Scott Ranks of Models of a Theory
Abstract.
The Scott rank of a countable structure is a measure, coming from the proof of Scott’s isomorphism theorem, of the complexity of that structure. The Scott spectrum of a theory (by which we mean a sentence of ) is the set of Scott ranks of countable models of that theory. In we give a descriptivesettheoretic classification of the sets of ordinals which are the Scott spectrum of a theory: they are particular classes of ordinals.
Our investigation of Scott spectra leads to the resolution (in ) of a number of open problems about Scott ranks. We answer a question of Montalbán by showing, for each , that there is a theory with no models of Scott rank less than . We also answer a question of Knight and Calvert by showing that there are computable models of high Scott rank which are not computably approximable by models of low Scott rank. Finally, we answer a question of Sacks and Marker by showing that is the least ordinal such that if the models of a computable theory have Scott rank bounded below , then their Scott ranks are bounded below .
1. Introduction
Scott [Sco65] showed that every countable structure can be characterized, up to isomorphism, as the the unique countable structure satisfying a particular sentence of the infinitary logic , called the Scott sentence of . Scott’s proof gives rise to a notion of Scott rank for structures; there are several different definitions, which we will discuss later in Section 2.1, but until then we may take the Scott rank of to be the least ordinal such that has a Scott sentence. This paper is concerned with the following general question: given a theory (by which we mean a sentence of ) what could the Scott ranks of models of be? This collection of Scott ranks is called the Scott spectrum of :
Definition 1.
Let be an sentence. The Scott spectrum of is the set
This is an old definition. For example, in 1981, Makkai [Mak81] defined the Scott spectrum of a theory in this way and showed that there is a sentence of without uncountable models whose Scott spectrum is unbounded below . In [Vää11, p. 151] a reference is made to gaps in the Scott spectrum—ordinals which are not in the Scott spectrum, but which are bounded above by some other in the Scott spectrum—but the only results proved about Scott spectra are about bounds below . This seems to be a general pattern: whenever Scott spectra are mentioned in the literature, it is to say that they are either bounded or unbounded below . This paper, to the contrary, is about the gaps, and about a classification of the sets of countable ordinals that can be Scott spectra. Our main result is a complete descriptivesettheoretic classification of the sets of ordinals which are Scott spectra. For this classification, we assume projective determinacy.
This work began with the following question, first asked by Montalbán at the 2013 BIRS Workshop on Computable Model Theory.
Question (Montalbán).
If is a sentence, must have a model of Scott rank two or less?
At the time, we knew very little about how to answer such questions. In this paper, we make a large step forward in our understanding of Scott spectra: not only do we answer the question negatively, but we also answer the generalization to any ordinal and we apply those techniques to solve other open problems about Scott ranks.
The paper is in two parts. The first part is a general construction in Section 3. Given a pseudoelementary class of linear orders, we build an sentence so that the Scott spectrum of is related to the set of wellfounded parts of linear orders in that class. The construction bears some similarity to work of Marker [Mar90]. In the second part, we apply the general construction to get various results about Scott spectra. We will describe these applications now.
1.1. theories with no models of low Scott rank
It follows easily from known results that for a given ordinal , there is a theory all of whose models have Scott rank at least . (We can, for example, take to be the Scott sentence of a model of Scott rank .) This is not very surprising, as the theory we get has quantifier complexity about ; complicated theories may have only complicated models. The interesting question is whether there is an uncomplicated theory all of whose models are complicated. Such theories exist.
