed by Shelah [8] (and subsequently many others) as an elaboration of Morley’s theory of stability; one of its main themes is identifying useful “dividing lines” between tame and wild theories. The dividing lines we are going to mention here are essentially of two kinds: first, variants of stability based on counting of types, and second, combinatorial properties based on the appearance of certain arrangements of points and definable sets in models.

While model theorists prefer to work with complete theories, the properties below are all stated in such a way that a theory  has a “tameness” property  iff every completion of  has property . Also, it will be generally the case that has a (tameness) property  iff every countable-language fragment of  has property .

For an overview of inclusions among the properties below, see Figure 1.

Theory  is -stable if for every and of size , there are at most complete types over . We say that is stable if it is -stable for some infinite cardinal , and superstable if it is -stable for all sufficiently large cardinals . An even stronger condition is that be totally transcendental; officially, this means that every formula  has Morley rank , but as we do not want to get into a definition of the rank, we can use the following characterization: is totally transcendental iff all countable-language fragments of  are -stable. For countable theories, -stability is equivalent to total transcendence, and implies -stability for all infinite cardinals ; the class of -stable theories includes uncountably categorical theories, which in turn include strongly minimal theories (meaning that for all , the only subsets of  definable with parameters are finite or cofinite).

Concerning the second kind of tameness properties, we will start with an alternative definition of stability as the first example. A theory has the order property () if there exists a formula (where and are tuples of the same length), a model , and a sequence of tuples in  such that

 M⊨φ(¯¯¯ai,¯¯¯aj)⟺i

for all ; otherwise, has the no-order property (). It turns out that has if and only if it is stable.

Beware of the terminological peculiarity that the base form of this condition on theories () is “negative” (wild), whereas the corresponding “positive” (tame) condition is denoted as its negation (). All properties below follow the same naming pattern.

A theory  has the independence property () if there is a formula , a model , and tuples and in  such that

 M⊨φ(¯¯¯ai,¯¯bI)⟺i∈I

for all and . Otherwise, is (also called dependent).

A theory  has the strict order property () if there is a formula , a model , and tuples in  such that

 M⊨∃¯¯¯x(φ(¯¯¯x,¯¯¯ai)∧¬φ(¯¯¯x,¯¯¯aj))⟺i

for all ; equivalently, is iff there is a formula that -provably defines a strict partial order, and there is a model in which the partial order defined by  has an infinite chain. Otherwise, is .

A theory is stable () if and only if it is both and . The class of theories that are but  includes o-minimal theories (i.e., theories featuring a linear order  such that every definable subset of the universe is a union of finitely many intervals, possibly degenerate).

Recall that denotes the set of finite sequences with entries from , ordered by the initial subsequence relation (which we write as ) to form an -branching tree; is the corresponding set of infinite sequences (which are branches of the tree). If , and , then is the initial subsequence of of length ; if and , then is extended with a new entry at the end. For clarity, we will write von Neumann numerals as .

A set of formulas is -inconsistent if each -element subset is inconsistent.

A theory  has the tree property () if there is a formula , a model , tuples in , and such that

• for each , the type is consistent, and

• for each , is -inconsistent.

Otherwise, is called or simple. Simplicity is officially defined in terms of properties of forking, but it is equivalent to ; there is a related stronger condition called supersimplicity, see e.g. [13]. Stable theories are simple, and simple theories are . Superstable theories are exactly the stable supersimple theories.

The tree property has two important variants. A theory  has the tree property if there is a formula , a model , and tuples in  such that

• for each , is consistent, and

• for each incomparable , is inconsistent.

has the tree property if there is a formula , a model , and tuples in  such that

• for each , is consistent, and

• for each such that , is inconsistent.

As usual, if is not , it is . A theory is if and only if it is both and . All theories are , and all theories are .

The region between simple and theories is further stratified by levels of the strong order property. For , a theory has the strong order property if there is a formula , a model , and tuples in  such that for all , but

 {φ(¯¯¯x0,¯¯¯x1),φ(¯¯¯x1,¯¯¯x2),…,φ(¯¯¯xk−2,¯¯¯xk−1),φ(¯¯¯xk−1,¯¯¯x0)} (7)

is inconsistent; otherwise, has . A theory has the strong order property if there are data as above such that (7) is inconsistent for all ; otherwise, has . For any theory , we have

 NTP1⟹NSO