Scott Rank Research Articles

THE STRUCTURAL COMPLEXITY OF MODELS OF ARITHMETIC

Abstract We calculate the possible Scott ranks of countable models of Peano arithmetic. We show that no non-standard model can have Scott rank less than $\omega $ and that non-standard models of true arithmetic must have Scott rank greater than $\omega $ . Other than that there are no restrictions. By giving a reduction via $\Delta ^{\mathrm {in}}_{1}$ bi-interpretability from the class of linear orderings to the canonical structural $\omega $ -jump of models of an arbitrary completion T of $\mathrm {PA}$ we show that every countable ordinal $\alpha>\omega $ is realized as the Scott rank of a model of T.

Games on AF-Algebras

Abstract We analyze $\textrm {C}^{\ast }$-algebras, particularly AF-algebras, and their $K_{0}$-groups in the context of the infinitary logic $\mathcal {L}_{\omega _{1} \omega }$. Given two separable unital AF-algebras $A$ and $B$, and considering their $K_{0}$-groups as ordered unital groups, we prove that $K_{0}(A) \equiv _{\omega \cdot \alpha } K_{0}(B)$ implies $A \equiv _{\alpha } B$, where $M \equiv _{\beta } N$ means that $M$ and $N$ agree on all sentences of quantifier rank at most $\beta $. This implication is proved using techniques from Elliott’s classification of separable AF-algebras, together with an adaptation of the Ehrenfeucht-Fraïssé game to the metric setting. We use moreover this result to build a family $\{ A_{\alpha } \}_{\alpha < \omega _{1}}$ of pairwise non-isomorphic separable simple unital AF-algebras which satisfy $A_{\alpha } \equiv _{\alpha } A_{\beta }$ for every $\alpha < \beta $. In particular, we obtain a set of separable simple unital AF-algebras of arbitrarily high Scott rank. Next, we give a partial converse to the aforementioned implication, showing that $A \otimes \mathcal {K} \equiv _{\omega + 2 \cdot \alpha +2} B \otimes \mathcal {K}$ implies $K_{0}(A) \equiv _{\alpha } K_{0}(B)$, for every unital $\textrm {C}^{\ast }$-algebras $A$ and $B$.

SCOTT COMPLEXITY OF COUNTABLE STRUCTURES

AbstractWe define the Scott complexity of a countable structure to be the least complexity of a Scott sentence for that structure. This is a finer notion of complexity than Scott rank: it distinguishes between whether the simplest Scott sentence is $\Sigma _{\alpha }$ , $\Pi _{\alpha }$ , or $\mathrm {d-}\Sigma _{\alpha }$ . We give a complete classification of the possible Scott complexities, including an example of a structure whose simplest Scott sentence is $\Sigma _{\lambda + 1}$ for $\lambda $ a limit ordinal. This answers a question left open by A. Miller.We also construct examples of computable structures of high Scott rank with Scott complexities $\Sigma _{\omega _1^{CK}+1}$ and $\mathrm {d-}\Sigma _{\omega _1^{CK}+1}$ . There are three other possible Scott complexities for a computable structure of high Scott rank: $\Pi _{\omega _1^{CK}}$ , $\Pi _{\omega _1^{CK}+1}$ , $\Sigma _{\omega _1^{CK}+1}$ . Examples of these were already known. Our examples are computable structures of Scott rank $\omega _1^{CK}+1$ which, after naming finitely many constants, have Scott rank $\omega _1^{CK}$ . The existence of such structures was an open question.

Bounds on Scott ranks of some polish metric spaces

If [Formula: see text] is a proper Polish metric space and [Formula: see text] is any countable dense submetric space of [Formula: see text], then the Scott rank of [Formula: see text] in the natural first-order language of metric spaces is countable and in fact at most [Formula: see text], where [Formula: see text] is the Church–Kleene ordinal of [Formula: see text] (construed as a subset of [Formula: see text]) which is the least ordinal with no presentation on [Formula: see text] computable from [Formula: see text]. If [Formula: see text] is a rigid Polish metric space and [Formula: see text] is any countable dense submetric space, then the Scott rank of [Formula: see text] is countable and in fact less than [Formula: see text].