Theorem 2.
Fix . There is a sentence whose models all have Scott rank .
1.2. Computable structures of high Scott rank
Nadel [Nad74] showed that if is a computable structure, then its Scott rank is at most . We say that a computable structure with noncomputable Scott rank, i.e. with Scott rank or , has high Scott rank. There are few known examples of computable structures of high Scott rank. Harrison [Har68] gave the first example of a structure of Scott rank : the Harrison linear order , which is a computable linear order of order type . The Harrison order is the limit of the computable ordinals in the following sense: given a computable ordinal, there is a computable ordinal such that . We say that such a structure is strongly computable approximable:
Definition 3.
A computable structure of noncomputable rank is weakly computably approximable if every computable infinitary sentence true in is also true in some computable . is strongly computably approximable if we require that have computable Scott rank.
Makkai [Mak81] gave the first example of an arithmetic structure of Scott rank , and Knight and Millar [KM10] modified the construction to get a computable structure. Calvert, Knight, and Millar [CKM06] showed that this structure is also strongly computably approximable. Calvert and Knight [CK06, Problem 6.2] asked the following question:
Question (Calvert and Knight).
Is every computable model of high Scott rank strongly (or weakly) computably approximable?
At the time, every known example of a computable structure of high Scott rank was strongly computably approximable. We show here that there are computable structures of Scott rank and which are not strongly computably approximable.
Theorem 4.
For or : There is a computable model of Scott rank and a sentence such that , and whenever is any structure and , has Scott rank .
1.3. Bounds on Scott Height
It follow from a general counting argument that there is a least ordinal such that if is a computable sentence whose Scott spectrum is bounded below , then the Scott spectrum of is bounded below . We call this ordinal the Scott height of , and we denote it .
Question (Sacks and Marker).
What is the Scott height of ?
Definition 5.
is the least ordinal which has no presentation.
Sacks [Sac83] showed that . Marker [Mar90] was able to resolve this question for pseudoelementary classes.
Definition 6.
A class of structures in a language is an pseudoelementary class (class) if there is an sentence in an expanded language such that the structures in are the reducts to of the models of . is a computable class if is a computable sentence.
We can define the Scott height of in a similar way to the Scott height of , except that now we consider all pseudoelementary classes which are the reducts of the models of a computable sentence. Marker [Mar90] showed that . Using our methods, we can expand this argument to .
Theorem 7.
.
We prove this theorem in Section 7.
1.4. Classifying the Scott spectra
Assuming projective determinacy, we will define a descriptive settheoretic class which will give a classification of the Scott spectra.
Definition 8.
A set of countable ordinals is a class of ordinals if it consists of the order types in for some class of linear orders on .
Note that and here are classes of presentations of ordinals as linear orders of . Frequently we will pass without comment between viewing a class as a collection of ordinals, i.e., of order types, and as a collection of presentations of linear orders.
Theorem 9 (Zfc + Pd).
The Scott spectra of sentences are the classes of ordinals with the property that, if is unbounded below , then either is stationary or is stationary.
We can also get an alternate characterization which is more tangible. To state this, we must define two ways to produce an ordinal from an arbitrary linear order.
Definition 10.
Let be a linear order. The wellfounded part of is the largest initial segment of which is wellfounded. The wellfounded collapse of , , is the order type of after we collapse the nonwellfounded part to a single element.
We can identify with the ordinal which is the order type of . We can also identify with its order type. If is wellfounded, with order type , then . If is not wellfounded, .
Theorem 11 (Zfc + Pd).
The Scott spectra of sentences are exactly the sets of the form:

