1 Introduction
Firstorder 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 standalone 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 betterknown arithmetic (e.g., is not finitely axiomatizable), it is distinguished by its interpretability properties—see Visser [12].
The abovementioned 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 quantifierfree sentences, with no universal quantifiers lurking behind.
We are going to prove that consistent theories with quantifierfree—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 quantifierfree or existential theory, it is weakly interpretable in for some , and the interpretation can be taken quantifierfree. 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 quantifierfree 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 noninterpretability 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 nostrictorder 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 nostrongorder 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 modeltheoretic 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 finitebutunspecified 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 firstorder 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 :
(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 onedimensional, parameterfree, and onepiece translations. In the full generality, we allow the following:

Translations may be multidimensional. 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 piecewise: 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 onepiece 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 onepiece, 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 (nonfunctional) pairing if there is a formula such that proves
If has an interpretation in a theory with pairing, it also has a onedimensional 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 parameterfree) 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 parameterfree onepiece 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 piecewise 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 representable^{1}^{1}1In the terminology of [9], definable. We reserve the latter word for something else, in accordance with current standard usage. predicates and functions in firstorder 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
for , .
A recursive predicate (rp) is represented in w.r.t. by a formula if
for all .
A partial recursive function (prf) is represented w.r.t. by a formula if
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
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
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 nonrecursive 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
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:nonrecall).
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
for ;
for each ary function , and such that ; and for each ary disjoint pair , the axioms
for , and
for . This definition also applies to rp using their identification with dprp .
Note that that the theory is axiomatized by open (= quantifierfree) 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 quantifierfree definitions, we will use as a proxy for .
For convenience, we also consider a finitelanguage formulation of . Let be the prf defined by
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
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 abovementioned 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 onepiece onedimensional parameterfree 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
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
(2)  
(3)  
(4) 
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 shorthands
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
One can check
(5) 
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 onepiece onedimensional parameterfree 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 nonmodeltheoretic 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 quantifierfree 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 quantifierfree 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 onesorted logic, the following construction is best thought of as yielding a multisorted 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 multisorted theory whose models are exactly the structures for . (Officially, and can be coded in a suitable onesorted 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 modelcomplete if all models are e.c. models of ; this implies the stronger condition that for all , implies . Equivalently, is modelcomplete 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 quantifierfree formula; it is enough to test this for existential formulas with only one quantifier. Any modelcomplete 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 modelcomplete 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:
(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 quantifierfree 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 selfmap 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 countablelanguage 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 countablelanguage 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
for all ; otherwise, has the noorder 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
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
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 ominimal 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
(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
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