We develop various Ehrenfeucht-Fra\"ıssé games for distances between metric structures. We study two forms of distances: pseudometrics stemming from mapping spaces onto each other with some form of approximate isomorphism, and metrics stemming from measuring the distances between two spaces isometrically embedded into a third space. Using an infinitary version of Henson's positive bounded logic with approximations, we form Scott sentences capturing fixed distances to a given space. The Scott sentences of separable spaces are in $\mathcal{L}_{ω_1ω}$ for 0-distances and in $\mathcal{L}_{ω_2ω}$ for positive distances.
Scott showed that for every countable structure $\mathcal{A}$, there is a sentence of the infinitary logic $\mathcal{L}_{ω_1ω}$, called a Scott sentence for $\mathcal{A}$, whose models are exactly the isomorphic copies of $\mathcal{A}$. Thus, the least quantifier complexity of a Scott sentence of a structure is an invariant that measures the complexity "describing" the structure. Knight et al.~have studied the Scott sentences of many structures. In particular, Knight and Saraph showed that a finitely generated structure always has a $Σ^0_3$ Scott sentence. We give a characterization of the fin…
We prove that every quasi-Hopfian finitely presented structure $A$ has a $d$-$Σ_2$ Scott sentence, and that if in addition $A$ is computable and $Aut(A)$ satisfies a natural computable condition, then $A$ has a computable $d$-$Σ_2$ Scott sentence. This unifies several known results on Scott sentences of finitely presented structures and it is used to prove that other not previously considered algebraic structures of interest have computable $d$-$Σ_2$ Scott sentences. In particular, we show that every right-angled Coxeter group of finite rank has a computable $d$-$Σ_2$ Scott sentence, as well a…
We give Scott sentences for certain computable groups, and we use index set calculations as a way of checking that our Scott sentences are as simple as possible. We consider finitely generated groups and torsion-free abelian groups of finite rank. For both kinds of groups, the computable ones all have computable $Σ_3$ Scott sentences. Sometimes we can do better. In fact, the computable finitely generated groups that we have studied all have Scott sentences that are "computable $d$-$Σ_2$" (the conjunction of a computable $Σ_2$ sentence and a computable $Π_2$ sentence). This was already shown fo…
The Hanf number for a set $S$ of sentences in $L_{ω_1,ω}$ (or some other logic) is the least infinite cardinal $κ$ such that for all $φ\in S$, if $φ$ has models in all infinite cardinalities less than $κ$, then it has models of all infinite cardinalities. S-D. Friedman asked what is the Hanf number for Scott sentences of computable structures. We show that the value is $\beth_{ω_1^{CK}}$. The same argument proves that $\beth_{ω_1^{CK}}$ is the Hanf number for Scott sentences of hyperarithmetical structures.
Given a countable scattered linear order $L$ of Hausdorff rank $α< ω_1$ we show that it has a $d\text{-}Σ_{2α+1}$ Scott sentence. Ash calculated the back and forth relations for all countable well-orders. From this result we obtain that this upper bound is tight, i.e., for every $α< ω_1$ there is a linear order whose optimal Scott sentence has this complexity. We further show that for all countable $α$ the class of Hausdorff rank $α$ linear orders is $\pmb Σ_{2α+2}$ complete.
This paper is part II of a study on cardinals that are characterizable by a Scott sentence, continuing the work from http://arxiv.org/abs/1007.2426v1. A cardinal $κ$ is characterized by a Scott sentence $ϕ_M$, if $ϕ_M$ has a model of size $κ$, but no model of $κ^+$. The main question in this paper is the following: Are the characterizable cardinals closed under the powerset operation? We prove that if $\aleph_β$ is characterized by a Scott sentence, then $2^{\aleph_{β+β_1}}$ is (homogeneously) characterized by a Scott sentence, for all $0<β_1<ω_1$. So, the answer to the above question is posit…
This is part I of a study on cardinals that are characterizable by Scott sentences. Building on [3], [6] and [1] we study which cardinals are characterizable by a Scott sentence $ϕ$, in the sense that $ϕ$ characterizes $κ$, if $ϕ$ has a model of size $κ$, but no models of size $κ^+$. We show that the set of cardinals that are characterized by a Scott sentence is closed under successors, countable unions and countable products (cf. theorems 2.3, 3.4, and corollary 3.6). We also prove that if $\aleph_α$ is characterized by a Scott sentence, at least one of $\aleph_alpha$ and $\aleph_alpha^+$ is …
Every countable structure has a sentence of the infinitary logic $\mathcal{L}_{ω_1 ω}$ which characterizes that structure up to isomorphism among countable structures. Such a sentence is called a Scott sentence, and can be thought of as a description of the structure. The least complexity of a Scott sentence for a structure can be thought of as a measurement of the complexity of describing the structure. We begin with an introduction to the area, with short and simple proofs where possible, followed by a survey of recent advances.