,

, or

where is a class of linear orders of .
Theorem 12 (Zfc + Pd).
Each Scott spectrum is the Scott spectrum of a sentence.
Theorem 13 (Zfc + Pd).
Every Scott spectrum of a class is the Scott spectrum of an sentence.
2. Preliminaries on Backandforth Relations and Scott Ranks
All of our structures will be countable structures in a countable language. The infinitary logic consists of formulas which allow countably infinite conjunctions and conjunctions; see [AK00, Sections 6 and 7] for background. We will use for the infinitary formulas and for the computable infinitary formulas (and similarly for and ).
2.1. Scott Rank
Let be a countable structure. There are a number of ways to define the Scott rank of , not all of which agree. We describe a number of different definitions before fixing one for the rest of the paper. For the most part, it does not matter, modulo some small changes, which definition we choose as our results are quite robust.
The first definition uses the symmetric backandforth relations which come from Scott’s proof of his isomorphism theorem [Sco65]. See, for example, [AK00, Sections 6.6 and 6.7].
Definition 14.
The standard symmetric backandforth relations on , for , are defined by:

if and satisfy the same quantifierfree formulas.

For , if for each and there is such that , and for all there is such that .
For each tuple , Scott proved that there is a least ordinal , the Scott rank of the tuple, such that if , then and are in the same automorphism orbit of . Equivalently, is the least ordinal such that if , then for all ordinals , or such that if , then and satisfy the same formulas. Then the Scott rank of is the least ordinal strictly greater than (or, in the definition used by Barwise [Bar75], greater than or equal to) the Scott rank of each tuple of . One can then define a Scott sentence for , that is, a sentence of which characterizes up to isomorphism among countable structures.
Another definition uses the nonsymmetric backandforth relations which have been useful in computable structure theory. See [AK00, Section 6.7].
Definition 15.
The standard (nonsymmetric) backandforth relations on , for , are defined by:

if for each quantifierfree formula with Gödel number less than the length of , if then .

For , if for each and there is such that .
Let if and ,
For , if and only if every formula true of is true of .
Then one can define the Scott rank of a tuple to be the least such that if , then and are in the same automorphism orbit of . The Scott rank of is then least ordinal strictly greater than the Scott rank of each tuple.
A third definition of Scott rank has recently been suggested by Montalbán based on the following theorem:
Theorem 16 (Montalbán [Mon]).
Let be a countable structure, and a countable ordinal. The following are equivalent:

has a Scott sentence.

Every automorphism orbit in is definable without parameters.

is uniformly (boldface) categorical without parameters.

Every type realized in is implied by a formula.

No tuple in is free.
Montalbán defines the Scott rank of to be the least ordinal such that has a Scott sentence. It is this definition which we will take as our definition of Scott rank. We write for the Scott rank of the structure . The free tuples which appear in the theorem above will also appear later.
Definition 17.
Let be a tuple of . Then is free if for each and , there are and such that and .
2.2. Scott Spectra
Recall that the Scott spectrum of an sentence is the set of countable ordinals
More generally, one can define the Scott spectrum of a class of countable structures . For each there is an sentence whose Scott spectrum is . For example, if is a structure of Scott rank ,^{1}^{1}1Such structures exist; for example, the results on linear orders in [AK00, Section 15] can be used to construct examples, or one can use the construction in [CFS13]. then we can take to be the Scott sentence for . However, the quantifier complexity of will be approximately . It is only as a result of our Theorem 2 that one can obtain such a theory of low quantifier complexity even when is very large.
We note some results about producing new Scott spectra by combining existing ones. The proofs are all simple constructions which we omit.
Proposition 18.
If are the Scott spectra of sentences, then is also the Scott spectrum of an sentence.
Proposition 19.
If is the Scott spectrum of an sentence and , then is also the Scott spectrum of an sentence.
Proposition 20.
Let and be sets of countable ordinals, and suppose that is the Scott spectrum of an sentence. If there is a countable ordinal such that
then is also the Scott spectrum of an sentence.
2.3. Nonstandard BackandForth Relations
Let be a linear order. We will consider to be a nonstandard ordinal, i.e., a linear ordering with an initial segment which is an ordinal, but whose tail may not necessarily be wellordered. Assume that has a smallest element .
Definition 21.
A sequence of equivalence relations are nonstandard backandforth relations on if they satisfy the definition of the standard backandforth relations (Definition 15), that is, if:

If is the smallest element of , if for each quantifierfree formula with Gödel number less than the length of , if then .