Bounds on continuous Scott rank

An analog of Nadel's effective bound for the continuous Scott rank of metric structures, developed by Ben Yaacov, Doucha, Nies, and Tsankov, will be established: Let $\mathscr{L}$ be a language of continuous logic with code $\hat{\mathscr{L}}$. Let $\Omega$ be a weak modulus of uniform continuity with code $\hat{\Omega}$. Let $\mathcal{D}$ be a countable $\mathscr{L}$-pre-structure. Let $\bar{\mathcal{D}}$ denote the completion structure of $\mathcal{D}$. Then $\mathrm{SR}_\Omega(\bar{D}) \leq \omega_1^{\hat{\mathscr{L}}\oplus\hat{\Omega}\oplus\mathcal{D}}$, the Church-Kleene ordinal relative to $\hat{\mathscr{L}}\oplus\hat{\Omega}\oplus\mathcal{D}$.

Coding in the automorphism group of a computably categorical structure

Using new techniques for controlling the categoricity spectrum of a structure, we construct a structure with degree of categoricity but infinite spectral dimension, answering a question of Bazhenov, Kalimullin and Yamaleev. Using the same techniques, we construct a computably categorical structure of non-computable Scott rank.

Scott ranks of models of a theory

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 Lω1ω) is the set of Scott ranks of countable models of that theory. In ZFC+PD we give a descriptive-set-theoretic classification of the sets of ordinals which are the Scott spectrum of a theory: they are particular Σ11 classes of ordinals.Our investigation of Scott spectra leads to the resolution (in ZFC) of a number of open problems about Scott ranks. We answer a question of Montalbán by showing, for each α<ω1, that there is a Π2in theory with no models of Scott rank less than α. 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 δ21 is the least ordinal α such that if the models of a computable theory T have Scott rank bounded below ω1, then their Scott ranks are bounded below α.

Some new computable structures of high rank

We give several new examples of computable structures of high Scott rank. For earlier known computable structures of Scott rank ω 1 C K \omega _1^{CK} , the computable infinitary theory is ℵ 0 \aleph _0 -categorical. Millar and Sacks asked whether this was always the case. We answer this question by constructing an example whose computable infinitary theory has non-isomorphic countable models. The standard known computable structures of Scott rank ω 1 C K + 1 \omega _1^{CK}+1 have infinite indiscernible sequences. We give two constructions with no indiscernible ordered triple.

Metric Scott analysis

We develop an analogue of the classical Scott analysis for metric structures and infinitary continuous logic. Among our results are the existence of Scott sentences for metric structures and a version of the López-Escobar theorem. We also derive some descriptive set theoretic consequences: most notably, that isomorphism on a class of separable structures is a Borel equivalence relation iff their Scott rank is uniformly bounded below ω1. Finally, we apply our methods to study the Gromov–Hausdorff distance between metric spaces and the Kadets distance between Banach spaces, showing that the set of spaces with distance 0 to a fixed space is a Borel set.

Strange Structures from Computable Model Theory

Let L be a countable language, let I be an isomorphism-type of countable L-structures, and let a∈2ω. We say that I is a-strange if it contains a computable-from-a structure and its Scott rank is exactly ω1a. For all a, a-strange structures exist. Theorem (AD): If C is a collection of ℵ1 isomorphism-types of countable structures, then for a Turing cone of a’s, no member of C is a-strange.