We study two complexity notions of groups - a computable Scott sentence and the index set of a group. Finding the exact complexity of one of them usually involves finding the complexity of the other, but this is not the case sometimes. J. Knight et al. determined the complexity of index sets of various structures. In this paper, we focus on finding the complexity of computable Scott sentences and index sets of various groups, generalizing methods that was previously used by J. Knight et al. We found computable Scott sentences for various different groups or class of groups, including nilpotent…
We give effective versions of some results on Scott sentences. We show that if $\mathcal{A}$ has a computable $Π_α$ Scott sentence, then the orbits of all tuples are defined by formulas that are computable $Σ_β$ for some $β<α$. (This is an effective version of a result of Montalbán.) We show that if a countable structure $\mathcal{A}$ has a computable $Σ_α$ Scott sentence and one that is computable $Π_α$, then it has one that is computable $d$-$Σ_β$ for some $β< α$. (This is an effective version of a result of A. Miller.) We also give an effective version of a result of D. Miller. Using the no…
We define and investigate HC-forcing invariant formulas of set theory, whose interpretations in the hereditarily countable sets are well behaved under forcing extensions. This leads naturally to a notion of cardinality ||Phi|| for sentences Phi of $L_{ω_1,ω}$, which counts the number of sentences of $L_{\infty,ω}$ that, in some forcing extension, become a canonical Scott sentence of a model of Phi. We show this cardinal bounds the complexity of (Mod(Phi), iso), the class of models of Phi with universe omega, by proving that (Mod(Phi),iso) is not Borel reducible to (Mod(Psi),iso) whenever ||Psi…
The Scott process of a relational structure $M$ is the sequence of sets of formulas given by the Scott analysis of $M$. We present axioms for the class of Scott processes of structures in a relational vocabulary $τ$, and use them to give a proof of an unpublished theorem of Leo Harrington from the 1970's, showing that a counterexample to Vaught's Conjecture has models of cofinally many Scott ranks below $ω_{2}$. Our approach also gives a theorem of Harnik and Makkai, showing that if there exists a counterexample to Vaught's Conjecture, then there is a counterexample whose uncountable models ha…
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 Lopez-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 $ω_1$. Finally, we apply our methods to study the Gromov-Hausdorff distance between metric spaces and the Kadets distance between Banach spaces, showing that the s…
A countable structure is said to be extendible if it has the same Scott sentence as some uncountable structure. Rigid structures are not extendible. We give an example of an extendible model with a rigid elementary extension.
We consider model-theoretic properties related to the expressive power of three analogues of $L_{ω_1, ω}$ for metric structures. We give an example showing that one of these infinitary logics is strictly more expressive than the other two, but also show that all three have the same elementary equivalence relation for complete separable metric structures. We then prove that a continuous function on a complete separable metric structure is automorphism invariant if and only if it is definable in the more expressive logic. Several of our results are related to the existence of Scott sentences for…
We give a formalism for approximate isomorphism in continuous logic simultaneously generalizing those of two papers by Ben Yaacov and by Ben Yaacov, Doucha, Nies, and Tsankov, which are largely incompatible. With this we explicitly exhibit Scott sentences for the perturbation systems of the former paper, such as the Banach-Mazur distance and the Lipschitz distance between metric spaces. Our formalism is simultaneously characterized syntactically by a mild generalization of perturbation systems and semantically by certain elementary classes of two-sorted structures that witness approximate isom…
A model checking computation checks whether a given logical sentence is true in a given finite structure. Provenance analysis abstracts from such a computation mathematical information on how the result depends on the atomic data that describe the structure. In database theory, provenance analysis by interpretations in commutative semirings has been rather succesful for positive query languages (such a unions of conjunctive queries, positive relational algebra, or datalog). However, it did not really offer an adequate treatment of negation or missing information. Here we propose a new approach…