If is not the smallest element of , if for each , for all there is such that .
While the standard backandforth relations are uniquely defined, this is not the case for nonstandard backandforth relations. However, they are uniquely determined on the wellfounded part of .
Remark 22.
Let be a linear order and a sequence of nonstandard backandforth relations on . The relations for are the same as the standard backandforth relations on .
For nonstandard , that is, , the backandforth relations hold only between tuples in the same automorphism orbit.
Lemma 23.
Let be a linear order and a sequence of nonstandard backandforth relations on . For , if , then there is an isomorphism of taking to .
Proof.
It is easy to see that
is a set of finite maps with the backandforth property. If for some , then and satisfy the same atomic sentences. Thus any such map extends to an automorphism. ∎
2.4. Admissible ordinals and Harrison linear orders
Given , is the least noncomputable ordinal. By a theorem of Sacks [Sac76], the countable admissible ordinals are all of the form for some set . For our purposes, we may take this as the definition of an admissible ordinal.
Harrison [Har68] showed that for each , there is an computable ordering which is not wellordered, but which has no hyperarithmetic descending sequence. Moreover, any such ordering is of order type for some computable ordinal . We call the Harrison linear order relative to . Note that the property of being the Harrison linear order relative to is : a linear order is the Harrison linear order relative to if:

it is computable,

for every computable ordinal and element , there is such that the interval has order type ,

it has a descending sequence, and

for every computable ordinal and index there is a jump hierarchy on which witnesses that is not a descending sequence.
Later we will use the fact that the set of admissible ordinals contains a club.
Definition 24.
A set is closed unbounded (club) if it is unbounded below and is closed in the order topology, i.e., if , then .
Definition 25.
A set is stationary if it intersects every club set.
Remark 26.
Given a set , the set of such that is an elementary substructure of is a club. Hence the set contains a club. (Recall also that every club is a stationary set.)
3. The Main Construction
In this section we will do the main work of this paper by giving the general construction used in the applications. Given an pseudoelementary class of linear orders, we will build a theory whose models have Scott ranks in correspondence with the linear orders in .
Theorem 27.
Let be a class of linear orders. Then there a an sentence such that
Moreover, suppose that is the class of reducts of a sentence . Then:

We can choose to be (or if is computable).

If is a computable model of with a computable successor relation, then there is a computable model of with .
With a little more work, we can replace the wellfounded collapse with the wellfounded part:
Theorem 28.
In Theorem 27, we can also get
3.1. Overview of the construction
Our structures will have two sorts, the order sort and the main sort. We will also treat elements of as if they are in the structure (e.g., we will talk about functions with codomain ). We can identify with an sentence in the language with a symbol for the ordering and possibly further symbols; is the class of reducts of models of to the language with just the symbol . Let be together with:

There are constants such that each element is equal to exactly one constant.

There is a partial successor function , and each nonmaximal element has a successor.

There is a sequence of subsets satisfying:

is not strictly bounded (i.e., there is no which is strictly greater than each element of ),

,

is the whole universe of the order sort.

If , then ,

For each and , there is a least element of with .