The countable admissible ordinal equivalence relation

Let Fω1 be the countable admissible ordinal equivalence relation defined on 2ω by xFω1y if and only if ω1x=ω1y. Some invariant descriptive set theoretic properties of Fω1 will be explored using infinitary logic in countable admissible fragments as the main tool. Marker showed Fω1 is not the orbit equivalence relation of a continuous action of a Polish group on 2ω. Becker stengthened this to show Fω1 is not even the orbit equivalence relation of a Δ11 action of a Polish group. However, Montalbán has shown that Fω1 is Δ11 reducible to an orbit equivalence relation of a Polish group action, in fact, Fω1 is classifiable by countable structures. It will be shown here that Fω1 must be classified by structures of high Scott rank. Let Eω1 denote the equivalence of order types of reals coding well-orderings. If E and F are two equivalence relations on Polish spaces X and Y, respectively, E≤aΔ11F denotes the existence of a Δ11 function f:X→Y which is a reduction of E to F, except possibly on countably many classes of E. Using a result of Zapletal, the existence of a measurable cardinal implies Eω1≤aΔ11Fω1. However, it will be shown that in Gödel's constructible universe L (and set generic extensions of L), Eω1≤aΔ11Fω1 is false. Lastly, the techniques of the previous result will be used to show that in L (and set generic extensions of L), the isomorphism relation induced by a counterexample to Vaught's conjecture cannot be Δ11 reducible to Fω1. This shows the consistency of a negative answer to a question of Sy-David Friedman.

COMPUTABLE STRUCTURES IN GENERIC EXTENSIONS

AbstractIn this paper, we investigate connections between structures present in every generic extension of the universe V and computability theory. We introduce the notion of generic Muchnik reducibility that can be used to compare the complexity of uncountable structures; we establish basic properties of this reducibility, and study it in the context of generic presentability, the existence of a copy of the structure in every extension by a given forcing. We show that every forcing notion making ω2 countable generically presents some countable structure with no copy in the ground model; and that every structure generically presentable by a forcing notion that does not make ω2 countable has a copy in the ground model. We also show that any countable structure ${\cal A}$ that is generically presentable by a forcing notion not collapsing ω1 has a countable copy in V, as does any structure ${\cal B}$ generically Muchnik reducible to a structure ${\cal A}$ of cardinality ℵ1. The former positive result yields a new proof of Harrington’s result that counterexamples to Vaught’s conjecture have models of power ℵ1 with Scott rank arbitrarily high below ω2. Finally, we show that a rigid structure with copies in all generic extensions by a given forcing has a copy already in the ground model.

ISOMORPHISM ON HYP

AbstractWe show that isomorphism is not a complete ${\rm{\Sigma }}_1^1$ equivalence relation even when restricted to the hyperarithmetic reals: If E1 denotes the ${\rm{\Sigma }}_1^1$ (even ${\rm{\Delta }}_1^1$) equivalence relation of [4] then for no Hyp function f do we have xEy iff f(x) is isomorphic to f(y) for all Hyp reals x,y. As a corollary to the proof we provide for each computable limit ordinal α a hyperarithmetic reduction of ${ \equiv _\alpha }$ (elementary-equivalence for sentences of quantifier-rank less than α) on arbitrary countable structures to isomorphism on countable structures of Scott rank at most α.

Improving a Bounding Result That Constructs Models of High Scott Rank

Let T be a theory in a countable fragment of Lω1,ω whose extensions in countable fragments have only countably many types. Sacks proves a bounding theorem that generates models of high Scott rank. For this theorem, a tree hierarchy is developed for T that enumerates these extensions. In this paper, we effectively construct a predecessor function for formulas defining types in this tree hierarchy as follows. Let Tγ ⊂ T with T - and Tγ -theories on level and γ, respectively. Then if p (Tγ) is a formula that defines a type for Tγ , our predecessor function provides a formula for defining its subtype in T . By constructing this predecessor function, we weaken an assumption for Sackss result.