For Theorem 27 (i.e. to get ) we will add

for all .
3 is a consequence of 1; moreover, 1 will make the trivial (see 7 below). 1 will only be used for the final computation of the Scott ranks of the models of , whereas 3 will be used in the construction itself. For Theorem 28, we will use a different axiom 1 instead of 1; 3 will also be a consequence of 1. The general construction will be the same, but 1 will give us a different computation of the Scott rank of the resulting models. Thus 3 is exactly that common part of 1 and 1 which is required for the construction, and the particulars of 1 and 1 are what give the Scott ranks. While reading through the construction for the first time, it might be helpful to assume that 1 is in effect. Each order type in is represented as a model of .
The order sort will be a model of . Our next step will be to define, for each model of , an sentence . The sentence will say that the order sort is a model of and the main sort is a model of . In defining , we will use quantifiers over , and will be uniform in .
For now, fix a particular model of . As a model of , will be a linear ordering, which we view as a nonstandard ordinal. The Scott rank of will be determined by ; in particular, if is actually an ordinal, then the Scott rank of will be its order type. If is a model of , then since by 1 each element of is named by a constant, the Scott rank of will be as low as possible, and so the Scott rank will be carried by . We will have
If has a least element and at least two elements, then for , will be (or in the case of Theorem 28). We can then modify slightly using Proposition 20 to get the theorem; we first modify so that every has a least element and at least two elements, and then we use Proposition 20 to add or , if desired, to the Scott spectrum. Since there are structures of Scott rank and which have Scott sentences which are and respectively, Proposition 20 gives the correct quantifier complexity.
will be constructed as follows. First, we will let be the class of finite structures satisfying the properties 16 and 17 below. We will show that has a Fraïssé limit. This is an ultrahomogeneous structure, and hence has very low Scott rank. We will add to the Fraïssé limit unary relations indexed by . will be a sentence of defining the Fraïssé limit of together with relations satisfying properties 1 and 2.
To see (1) of Theorem 27, we can take the Morleyization of . This will be a sentence which defines the same class of linear orders. The construction of relative to is , so if we define in the same way as above but replacing by its Morleyization , will now be . Since in each model of , each element of is named by a constant, we still have
If is actually a computable formula, then its Morleyization is computable, and this will be computable.
3.2. The definition of
Fix . We begin by constructing the age of our Fraïssé limit. Let be be the class of finite structures satisfying 16 and 17 below. Structures in should be viewed as trees.

a partial treeordering, that is, the set of predecessors of any element is linearly ordered.

is the unique smallest element.

Each element other than has a unique predecessor, and is a unary function picking out that predecessor.

Each element has finite length, i.e., there is a finite chain of successors starting at and ending at that element.

and are unary functions.

If , then .
The properties 16 that we have introduced so far already define the age of a Fraïssé limit in the restricted language . In reading the properties 17 below, it will helpful to have this model in mind.
Lemma 29.
The class of finite structures in the language satisfying 16 has the hereditary property (HP), the amalgamation property (AP), and the joint embedding property (JEP). The Fraïssé limit is (isomorphic to) the following structure .
Fix an infinite set . The domain of is the set of all finite sequences
with , , and , such that . We interpret the relations in the natural way: is the standard ordering of extensions of sequences, is the standard predecessor function, , and .
Proof.
Given an element
of this structure, write for and for . Write for the length of .
We will now add an additional function whose properties are axiomatized by 17. is a function from to . is defined only on those pairs with , , and . Note that the domain of is an equivalence relation, for which we write . For convenience, when we talk about for some and we will often implicitly assume that . We view the range of as a totally ordered set via the lexicographic ordering on , with smaller than every element of . Given with , let be the first coordinate of , i.e., the coordinate in , and let be the second coordinate. If , then we let .
One can view as a nested sequence of relations on , defined by if . If , then and are not at all related. It will follow from 1, 2, and 3 that these are equivalence relations. These equivalence relations are nested and continuous (i.e., ). The most important relations are the relations which we will denote by . The relations will be nonstandard backandforth relations (see Lemma 35). The definition of the backandforth relations is not , so we cannot just ask that satisfy the definition of the backandforth relations. This is where we use and the in ; their role is to convert an existential quantifier into a universal quantifier by acting as a sort of Skolem function.
If is not a dead end, the children of are divided into infinitely many subsets indexed by via the function . If , then for every child of , there will be a child of with ; this is in keeping with the idea of making the equivalence relations agree with the backandforth relations. If , then this will not be true for all . However, it will be true for exactly those with . Rather than saying that there is a child of such that no child of has , we can say that for all children of with , there is no child of with . This is of lower quantifier complexity. (Note that we cannot say that for all and children of and , . This is for the same reason as the following fact: if and are such that for all and , , then .)
For all , , and with :

,

,

,

.