Three red herrings around Vaught’s conjecture

We give a model theoretic proof that if there is a counterexample to Vaught’s conjecture there is a counterexample such that every model of cardinality א1 is maximal (strengthening a result of Hjorth’s). We also give a new proof of Harrington’s theorem that any counterexample to Vaught’s conjecture has models in א1 of arbitrarily high Scott rank below א2. The three red herrings1 are false leads towards solving Vaught’s conjecture. Here is one strategy for establishing Vaught’s conjecture that there is no sentence of Lω1,ω that has exactly א1 countable models. Hjorth [9, 8], using descriptive set theoretic results of Mackey and others, has established that if there is a counterexample then there is one that has no model in א2. On the other hand, unpublished results of Harrington show that every counterexample has models with arbitrarily large Scott ranks below א2. This supports the notion that one might construct a model of an arbitrary counterexample that has cardinality א2. The resulting contradiction would yield the conjecture. In the introduction we give some background on the conjecture and describe the three red herrings. Recall that Vaught’s conjecture [19] concerns the number of countable models of a countable first-order theory, or more generally, of a sentence in the infinitary logic Lω1ω , where countable conjunctions and disjunctions but only finite strings of quantifiers are allowed for some countable vocabulary τ . The conjecture states: Vaught Conjecture. If φ is a sentence of Lω1ω then φ either has countably many or continuum-many countable models up to isomorphism. The more “absolute” version replaces “continuum-many” by “a perfect set of ” in the conclusion, where a perfect set of countable models is a perfect set of reals, each of which codes a countable model, such that distinct reals in the perfect set code non-isomorphic models. We say a sentence φ of Lω1ω is scattered if it does not have a perfect set of countable models. Morley [20] defined scattered as: for every countable ∗Research partially supported by Simons travel grant G5402 †Research supported by FWF (Austrian Science Fund) Grant P24654-N25. ‡Research supported by the Austrian Science Fund (FWF) Lise Meitner Grant M1410-N25 §Partially supported by NSF grant DMS-1308546 1See http://en.wikipedia.org/wiki/Red_herring for a fairly comprehensive account of the meaning of the red herring idiom.

A robuster Scott rank

We give a new definition of Scott rank motivated by our main theorem: For every countable structure A and ordinal α < ω1, we have that: every automorphism orbit is Σ α -definable without parameters if and only if A has a Π α+1 Scott sentence, if and only if A is uniformly boldface ∆α-categorical. As a corollary, we show that a structure is computably categorical on a cone if and only if it is the model of a countably categorical Σ 3 sentence.

Scott rank of Polish metric spaces

Following the work of Friedman, Koerwien, Nies and Schlicht we positively answer their question whether the Scott rank of Polish metric spaces is countable.

Classes of Ulm type and coding rank-homogeneous trees in other structures

AbstractThe first main result isolates some conditions which fail for the class of graphs and hold for the class of Abelianp-groups, the class of Abelian torsion groups, and the special class of “rank-homogeneous” trees. We consider these conditions as a possible definition of what it means for a class of structures to have “Ulm type”. The result says that there can be no Turing computable embedding of a class not of Ulm type into one of Ulm type. We apply this result to show that there is no Turing computable embedding of the class of graphs into the class of “rank-homogeneous” trees. The second main result says that there is a Turing computable embedding of the class of rank-homogeneous trees into the class of torsion-free Abelian groups. The third main result says that there is a “rank-preserving” Turing computable embedding of the class of rank-homogeneous trees into the class of Boolean algebras. Using this result, we show that there is a computable Boolean algebra of Scott rank.

COMPUTABLE STRUCTURES OF RANK $\omega_{1}^{{\rm CK}}$

For countable structure, "Scott rank" provides a measure of internal, model-theoretic complexity. For a computable structure, the Scott rank is at most [Formula: see text]. There are familiar examples of computable structures of various computable ranks, and there is an old example of rank [Formula: see text]. In the present paper, we show that there is a computable structure of Scott rank [Formula: see text]. We give two different constructions. The first starts with an arithmetical example due to Makkai, and codes it into a computable structure. The second re-works Makkai's construction, incorporating an idea of Sacks.

Model-theoretic complexity of automatic structures

We study the complexity of automatic structures via well-established concepts from both logic and model theory, including ordinal heights (of well-founded relations), Scott ranks of structures, and Cantor–Bendixson ranks (of trees). We prove the following results: (1) The ordinal height of any automatic well-founded partial order is bounded by ω ω . (2) The ordinal heights of automatic well-founded relations are unbounded below ω 1 C K , the first non-computable ordinal. (3) For any computable ordinal α , there is an automatic structure of Scott rank at least α . Moreover, there are automatic structures of Scott rank ω 1 C K , ω 1 C K + 1 . (4) For any computable ordinal α , there is an automatic successor tree of Cantor–Bendixson rank α .