If and are successors of and with , .

If , then for every a successor of with , there are no successors of with .

If , then .
While was not in the domain of , we will consider to be , i.e., to be greater than each element of .
1, 2, and 3 are just saying that the relations defined above are reflexive, symmetric, and transitive respectively (and hence equivalence relations). 6 is the axiom which is doing most of the work.
The intuition behind 7 will be explained in Subsection 3.5. For now, the reader can simply imagine that for each (as it will be for Theorem 27), so that 7 is a vacuous condition.
Lemma 30.
Proof.
It is easy to see that has the hereditary property. Note that every finite structure in contains, via an embedding, the structure with one element . So the joint embedding property will follow from the amalgamation property.
For the amalgamation property, let be a structure in which embeds into and . Identify with its images in and , and assume that the only elements common to both and are the elements of . By amalgamating one element at a time, we may assume that contains only a single element not in . The element is the child of some element of , and has no children in . We will define a structure whose domain is and then show that is in .
First, we can take the amalgamation of the structures in the language as in Lemma 29; we just add to , setting , , and to be the same as in . Set . We need to define when is an element from with . Define to be the maximum of over all with . If there are no such , set . By 3 this is welldefined, that is, for , is the maximum of over all .
For 3, we have two new cases to check. For the first case, fix with ; we will show that . If there is no with , then , and so we have . Otherwise, let be such that . By definition,
Now for the second case, again fix with ; now we will show that . If there is no with , then , and so we have
Otherwise, let be such that , and let be such that . Then
For 4, suppose that has . Then either , in which case there is nothing to check, or for some . In the second case, either or .
Now we check 5. Let be the parent of . Fix and let be the parent of . We must show that . Choose such that . (If there is no such , then we can immediately see that 5 holds.) Let be the parent of . Then we have and so that
Next we check 6. Since has no children in , the only new case to check is as follows. Let be the parent of , and let be such that . Suppose that and let . Suppose to the contrary that there is a child of with . Then, by definition there is such that . Let be the parent of . Since and , and . Hence . This is a contradiction.
Finally, for 7, if we are done. So we may suppose that for some . Then since and are both in , the same is true of . ∎
Lemma 31.
The reduct of the Fraïssé limit of to the language is the structure from Lemma 29.
Proof.
For a fixed , let be the sentence describing the Fraïssé limit of , and to which we add unary relations satisfying 1 and 2 below. The relations will name the equivalence classes , so that while the Fraïssé limit is ultrahomogeneous, the models of will not be. The Fraïssé limit is axiomatizable by a formula. If is computable with a computable successor relation, then is a computable age, and hence the Fraïssé limit is axiomatizable by a formula. Since 16 and 17 are all , they hold in the Fraïssé limit. Since 1 and 2 are also formulas, is axiomatizable.

For each , for exactly one .

For all and , if and only if for all , .
Lemma 32.
If is computable with a computable successor relation, then has a computable model.
Proof.
By Theorem 3.9 of [CHMM11], there is a computable Fraïssé limit of . Then we can add the relations in a computable way. ∎
3.3. Computation of the Scott rank
Fix a model of . Let be a countable model of . The remainder of the proof is a computation of . As remarked earlier, for this section we will assume that has a least element and has at least two elements. We will show that for such an . Let be the wellordered part of . Recall that we identify elements of with ordinals in the natural way.
Lemma 33.
Fix . Suppose that and , , and are tuples from such that where ranges among all of these elements and their predecessors, and such that:

,

for each , .
Suppose moreover that and are closed under the predecessor relation . Then there is such that and satisfy (i) and (ii).
Proof.
We may assume that is not one of , as if then we could take . Thus for no is . By repeated applications of the claim, we may also assume that is among the .
Let be the predecessor of , and let be those with . Let be the predecessors of the . Let , , and be the corresponding . We will define a finite structure with domain consisting of , , , and a new element